Quantum Physics with Deeper Questions with ChatGPTPermalink

IntroductionPermalink

I was a physics student during my college years, and I’ve always loved the subject—even today. However, Quantum Physics, despite all my reading and learning, remains a mystery to me. At this stage, who can teach me Quantum Physics effectively? Finding an able professor, aligning my availability with theirs, and hoping they’d be willing to teach me for free seems impossible. So, I turned to a Large Language Model (LLM) for help. I could have explored other LLMs like Grok, Claude, DeepSeek, and many more. I’m not saying the others are bad, nor am I claiming that ChatGPT is the best at providing plausible answers. Whether an answer is correct or plausible also depends on the learner’s ability.

This extensive session with ChatGPT has been a great learning experience for me. It has taught me so much about Quantum Physics, while simultaneously revealing how powerless LLMs can be when we don’t ask the right questions. We might criticize these LLMs as mere “next-word predictors,” but their power is undeniably immense. At the same time, I realize that without great teachers and researchers, we cannot advance science relying solely on LLMs. Yes, these models are more powerful than most humans in certain ways, but among humans, there are exceptional minds like Thomas Edison, Albert Einstein, Newton, Galileo, Ramanujan, and Aryabhata. It’s true that not all humans match their caliber, but we must recognize that creating such intellects requires a supportive society, systems, culture, and upbringing—conditions that allow such extraordinary individuals to emerge and challenge these AI systems. This challenge isn’t about discarding AI but about improving it and using it for the betterment of humanity.

Questions I askedPermalink

1. How come a quantum bit be 0 and 1 at the same time?
2. There is fault in “coin in air” analogy.
3. What are the foundations of quantum physics?
4. Why it is impossible to determine position and momentum at the same time?
5. Why do we say electron has wave particle duality?
6. What is the meaning that electron has wave nature?
7. Can you help me understanding wavefunction?
8. What path electron follow while traveling?
9. What is the meaning of electron is spread out during journey?
10. Does wavefunction keep changing during the travel?
11. How can a free electron with a constant energy moving through space?
12. How an electron can collide with atom?
13. Why electrons are negatively charged?
14. What is the smallest measurable charge?
15. What is coulomb and how to measure that?
16. There is a Gap in 1 Coulomb calculation value
17. How do we count that the number of electron is 6.242×10 ^18
18. How did we reach to this conclusion that 1ev or electron’s charge is exactly this?
19. How much is 1 coulomb charge?
20. How to measure how many electron has moved?
21. If I want to measure number of electron moved in 10 watt bulb in 1 hour then how to do that?
22. How to measure 1 joule?
23. There is confusion in 1 joule calculation.
24. Conservation of energy is a fundamental principle in physics
25. How come massless gives rise to mass?
26. How can a photon achieve high energy?
27. How photon can have momentum when mass is zero?
28. What is the size of photon and electron that they collide and effect each other?
29. We don’t know the size of a light particle of 500 nm wavelength!
30. When electron don’t have definite size then how do we measure their mass?
31. Particles in Classical vs Quantum Physics.
32. What Does “Finding” an Electron Mean?
33. Water wave - Electron wave analogy is not correct.
34. The meaning of electron’s wavefunction can exist in a superposition of states.
35. I can find a particular electron anywhere in the universe!
36. Can I go thousands of mile away and find that electron there?
37. What are different possible location of an electron?
38. A wavefunction is function of multiple functions, right?
39. How to construct a wavefunction?
40. Refine this wavefunction for 3D space.
41. With a given wavefunction, can we location particle at time t?
42. If I give you radius and wavefunction can you locate electron?
43. It is crazy, I need some expert now!

How come a quantum bit be 0 and 1 at the same time?Permalink

When they say a quantum bit (qubit) can be “zero and one at the same time,” they are referring to superposition, which allows a qubit to exist in a combination of both states until it is measured. This is a unique feature of quantum systems, making quantum computing different from classical computing.

Classical vs. Quantum Bits

• Classical bits (the ones in traditional computing) can be in one of two states at any time: either 0 or 1.
• Quantum bits (qubits), on the other hand, can exist in a superposition of both 0 and 1 states simultaneously, meaning that a qubit can be in a state that is a combination of both 0 and 1.

What is Superposition?

• In classical computing, you would have a bit that is either in state 0 or state 1. But in quantum computing, a qubit is described by a quantum state that can be a linear combination (or superposition) of both states.

For example, a qubit might be in a state that is 50% 0 and 50% 1, or any other combination. Mathematically, this is represented as: $$\text{qubit state} = \alpha |0\rangle + \beta |1\rangle$$

where $(\alpha )$ and $(\beta )$ are complex numbers that define the probability amplitudes of measuring the qubit in state 0 or state 1.

The Key Points:

• Superposition: A qubit can be in a combination of both states at once, unlike classical bits that are always in one state.
• Measurement: When you measure a qubit, it “collapses” into one of the two possible states (0 or 1), with probabilities determined by the coefficients $(\alpha )$ and $(\beta )$. Before measurement, however, it truly exists in both states.
• Interference: This simultaneous existence of both states can lead to quantum interference, which is exploited in quantum computing algorithms to solve certain problems faster than classical computers.

Example: Think of a coin spinning in the air. While it’s spinning, it’s not strictly heads or tails, but a mixture of both. Only when the coin lands (which is like measurement) do you know whether it’s heads or tails.

Similarly, a qubit can represent a state that is “both 0 and 1” until you measure it, at which point it will collapse into one of the two states, just like the coin lands on heads or tails.

There is fault in “coin in air” analogy.Permalink

In your example of the coin at any point of time when coin is in the air by taking the photograph with a high frame rate camera we know whether coin is had a tail at that time so it is not correct to say that it is in both the position.

When you spin a coin, it is indeed in a specific state at any given moment—either heads or tails—though we may not know that state until we measure it by looking at it. A high-frame-rate camera could capture that exact moment and reveal whether the coin is closer to heads or tails, but the coin is never in both states simultaneously in the classical sense.

So, the coin example is not a perfect analogy for quantum superposition, as the coin only takes one state (heads or tails) at any moment, regardless of whether we observe it. It’s just that we don’t know the state until we observe it.

Superposition is not about “both states at once” in a physical sense; it’s a deeper probabilistic reality unique to quantum systems.

What are the foundations of quantum physics?Permalink

The foundations of quantum physics reveal a universe that is fundamentally different from our everyday classical experiences. Quantum mechanics governs the microscopic realm with principles like superposition, entanglement, and uncertainty, enabling groundbreaking technologies while challenging our understanding of reality. The foundations of quantum physics are the fundamental principles, concepts, and theories that describe the behavior of matter and energy at the atomic and subatomic levels. Quantum physics challenges the classical worldview, introducing concepts that are often counterintuitive. Here are the key foundations:

1. Wave-Particle DualityPermalink

• Principle: Particles such as electrons and photons exhibit both particle-like and wave-like behavior, depending on how they are observed.
• Key Experiments:
  • Double-Slit Experiment: Demonstrates that particles like electrons can create an interference pattern, a hallmark of wave behavior, when not observed. When observed, they behave like particles.
• Implication: Matter and energy cannot be strictly classified as waves or particles but exist in a dual nature.

2. Quantum SuperpositionPermalink

• Principle: A quantum system can exist in multiple states simultaneously until measured. For example, a qubit can be in a combination of $(\Vert 0⟩)$ and $(\Vert 1⟩)$.
• Key Concept:
  • The state of a quantum system is represented by a wavefunction $(\Vert \psi ⟩)$, which contains all possible states and their probabilities.
• Example: Schrödinger’s cat thought experiment illustrates this principle on a macroscopic scale.

3. QuantizationPermalink

• Principle: Certain properties, such as energy, angular momentum, and charge, are discrete (quantized) rather than continuous.
• Key Concept:
  • Energy levels in an atom are quantized, meaning electrons can only occupy specific energy states.
• Key Equation:
  • Planck’s relation: $(E=h\nu )$, where $(E)$ is energy, $(h)$ is Planck’s constant, and $(\nu )$ is the frequency.

4. Uncertainty PrinciplePermalink

• Principle: It is impossible to simultaneously know both the exact position and momentum of a particle. The more precisely one is known, the less precisely the other can be determined.
• Key Equation: $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$ where $(\mathrm{\Delta }x)$ is the uncertainty in position, $(\mathrm{\Delta }p)$ is the uncertainty in momentum, and $(\hslash )$ is the reduced Planck’s constant.
• Implication: This limits the precision of measurements and reflects a fundamental property of quantum systems.

5. Quantum EntanglementPermalink

• Principle: Particles can become entangled, meaning their properties are correlated in such a way that the state of one particle instantly influences the state of the other, regardless of the distance between them.
• Key Experiment:
  • Bell’s theorem and subsequent experiments show that entangled particles violate classical notions of locality and realism.
• Implication: Challenges classical ideas of causality and suggests “non-local” connections in quantum systems.

6. Measurement and Wavefunction CollapsePermalink

• Principle: The act of measurement causes the wavefunction of a quantum system to collapse into a single state.
• Key Concepts:
  • Before measurement, a quantum system exists in a superposition of states.
  • Measurement “chooses” one state, seemingly at random, according to probabilities dictated by the wavefunction.
• Interpretations:
  • Copenhagen Interpretation: Measurement causes a real physical collapse.
  • Many-Worlds Interpretation: All possible outcomes occur, each in a separate parallel universe.

7. Probability and the Born RulePermalink

• Principle: The probability of finding a particle in a particular state is given by the square of the amplitude of its wavefunction.
• Key Equation: $P=\Vert \psi {\Vert }^2$ where $(P)$ is the probability and $(\Vert \psi {\Vert }^2)$ is the square of the wavefunction’s amplitude.
• Implication: Quantum mechanics is inherently probabilistic.

8. Schrödinger EquationPermalink

• Principle: The time evolution of a quantum system is governed by the Schrödinger equation.
• Key Equation: $i\hslash \frac{\partial }{\partial t}\Vert \psi (t)⟩=\hat{H}\Vert \psi (t)⟩$ where $(\hat{H})$ is the Hamiltonian operator, $(\Vert \psi (t)⟩)$ is the wavefunction, and $(i)$ is the imaginary unit.
• Implication: Describes how the quantum state of a system evolves over time.

9. ComplementarityPermalink

• Principle: Properties like position and momentum, or wave and particle behavior, are complementary. You can observe one property, but not both simultaneously.
• Implication: There is no single classical analogy for a quantum system.

10. Quantum Field Theory (QFT)Permalink

• Principle: Quantum mechanics combined with special relativity leads to QFT, where particles are seen as excitations of underlying fields.
• Key Concept:
  • Photons are excitations of the electromagnetic field.
  • Quantum electrodynamics (QED) and quantum chromodynamics (QCD) are key QFTs describing fundamental forces.

Key Figures in Quantum PhysicsPermalink

1. Max Planck: Introduced the idea of quantized energy.
2. Albert Einstein: Explained the photoelectric effect and introduced the concept of photons.
3. Niels Bohr: Developed the Bohr model of the atom and the concept of complementarity.
4. Erwin Schrödinger: Formulated the Schrödinger equation.
5. Werner Heisenberg: Developed the uncertainty principle.
6. Paul Dirac: Combined quantum mechanics and special relativity.

Applications and ImplicationsPermalink

Quantum physics forms the basis of technologies like:

• Semiconductors and transistors.
• Lasers and LEDs.
• Quantum computing.
• Cryptography.
• MRI and other medical imaging techniques.

Why it is impossible to determine position and momentum at the same time?Permalink

The statement that it is impossible to simultaneously know both the exact position and momentum of a particle is a cornerstone of quantum physics, known as Heisenberg’s Uncertainty Principle. To understand this principle and the concept of momentum, let’s break it down:

What is Momentum?Permalink

• Momentum $(p)$ is a measure of the motion of an object. It is defined as: $p=mv$ where:
  • $(m)$: mass of the object
  • $(v)$: velocity of the object
• Momentum is a vector quantity, meaning it has both magnitude (how fast it’s moving) and direction (the direction of motion).

In quantum mechanics, momentum is related to the wavelength of a particle’s wave-like behavior: $p=\frac{h}{\lambda }$ where:

• $(h)$: Planck’s constant $(6.626\times {10}^{-34}\text{Js})$
• $(\lambda )$: wavelength of the particle

Heisenberg’s Uncertainty PrinciplePermalink

The uncertainty principle states: $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$ where:

• $(\mathrm{\Delta }x)$: uncertainty in position
• $(\mathrm{\Delta }p)$: uncertainty in momentum
• $(\hslash )$: reduced Planck’s constant $(\hslash =\frac{h}{2\pi })$

This means there is a fundamental limit to how precisely we can know both the position $(x)$ and momentum $(p)$ of a particle simultaneously.

Why is This True?Permalink

The principle arises because particles in quantum mechanics exhibit wave-particle duality, meaning they have both wave-like and particle-like properties.

1. Wave-Particle Duality:
   • A particle, such as an electron, behaves like a wave. The wave’s spread represents the range of possible positions for the particle.
   • A sharply localized wave packet (to pinpoint the position) is made by combining waves of many different wavelengths. This broadens the uncertainty in momentum.
2. Measurement Disturbance:
   • Measuring the position of a particle requires interacting with it, for instance, by using light (photons).
   • The interaction disturbs the particle’s momentum, introducing uncertainty.

Example to Understand ThisPermalink

Imagine shining a laser (light) on an electron to determine its position:

1. Determining Position:
   • To measure the position precisely, you use light with a very short wavelength (high-energy photons). The shorter the wavelength, the better the resolution.
   • However, the high-energy photon disturbs the electron, changing its momentum significantly.
2. Determining Momentum:
   • If you use light with a longer wavelength (low-energy photons), the disturbance to the electron’s momentum is small, allowing for a more accurate measurement of momentum.
   • But now the position is less certain because the longer wavelength makes the electron’s exact position blurry.

This trade-off between position and momentum is a direct manifestation of the uncertainty principle.

Key IntuitionPermalink

If you visualize the particle as a wave:

• A wave with a very specific position (like a sharp spike) is made up of many overlapping waves of different momenta.
• A wave with a very specific momentum (single wavelength) is spread out over space, making the position uncertain.

ImplicationsPermalink

The uncertainty principle is not due to limitations in our measurement instruments but a fundamental property of nature. It explains:

• Why electrons in an atom don’t spiral into the nucleus: Confined to a small region, their momentum becomes highly uncertain, preventing them from “falling in.”
• The behavior of particles at microscopic scales, leading to technologies like semiconductors and quantum computing.

In essence, nature forbids particles from being pinned down completely in both their motion and location at the same time.

Why do we say electron has wave particle duality?Permalink

When we say an electron has a wave property, it means that the behavior of the electron cannot be fully described as that of a small particle moving in a straight line. Instead, it exhibits properties associated with waves, such as interference and diffraction, in addition to its particle-like behavior. This is a fundamental aspect of wave-particle duality, a concept in quantum mechanics. Electrons don’t “wiggle” or physically move like water waves, but they do exhibit behaviors associated with waves, such as interference and diffraction. This wave-like property describes the probabilities of where the electron might be, not its actual physical motion.

Let’s break this down:

1. Wave Properties of ElectronsPermalink

Key Characteristics:Permalink

• Interference:
  • Like waves on water, electrons can create interference patterns when they pass through slits or obstacles. This is famously demonstrated in the double-slit experiment.
• Diffraction:
  • Electrons bend around edges or through small openings, just like light waves.
• Wavelength:
  • The wave-like behavior of an electron is characterized by its de Broglie wavelength, given by: $\lambda =\frac{h}{p}$ where:
    • $(\lambda )$: wavelength
    • $(h)$: Planck’s constant
    • $(p)$: momentum of the electron

2. Does This Mean an Electron Doesn’t Travel in a Straight Line?Permalink

Not exactly. The electron’s wave-like behavior doesn’t mean it zigzags or physically “oscillates” like a wave on a string. Instead, the wave represents a probability distribution:

• The electron’s wave-like nature reflects the probability of where it might be found.
• Its exact trajectory cannot be determined because of the uncertainty principle.

Trajectory Explanation:Permalink

• In some contexts, like classical mechanics, we model an electron traveling in a straight line (e.g., in an electric field).
• But when quantum effects are significant (e.g., at atomic scales), the “straight-line” concept breaks down, and its behavior is better described by a wavefunction (a mathematical function describing the probability of its position and momentum).

3. The Double-Slit ExperimentPermalink

This experiment is the clearest demonstration of the wave property of electrons:

Setup:Permalink

• Electrons are shot one at a time toward a barrier with two slits.
• A detection screen is placed on the other side to record where electrons hit.

Observation:Permalink

• Instead of forming two clusters corresponding to the slits (as particles would), the electrons create an interference pattern of alternating bright and dark bands, like waves.

Explanation:Permalink

• Each electron passes through both slits simultaneously (in a wave-like manner) and interferes with itself.
• When detected, however, the electron behaves like a particle, landing at a specific point on the screen.

4. Key Insight: Wave-Particle DualityPermalink

Electrons (and other quantum particles) are neither purely waves nor purely particles:

• Wave-Like Behavior:
  • Exhibited in phenomena like interference, diffraction, and de Broglie wavelengths.
• Particle-Like Behavior:
  • Exhibited when electrons are detected as discrete entities, such as hitting a screen or forming atoms.

5. Analogy: A CloudPermalink

To visualize this, think of the electron as a cloud of probability rather than a point particle or a literal wave:

• The “cloud” spreads out like a wave, showing where the electron is likely to be found.
• When you measure it, the “cloud collapses,” and the electron appears as a particle in a specific location.

What is the meaning that electron has wave nature?Permalink

I want to understand this further, let’s say a single leaves the electron gun. And we are saying it has wave nature it means what? We know at what speed electron travels we can know know exact where it is after 1256 pico second.

This is what makes quantum mechanics fundamentally different from classical mechanics—it deals in probabilities, not certainties.

1. What Does Wave Nature Mean for a Single Electron?Permalink

When a single electron is emitted from the gun, it behaves both as a particle and as a wave. The wave nature means that the electron’s position and trajectory are not deterministic (like a classical particle), but instead described by a wavefunction. This wavefunction:

• Spreads out in space.
• Represents the probability of finding the electron at a particular location if we measure it.

Key Point:Permalink

• The wave nature doesn’t mean the electron is physically spread out like a water wave.
• Instead, it means we can’t know precisely where the electron is at any given time—only the likelihood of finding it at different places.

2. Can We Predict Where the Electron Will Be After 1256 Picoseconds?Permalink

If we know the initial speed of the electron and the time elapsed, classical mechanics would let us calculate its position exactly. However, in quantum mechanics, we cannot do this due to the Uncertainty Principle: $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$

Why We Can’t Predict Exactly:Permalink

• Initial Position Uncertainty: When the electron is emitted, we already have some uncertainty $(\mathrm{\Delta }x)$ in its position.
• Momentum Uncertainty: If we try to measure its momentum $(p=mv)$ very precisely, this increases the uncertainty in its position.
• Over time, this initial uncertainty causes the wavefunction to spread out, making it harder to predict the electron’s exact location.

Instead of an Exact Position:Permalink

What we get is a probability distribution:

• After 1256 picoseconds, the wavefunction has evolved, and it tells us the likelihood of finding the electron at various positions if we were to measure it.

3. Visualizing the Wavefunction of a Single ElectronPermalink

Think of the electron’s wavefunction like a cloud that spreads out over time:

• Initially, when the electron is emitted, the wavefunction is concentrated near the gun.
• As time progresses, the cloud expands, representing the growing uncertainty in the electron’s position.

If you were to measure the electron’s position at 1256 picoseconds:

• You wouldn’t know in advance exactly where it would be.
• But you’d find it in one specific location, consistent with the probabilities given by the wavefunction.

4. Is This Measurable?Permalink

Yes, and this has been experimentally verified. For instance:

• In the double-slit experiment, if you fire electrons one at a time, each electron hits the detection screen at a specific point (particle behavior).
• But over many electrons, an interference pattern emerges (wave behavior), showing the probability distribution of where electrons are likely to land.

5. Why Speed Doesn’t Help Predict the Exact PositionPermalink

Even if we know the electron’s speed $(v)$:

• Position Uncertainty: The position is fundamentally uncertain due to the quantum nature of the particle.
• Wave-Like Spread: The wavefunction spreads out as the electron travels, increasing the uncertainty over time.

Analogy:Permalink

Imagine a wave traveling in water. You know the speed of the wave, but you cannot say “this molecule of water is here at this time.” Instead, you describe the wave as a whole, knowing that any individual molecule’s position is uncertain.

6. Practical Implication: Why This MattersPermalink

In quantum mechanics, the inability to know exact position and momentum simultaneously is not a limitation of our tools; it’s a fundamental property of nature. This is why:

• Quantum particles like electrons behave probabilistically.
• Their behavior cannot always be intuitively understood in classical terms.

SummaryPermalink

For a single electron:

• Its wave nature means it is described by a probability wave.
• We can calculate the probability of where it might be after 1256 picoseconds, but not its exact position.
• Even if we know its speed, quantum uncertainty makes the position inherently unpredictable.

Can you help me understanding wavefunction?Permalink

1. Water Waves vs. Electron WavesPermalink

Water Waves:Permalink

• In water waves, individual water molecules don’t travel far. Instead:
  • Each molecule oscillates around a fixed position.
  • The wave energy moves through the water, pushing adjacent molecules.

Electron Waves:Permalink

• Unlike water molecules in a wave, an electron itself travels from the electron gun to the slit and then to the detection screen.
• The “wave” of an electron doesn’t describe a physical oscillation. Instead, it describes the probabilities of where the electron might be found at any given moment.

Key Difference:Permalink

• For water, the wave describes the motion of many molecules working together.
• For an electron, the “wave” is not a physical object but a mathematical description (its wavefunction).

2. What Happens to a Single Electron Traveling from the Gun?Permalink

When a single electron is emitted:

1. Particle Behavior:
   • The electron is a physical object, and we can detect it as a tiny particle when it interacts with something (like the detection screen).
   • It travels the distance between the electron gun and the screen.
2. Wave-Like Behavior:
   • The electron doesn’t take a single, fixed path. Instead, its wavefunction spreads out, describing all the possible paths it might take.
   • The wavefunction evolves as it moves, meaning the probability distribution for where the electron could be is constantly changing.

3. Why Is the Wavefunction Important?Permalink

The wavefunction doesn’t mean the electron is “smeared out” like a water wave. Instead, it tells us:

• The probability of finding the electron at a particular place if we were to measure it.
• The electron travels as a single particle, but its potential paths are described by the wave.

Double-Slit Example:Permalink

• When the electron passes through the slits:
  • It behaves as though it travels through both slits simultaneously (a wave property).
  • When it hits the detection screen, it appears at one specific spot (a particle property).
• Over many electrons, the resulting interference pattern shows the wave nature.

4. Does the Electron “Really Travel”?Permalink

Yes, the electron physically moves from the gun to the detection screen. However:

• Its exact path is undefined until we measure it.
• Its wave nature means that it doesn’t follow a single straight line like a classical particle would. Instead, it explores multiple possible paths simultaneously, which are described by its wavefunction.

5. Clarification of TermsPermalink

To avoid confusion:

• The electron is a particle but has a wavefunction that describes its behavior.
• The wavefunction is not a physical wave like water but a mathematical tool to predict probabilities.

SummaryPermalink

1. A water wave describes how energy propagates through a medium (like water), but the molecules don’t travel far.
2. An electron does travel, but its behavior is governed by quantum mechanics:
   • Its motion is probabilistic and described by a wavefunction.
3. The wavefunction reflects where the electron might be at any given time, but the electron itself will appear as a particle when measured.

What path electron follow while traveling?Permalink

When we say “The electron doesn’t have a single, definite path.” Then what path it follows?

When we say “the electron doesn’t have a single, definite path,” it means that in the quantum world, the electron’s motion is not deterministic (like a classical particle following a straight line trajectory). Instead, it is described by probabilities. Let me break this down for you in detail.

1. Classical Path vs. Quantum PathPermalink

• Classical Path:
  • In classical mechanics (like throwing a ball), the trajectory of the ball is well-defined and can be calculated precisely given its initial position, speed, and direction.
  • The ball moves in a predictable, straight line or a curved trajectory (like a parabola in projectile motion).
• Quantum Path:
  • For a quantum particle like an electron, it’s not accurate to say it takes a single, specific path from point A (e.g., the electron gun) to point B (e.g., the detection screen).
  • Instead, the electron’s motion is described by its wavefunction, which spreads out and assigns probabilities to all possible paths it might take.

2. Does the Electron Move in Random Directions?Permalink

Sort of, but it’s not random in the everyday sense. Here’s what happens:

• The electron’s position and momentum are uncertain (as per Heisenberg’s Uncertainty Principle), meaning we can’t precisely track its path.
• It behaves like a wave and “explores” many possible routes simultaneously.
• If we were to measure it during its journey, we’d detect it in a specific location, but until then, all possible paths contribute to its behavior.

Visualization:Permalink

Imagine the electron “spreading out” like a ripple on a pond:

• The wavefunction represents all possible positions the electron could be in as it travels.
• Some paths (e.g., straight-line paths) may be more probable, while others (e.g., zig-zag or looping paths) are less likely but still possible.

3. Example: Double-Slit ExperimentPermalink

In the famous double-slit experiment, the electron exhibits this behavior clearly:

1. Without a detector at the slits:
   • The electron passes through both slits simultaneously as a wave.
   • The wave interferes with itself, creating an interference pattern on the detection screen.
   • This shows the electron’s path is not fixed; it “explores” all possible paths.
2. With a detector at the slits:
   • If we measure which slit the electron goes through, it behaves like a particle, taking a single path through one slit.
   • The interference pattern disappears because the wavefunction collapses to a specific path.

4. Does the Electron Move in Zig-Zags?Permalink

While we can’t directly observe the electron’s exact motion, the probabilistic nature of its path means:

• It might follow a path that’s not a straight line.
• Mathematically, we account for this using Feynman’s Path Integral approach:
  • This method sums up all possible paths (straight, zig-zag, looping) the electron might take from the source to the detector.
  • Each path contributes to the probability amplitude of the electron reaching the detection screen.

In practical terms:

• If you imagine the electron’s motion, it might “wiggle” or “spread out” in its possible positions during the journey.
• However, when you detect it, it shows up at one specific location.

5. Why Doesn’t the Electron Just Move in a Straight Line?Permalink

The deviation from a straight path happens because:

1. Wave Nature: The electron behaves like a wave, so its motion is governed by probabilities, not a single trajectory.
2. Uncertainty Principle: We can’t precisely define both its position and momentum, so its motion inherently has uncertainty.
3. Interactions: As the electron interacts with its environment (e.g., electromagnetic fields, particles in the medium), these interactions influence its path.

6. SummaryPermalink

• The electron doesn’t travel in a perfectly straight line like a classical particle.
• Instead, its motion is a superposition of all possible paths, with some paths being more probable than others.
• This behavior is not random but dictated by quantum rules, such as the wavefunction and probabilities.

What is the meaning of electron is spread out during journey?Permalink

You said “If you imagine the electron’s motion, it might “wiggle” or “spread out” in its possible positions during the journey.” Without measuring, how can we say this? After measuring anyway it is at one specific location!

1. Wavefunction: The Mathematical DescriptionPermalink

In quantum mechanics, the state of a particle (like an electron) is described by its wavefunction $(\psi (x,t))$. This wavefunction:

• Encodes all the information about the electron’s position, momentum, and other properties.
• Is a mathematical tool that predicts the probability distribution of where the electron might be found if measured.

The “wiggle” or “spread” we refer to is not the physical motion of the electron like a classical particle but rather the spread of the wavefunction.

2. Indirect Evidence from ExperimentsPermalink

Even though we can’t directly observe the “spread” of the electron before measuring, experiments show that this spread exists.

Example: Double-Slit ExperimentPermalink

1. When we fire a single electron at a double-slit setup without a detector:
   • The electron interferes with itself and produces an interference pattern on the detection screen.
   • This pattern only makes sense if the electron’s wavefunction passed through both slits simultaneously, spreading out and interacting like a wave.
2. When we add a detector to measure which slit the electron goes through:
   • The interference pattern disappears, and the electron behaves like a particle.
   • This suggests that the wavefunction collapsed to a specific path upon measurement.

Key Insight:Permalink

The interference pattern demonstrates that the electron behaved as if it explored multiple paths before being measured, consistent with the spread of the wavefunction.

3. Why Does Measurement Collapse the Wavefunction?Permalink

Before measurement, the electron’s wavefunction describes a superposition of possibilities:

• The electron might be here, there, or anywhere in between, with probabilities assigned to each position.

When we measure the electron:

• The wavefunction collapses to a single position, and we observe the electron at one specific location.

However, the interference patterns and probabilities observed across repeated experiments strongly suggest the wavefunction’s spread existed before measurement.

4. Why Trust the Wavefunction?Permalink

The wavefunction is a model, but it’s supported by decades of experimental evidence:

• Quantum mechanics correctly predicts the results of experiments like the double-slit test, tunneling in semiconductors, and atomic spectroscopy.
• Every successful application of quantum mechanics (e.g., designing semiconductors or lasers) confirms the reality of the wavefunction.

5. What Does “Wiggle” Mean?Permalink

The “wiggle” of the electron refers to the oscillating nature of its wavefunction:

• For example, a wavefunction might look like $(\psi (x)=A\sin (kx-\omega t))$, where the sine wave describes oscillations in probability amplitudes.
• These oscillations can interfere constructively or destructively, influencing where the electron is more or less likely to be detected.

6. Measurement Does Not Disprove the SpreadPermalink

When we measure an electron, we find it in one specific place. But:

• The interference patterns observed in experiments tell us that before measurement, the electron’s wavefunction described many possibilities.
• The act of measurement is what forces the electron to “choose” a location.

Think of it like this:

• A flashlight beam spreads out in all directions (wave nature), but if you place a detector in one spot, you only detect photons there.

7. Why Can’t We See the Wave Directly?Permalink

The wave nature of the electron is not something we can observe directly, but we observe its effects:

• Interference patterns, quantum tunneling, and energy quantization all point to the existence of the wavefunction.
• The spread of the wavefunction is a mathematical and conceptual representation of the electron’s possible states.

SummaryPermalink

We say the electron “spreads out” or “wiggles” because its wavefunction spreads across multiple possible paths, as confirmed by indirect evidence like interference patterns. When measured, the electron collapses to a specific location, but this doesn’t negate the prior spread of its wavefunction.

Does wavefunction keep changing during the travel?Permalink

Yes, the wavefunction changes during the “travel” of the electron! Let’s break this down clearly step by step to make sense of what’s happening:

1. Wavefunction Represents the Electron’s StatePermalink

The wavefunction $(\psi (x,t))$:

• Is a mathematical object that encodes all the information about an electron’s state.
• Depends on position $(x)$ and time $(t)$, so it evolves as the electron moves.

For instance:

• A wavefunction could look like $(\psi (x,t)=A\sin (kx-\omega t))$ (a sine wave).
• Or it could look like $(\psi (x,t)=A{e}^{-x^2})$ (a Gaussian wave packet).
• The exact shape of the wavefunction depends on the situation, like the forces acting on the electron or the boundaries it moves through.

2. Wavefunction Evolution (Changing During Travel)Permalink

The wavefunction doesn’t stay static. Instead:

• The Schrödinger equation governs how the wavefunction evolves in time and space.
• Forces, potentials, and obstacles (like slits in the double-slit experiment) influence the wavefunction’s evolution.

3. Gaussian and Sine Wave – Why Are They Different?Permalink

Gaussian Wave Packet (localized wave):Permalink

• Represents a particle that is mostly concentrated in one region, but with some spread.
• Example: A single electron fired from an electron gun.

Sine Wave (plane wave):Permalink

• Represents a particle that has a definite momentum but is spread out over all space.
• Example: A free electron with a constant energy moving through space.

Why Different?Permalink

The wavefunction’s shape depends on how the electron was prepared. A Gaussian wave packet is often used to describe a localized electron, while a sine wave represents an idealized, infinite traveling wave.

4. What Happens During Travel?Permalink

Imagine the electron’s wavefunction starting as a Gaussian wave packet:

1. Spreading: As the electron travels, its wavefunction spreads out. The uncertainty in position increases.
   • This happens because different parts of the wave move with slightly different momenta, causing the packet to spread.
   • Mathematically, the wavefunction evolves into a broader Gaussian over time.
2. Interference: If the electron encounters slits or barriers:
   • The wavefunction splits, creating overlapping regions where constructive and destructive interference occur.
   • The shape of the wavefunction becomes more complex but is still governed by the Schrödinger equation.
3. Collapse on Measurement:
   • Before measurement, the wavefunction represents a superposition of all possible positions.
   • When you measure the electron’s position, the wavefunction collapses, and you find the electron at a single location.

5. Wavefunction vs. Probability DistributionPermalink

The wavefunction $(\psi (x,t))$ is not a probability distribution itself:

• $(\psi (x,t))$ can be complex (have real and imaginary parts).
• The probability distribution is given by $(\Vert \psi (x,t){\Vert }^2)$, the square of the wavefunction’s magnitude.

For example:

• A Gaussian wavefunction $(\psi (x,t)=A{e}^{-x^2})$ leads to a Gaussian probability distribution $(\Vert \psi (x,t){\Vert }^2=\Vert A{\Vert }^2{e}^{-2x^2})$.
• A sine wave $(\psi (x,t)=A\sin (kx-\omega t))$ leads to a sinusoidal probability distribution.

The wavefunction itself can oscillate (e.g., a sine wave) or spread (e.g., a Gaussian broadening), and this influences the resulting probabilities.

6. Wavefunction Behavior During TravelPermalink

Yes, the wavefunction changes during travel because:

• The electron’s environment changes (e.g., slits, barriers, fields).
• The Schrödinger equation dictates how the wavefunction evolves over time.

Think of it like ripples in a pond:

• At first, the ripples might be localized to one area (like a Gaussian).
• As time goes on, the ripples spread and interact with the boundaries of the pond.

7. SummaryPermalink

• The wavefunction $(\psi (x,t))$ evolves dynamically as the electron moves.
• Its shape and evolution depend on the electron’s initial state, forces acting on it, and any interactions it experiences.
• A Gaussian wavefunction might start narrow but spread over time, while a sine wave remains periodic but can interact and interfere.

How can a free electron with a constant energy moving through space?Permalink

“A free electron with a constant energy moving through space.” How energy remains constant while traveling, it should reduce when reaching towards the target

1. Energy of a Free ElectronPermalink

When I mentioned “a free electron with constant energy,” I was referring to a specific case where we assume the electron is not interacting with any forces that would cause its energy to change. In this case, the energy of the electron remains constant as it travels. This is in line with the idea of conservation of energy in a closed system.

Energy of a free electron consists of:Permalink

• Kinetic Energy: This is the energy the electron has due to its motion. $E_{\text{kinetic}}=\frac{1}{2}m{v}^2$ where $(m)$ is the mass of the electron, and $(v)$ is its velocity.

When there are no external forces (such as electric or magnetic fields) acting on the electron, its kinetic energy stays the same as it travels. So, as long as no work is done on the electron, its energy doesn’t change.

2. Does Energy Change When the Electron Moves?Permalink

When you mentioned that energy should “reduce” as the electron moves towards a target, that suggests you’re thinking of a situation where energy is being transferred to something else—such as friction, collisions, or other interactions with the environment. However, in the idealized case of a “free electron” (one that is not interacting with any external forces), its energy remains constant as it moves.

This is similar to how an object traveling in deep space with no external forces acting on it would keep moving at a constant speed without losing energy.

3. Why Does Energy Stay Constant in this Case?Permalink

• In Free Space: When we say the electron is “free,” we mean it isn’t experiencing forces like:
  • Electric fields (which would change its velocity, and hence its kinetic energy),
  • Magnetic fields (which could alter the direction of motion, but wouldn’t change its speed or kinetic energy directly),
  • Friction or drag (which is only relevant in a medium like air or water, where the electron would lose energy to the medium).

Thus, in free space, as long as there’s nothing to slow the electron down or accelerate it, its kinetic energy remains constant.

4. What About Energy Loss in Real-World Situations?Permalink

In real-world situations (like when electrons travel through a conductor or interact with other particles), energy loss or gain can happen:

• In a conductor, electrons might lose energy due to collisions with atoms in the material (which is why materials have resistance).
• In a vacuum tube (for example, in an electron gun), electrons could also lose energy through collisions or other interactions with the surrounding environment.

But in the idealized quantum picture of a “free electron,” these interactions are neglected, so the energy remains constant.

5. Momentum and Energy Relationship in Free ElectronPermalink

For a free electron:

• The momentum of the electron is related to its velocity by $(p=mv)$.
• The energy (kinetic energy) is related to its momentum by $(E=\frac{p^2}{2m})$.

As long as the velocity (and thus the momentum) of the electron doesn’t change, its energy stays constant.

SummaryPermalink

• Energy of a free electron remains constant when no external forces are acting on it (it’s moving in free space with no interaction).
• In real-life situations, like in a conductor or interacting with other particles, electrons can lose energy due to collisions, resistance, or other interactions.
• In the idealized case, where the electron is free, there’s no reason for its energy to change as it moves toward the target.

How an electron can collide with atom?Permalink

electron is extremely small, how come they collide with atom and then lose energy due to collisions with atoms

You’re right that atoms are mostly empty space, but there’s a bit more happening at the microscopic level that helps explain how an electron can lose energy when moving through a conductor. Let’s break it down step by step:

1. Structure of a ConductorPermalink

While atoms are indeed mostly empty space, they still have a nucleus (made up of protons and neutrons) and electrons surrounding them in electron clouds. The electrons in a conductor, like copper or aluminum, are delocalized—meaning they are not tightly bound to a specific atom. These free electrons form what’s called the electron sea, and they can move through the material (this is the electrical current).

Even though the atoms themselves are mostly empty space, the nucleus and the electron cloud still exert forces on the moving electrons.

2. Collisions with AtomsPermalink

When an electron moves through a conductor, it’s not like it’s traveling through an empty vacuum. The electrons in the material constantly interact with the atomic nuclei and the other delocalized electrons. These interactions are primarily of two types:

• Electron-Atom Collisions: The moving free electrons can collide with the positively charged atomic nuclei (or ions) in the material. While atoms are mostly empty space, the nucleus is dense and carries a positive charge, and the electron cloud can exert forces on the moving electrons as well.
  • When the electron collides with these nuclei, it can lose some energy in the process. This is because the collision transfers some of the electron’s energy to the atomic lattice (the structure of atoms in the material), causing the atoms to vibrate more (this is what we perceive as heat).
• Electron-Electron Interactions: Electrons moving through the conductor can also interact with other free electrons in the material. These interactions can also cause the electron to scatter, losing some of its momentum and energy.

3. What Happens During the Collision?Permalink

When the electron collides with an atom (or ion) in the conductor, it doesn’t completely “stop” or lose all its energy. Instead, it loses some of its kinetic energy and changes direction. This process is scattering, and it’s a key reason why conductors have resistance.

The energy lost by the electron during a collision:

• Increases the vibration of atoms in the conductor (which corresponds to an increase in temperature).
• Slows down the electron’s movement slightly (since it loses some energy).

This resistance to the flow of electrons is what gives rise to the phenomenon we call electrical resistance in materials.

4. Why Doesn’t the Electron Go Straight Through?Permalink

Even though the atom itself is mostly empty space, the electron cloud and nucleus interact with the moving electron. Electrons are negatively charged, and the positively charged nucleus can create a force field that repels or attracts the moving electron, causing it to scatter.

In quantum mechanics, electrons don’t travel in neat, straight lines; their positions and momenta are uncertain, and they behave like waves. The wave nature of the electron also means that when it encounters the atomic lattice, it is likely to be scattered in different directions, further slowing down the movement.

5. Effect of Temperature on CollisionsPermalink

The temperature of the conductor also plays a role. At higher temperatures, the atoms in the conductor vibrate more due to increased thermal energy, which leads to more frequent collisions between the free electrons and the atoms. This is why resistance increases with temperature: more collisions = more energy loss.

6. A Simple AnalogyPermalink

Imagine you’re in a crowded room, walking through a group of people. While most of the room might be “empty” space, you still bump into people, and your movement gets slowed down. The more crowded the room (or the higher the temperature), the more often you bump into people, and the slower you move.

Similarly, an electron moving through a conductor is like a person moving through a crowded room of atoms. The more it “bumps into” the atoms, the more energy it loses, and the harder it is for the electron to move smoothly.

SummaryPermalink

• Atoms in a conductor may be small, but they still exert forces on moving electrons due to their nucleus and electron cloud.
• Electrons in conductors interact with these atoms through collisions, leading to energy loss in the form of heat.
• This energy loss is what gives rise to electrical resistance, and it’s also why electrons don’t move in straight lines but instead get scattered and lose energy.

Why electrons are negatively charged?Permalink

You said “Electrons are negatively charged,” Why do you say it is negative charged, who not to say it is positive charged? What there is the charge of electron which makes it negative?

1. The Convention of Positive and Negative ChargesPermalink

When Benjamin Franklin first investigated electricity in the 18th century, he arbitrarily decided to label the type of charge that accumulated on a glass rod when rubbed with a piece of silk as positive. The opposite charge, which accumulated on the silk, was then labeled as negative. This was just a convention, and no one knew at the time that the actual particles responsible for these charges were atoms or subatomic particles like electrons.

2. Why Is the Electron Called Negative?Permalink

After the discovery of the electron by J.J. Thomson in 1897, it was found to be a subatomic particle that carried an electric charge. Based on the earlier convention, the charge on this particle was assigned the label negative, because it was opposite to the type of charge accumulated on the glass rod (which was labeled positive).

So, the electron’s charge is actually negative, but this is purely due to the historical convention, not any inherent property of the electron itself. It is just a way of distinguishing it from the positive charge, which is carried by protons.

3. Charge of an ElectronPermalink

The charge of an electron is a fundamental property. It is: $q_{\text{electron}}=-1.602\times {10}^{-19}\text{Coulombs}$ This is a very small amount of charge, but it’s a basic unit of charge that is always associated with the electron.

• The negative sign indicates that the electron has an opposite charge relative to the proton, which carries a positive charge.

4. Why Negative?Permalink

• The reason we call the electron negative is based on historical convention and the fact that, when you place an electron in an electric field, it moves in a direction opposite to that of a positive charge.
• In an electric field, positive charges move toward the negative side, and negative charges (like electrons) move toward the positive side.

This oppositional behavior is why electrons are considered negatively charged.

5. Can We Imagine Electrons as Positive?Permalink

From a physics standpoint, the electron’s negative charge is an inherent property of the electron itself. We could have used different terminology and called the electron positive, but the behavior of the electron in electric fields would remain the same—electrons would still move in the same direction in a given field. The important thing is that electrons have a charge opposite to that of protons (which are positively charged), and that fundamental difference is what defines their interactions in physics.

6. Positive Charge: ProtonsPermalink

The proton, which is the particle found in the nucleus of an atom, carries a positive charge. Its charge is equal in magnitude to that of the electron, but with the opposite sign: $q_{\text{proton}}=+1.602\times {10}^{-19}\text{Coulombs}$

SummaryPermalink

• The negative charge of the electron is based on an historical convention and is not something inherent to the electron itself. It is opposite the positive charge of protons.
• The electron’s charge is a fundamental property and is represented as -1.602 x 10^-19 Coulombs.
• Electrons are called negative because of how they behave in an electric field relative to positively charged particles like protons.

What is the smallest measurable charge?Permalink

The smallest measurable charge is the elementary charge, denoted by e, and it represents the charge carried by a single electron or proton. The value of the elementary charge is:

This is the smallest unit of charge that can exist in isolation.

Why is it the smallest measurable charge?Permalink

In nature, charge always appears as integer multiples of this elementary charge. You won’t find a particle with a charge smaller than the elementary charge in isolation. This is because charge is quantized, meaning it only comes in discrete amounts, and the elementary charge is the smallest discrete unit.

• For example:
  • The electron carries a charge of -e (negative charge).
  • The proton carries a charge of +e (positive charge).

Measuring ChargePermalink

When we measure charge in experiments (like in a Millikan oil drop experiment), we observe that charge comes in discrete packets, each corresponding to the value of the elementary charge. This was confirmed by Millikan’s famous experiment in 1909, which measured the charge of the electron.

SummaryPermalink

• The smallest measurable charge is the elementary charge $(e=1.602\times {10}^{-19}\text{Coulombs})$.
• Charge is quantized, meaning it can only exist in integer multiples of the elementary charge.

What is coulomb and how to measure that?Permalink

The Coulomb (C) is the SI (International System of Units) unit of electric charge. It measures the quantity of electric charge.

1 Coulomb is defined as the amount of charge transported by a current of one ampere in one second.

$1\text{C}=1\text{A}\times 1\text{s}$ Where:

• A stands for amperes (unit of electric current)
• s stands for seconds

Relation to Elementary ChargePermalink

• 1 Coulomb is the charge of about $(6.242\times {10}^{18})$ electrons or protons.
• Since the charge of a single electron is $(-1.602\times {10}^{-19}\text{C})$, this means that one Coulomb contains a huge number of individual electron or proton charges.

How is Charge Measured in Coulombs?Permalink

Measuring charge in Coulombs involves determining the amount of electric charge passing through a conductor or electric circuit. There are various methods and instruments for measuring electric charge. Here are some common ways to measure charge:

1. Using an Electrometer:Permalink

An electrometer is an instrument used to measure electrical charge and electric potential. It is highly sensitive and can measure small amounts of charge.

• The electrometer can measure the voltage between two points, and using Ohm’s Law, you can compute the charge if you know the capacitance.
• A charge amplifier in electrometers can convert charge into a measurable voltage, allowing scientists to measure small charges accurately.

2. Using a Capacitor:Permalink

A capacitor is a device that stores electric charge. By charging a capacitor and measuring the voltage across it, you can determine the amount of charge stored in the capacitor.

Where:

• $(Q)$ is the charge in Coulombs.
• $(C)$ is the capacitance of the capacitor (in Farads).
• $(V)$ is the voltage across the capacitor (in Volts).

3. Using a Faraday Cup:Permalink

A Faraday cup is an instrument used to measure the amount of charge that hits its surface. It consists of a conductive cup, and the electric charge collected by the cup is measured, usually by converting the charge into a measurable current.

• The Faraday cup is commonly used in particle physics to measure the charge of particles like electrons and ions.

4. Using Current and Time (for Continuous Measurement):Permalink

If you know the current (I) in a circuit and the amount of time (t) the current flows, you can calculate the charge using the relationship:

Where:

• $(Q)$ is the total charge in Coulombs.
• $(I)$ is the current in amperes.
• $(t)$ is the time in seconds.

For example, if a current of 1 ampere flows for 1 second, then:

Summary of Units and Relationships:Permalink

• 1 Coulomb (C) = the amount of charge transported by 1 ampere of current in 1 second.
• The Coulomb is a large unit, so we often talk about much smaller units like the elementary charge $(e=1.602\times {10}^{-19}\text{C})$ for individual particles like electrons.

ConclusionPermalink

The Coulomb is the standard unit of electric charge, and we measure it through various methods like using electrometers, capacitors, and instruments like the Faraday cup. Charge is often measured as the flow of current over time or by observing the potential difference across capacitors.

There is a Gap in 1 Coulomb calculation valuePermalink

is indeed very close to 1, but not exactly 1. This small difference is due to the fact that the value of the elementary charge (the charge of a single electron or proton) isn’t perfectly aligned with the number of electrons that would make up exactly 1 Coulomb.

In reality, 6.242 × 10¹⁸ electrons is approximately equal to 1 Coulomb, but slight differences arise because:

• The exact value of the elementary charge is $(1.602176634\times {10}^{-19}\text{C})$, and in previous calculations, I had rounded it off to $(1.602\times {10}^{-19}\text{C})$.
• There are also experimental uncertainties when measuring such tiny quantities, leading to small deviations from theoretical perfection.

To summarize:Permalink

• The number $(6.242\times {10}^{18})$ electrons represents the approximate number of electrons that contribute to 1 Coulomb of charge.
• The value $(0.9999684)$ is extremely close to 1, but it indicates that the number is an approximation due to rounding and the practical measurement of quantities on the atomic scale.

This slight difference doesn’t significantly affect most practical calculations and experiments where we’re dealing with charge at macroscopic scales.

How do we count that the number of electron is 6.242×10 ^18Permalink

The value $(6.242\times {10}^{18})$ electrons per Coulomb is based on the elementary charge of an electron, which is $(1.602\times {10}^{-19}\text{C})$. To understand how this number is determined, let’s walk through the process:

Step-by-step Breakdown:Permalink

1. Elementary Charge (e):
   • The charge of a single electron is $(e=-1.602\times {10}^{-19}\text{C})$. This is a fundamental constant in physics, representing the smallest unit of electric charge.
2. Total Charge (Q) for 1 Coulomb:
   • One Coulomb (C) is the total charge that flows through a conductor when a current of 1 ampere is sustained for 1 second.
   • Therefore, the total charge we want to consider is $(1\text{C})$.
3. Calculating the Number of Electrons in 1 Coulomb:
   • To find how many electrons (or protons) are needed to make up 1 Coulomb of charge, we divide the total charge $(1\text{C})$ by the charge of a single electron $(e)$:

$\text{Number of electrons}=\frac{\text{Total charge}}{\text{Charge of one electron}}$ $=\frac{1\text{C}}{1.602\times {10}^{-19}\text{C/electron}}=6.242\times {10}^{18}\text{electrons}$

So, 1 Coulomb corresponds to about $(6.242\times {10}^{18})$ electrons, based on the known value of the elementary charge.

Understanding the Process:Permalink

• This calculation assumes that the charge of each electron is consistent (and known) at $(1.602\times {10}^{-19}\text{C})$.
• By dividing the total charge (1 Coulomb) by the charge of a single electron, we obtain the number of electrons required to produce that total charge.

Why this Calculation Works:Permalink

This relationship is fundamental to electricity and electronics, and it allows us to relate macroscopic quantities of charge (like Coulombs) to the behavior of microscopic particles (like electrons). It provides a bridge between atomic-scale properties and the measurable quantities in everyday life.

Conclusion:Permalink

• We arrive at $(6.242\times {10}^{18})$ electrons per Coulomb by dividing the total charge (1 Coulomb) by the elementary charge of a single electron.
• This value is an approximation based on the known value of the elementary charge, and it helps us understand how many electrons are required to make up 1 Coulomb.

How did we reach to this conclusion that 1ev or electron’s charge is exactly this?Permalink

The value of the electron’s charge, $(1.602\times {10}^{-19}\text{C})$, was not just determined through theory but through careful experimental measurements and scientific experiments. Let me explain how scientists arrived at this conclusion step by step.

How the Charge of an Electron Was DeterminedPermalink

1. Millikan’s Oil Drop Experiment (1909)Permalink

One of the most famous experiments that determined the charge of the electron was Robert Millikan’s Oil Drop Experiment. In this experiment, Millikan measured the force on tiny oil droplets suspended between two charged plates. Here’s a brief overview of the experiment:

• Millikan’s Setup: He sprayed tiny droplets of oil between two parallel charged plates. He adjusted the electric field between the plates to suspend individual droplets in mid-air (in equilibrium), balancing the force due to gravity with the force due to the electric field.
• The Key Idea: The oil droplets were negatively charged (due to friction), and by adjusting the electric field, Millikan could precisely determine the charge on individual droplets.
• Data Collection: By carefully measuring the velocity of oil droplets in different positions and the strength of the applied electric field, he was able to determine the discrete charges on the droplets.

2. Key Finding:Permalink

• Millikan observed that the charges on the droplets were always multiples of a smallest unit. This unit was consistent across many trials, and it turned out to be the charge of a single electron.
• From this experiment, Millikan deduced that the charge of a single electron is approximately $(1.602\times {10}^{-19}\text{C})$.

3. Why Millikan’s Experiment Worked:Permalink

• Millikan used very precise equipment and methods to suspend and measure the droplets. The balance of forces acting on the droplets allowed for an extremely accurate determination of the charge, and by observing that the charges were quantized (appeared in whole multiples of a single value), he concluded that this was the fundamental unit of electric charge, which we now call the elementary charge.

4. Modern Refinements:Permalink

• Since Millikan’s time, scientists have refined the methods used to measure the charge of the electron, but his experiment was the most direct and influential. Modern techniques, such as using Josephson junctions and quantum Hall effect measurements, have confirmed Millikan’s result with much higher precision.

Understanding the Value of $(1.602\times {10}^{-19}\text{C})$Permalink

• The value $(1.602\times {10}^{-19}\text{C})$ is the fundamental charge carried by a single electron (or proton), and it is the smallest charge that can be isolated and measured in nature.
• This value is so small that it’s hard to visualize, but it is extremely significant because it defines the quantization of charge in the universe. It means that charges in nature can only exist in multiples of this fundamental value.

How We Use This Charge TodayPermalink

• The charge of the electron is fundamental to understanding electricity, magnetism, semiconductors, and a host of other fields.
• This value is used in equations like Coulomb’s Law, which describes the force between charged particles, and in calculating currents and electric fields.

Summary:Permalink

• The value of $(1.602\times {10}^{-19}\text{C})$ (the charge of an electron) was experimentally determined by Robert Millikan in his famous oil drop experiment.
• Through careful measurement, Millikan demonstrated that the charge on oil droplets was always a multiple of a small unit, which he identified as the elementary charge (the charge of a single electron).
• Modern experiments have confirmed and refined this value, which is now a cornerstone of quantum mechanics, electrodynamics, and electrical engineering.

How much is 1 coulomb charge?Permalink

If 6.242×10 ^18 electrons can move in any medium for one second then they will produce 1 C charge?

Concept: Current and ChargePermalink

• Charge (Q) is measured in Coulombs (C). One Coulomb represents the total charge that moves through a conductor when a current of 1 Ampere (A) flows for 1 second.

• Current (I) is the rate at which charge flows, and it’s measured in Amperes (A). One Ampere is defined as one Coulomb of charge flowing per second:

where:

• $(I)$ = current (in Amperes, A),
• $(Q)$ = charge (in Coulombs, C),
• $(t)$ = time (in seconds, s).

Connecting Electrons and CoulombsPermalink

• The charge of a single electron is $(e=1.602\times {10}^{-19}\text{C})$.
• 1 Coulomb (C) of charge corresponds to the movement of $(6.242\times {10}^{18})$ electrons.

So, if $(6.242\times {10}^{18})$ electrons move through a conductor in 1 second, they will carry a total charge of 1 Coulomb (C).

To Summarize:Permalink

• If 6.242 × 10¹⁸ electrons move through a conductor in 1 second, they will create a 1 Coulomb (C) of charge.
• This is because $(1\text{C})$ of charge corresponds to exactly $(6.242\times {10}^{18})$ electrons (since each electron has a charge of $(1.602\times {10}^{-19}\text{C})$.

In practical terms:

• When you have a current of 1 Ampere, 1 Coulomb of charge flows through a conductor in 1 second, which corresponds to about $(6.242\times {10}^{18})$ electrons moving through that conductor.

How to measure how many electron has moved?Permalink

Measuring the number of electrons that have moved through a conductor involves understanding the current and time, and then using the relationship between charge and the number of electrons. Here’s how this works step-by-step:

1. Measure the CurrentPermalink

To measure the number of electrons that have moved, you first need to measure the electric current flowing through the conductor. The current, $(I)$, is the flow of charge per unit time and is measured in Amperes (A). One Ampere is defined as one Coulomb per second:

where:

• $(I)$ = current (in Amperes, A),
• $(Q)$ = total charge (in Coulombs, C),
• $(t)$ = time (in seconds, s).

2. Calculate the Total ChargePermalink

Once you measure the current, you can calculate the total charge, $(Q)$, that has passed through the conductor in a given time $(t)$.

• If the current is $(1\text{A})$ and the time is $(1\text{s})$, the total charge is $(1\text{C})$.
• If the current is $(2\text{A})$ and the time is $(1\text{s})$, the total charge is $(2\text{C})$, and so on.

3. Relating Charge to the Number of ElectronsPermalink

To find the number of electrons that correspond to the total charge $(Q)$, you use the charge of an electron, which is $(1.602\times {10}^{-19}\text{C})$.

The total number of electrons, $(N_e)$, that have moved is:

For example, if 1 Coulomb of charge passes through a conductor:

Step-by-Step Example:Permalink

• Suppose you have a current of $(2\text{A})$ flowing for $(3\text{seconds})$.
• The total charge that has passed through the conductor is: $Q=2\text{A}\times 3\text{s}=6\text{C}$
• Now, to find the number of electrons that correspond to this charge: $N_e=\frac{6\text{C}}{1.602\times {10}^{-19}\text{C/electron}}=3.745\times {10}^{19}\text{electrons}$

So, in this case, $(3.745\times {10}^{19})$ electrons would have moved through the conductor in those 3 seconds.

How to Actually Measure It:Permalink

In practice, measuring the exact number of electrons directly is difficult, but you can measure the current (in Amperes) using a multimeter or ammeter, and then calculate the number of electrons based on the current and time. Alternatively, advanced instruments like coulombmeters or integrated circuits could be used to directly measure the amount of charge passing through a conductor.

Summary:Permalink

• Step 1: Measure the current (in Amperes) and the time it flows.
• Step 2: Calculate the total charge that has passed using $(Q=I\times t)$.
• Step 3: Calculate the number of electrons using the relationship $(N_e=\frac{Q}{\text{charge of an electron}})$.

This is how we estimate the number of electrons that have moved in a given period based on the flow of current.

If I want to measure number of electron moved in 10 watt bulb in 1 hour then how to do that?Permalink

To calculate the number of electrons that moved through a 10-watt bulb in 1 hour, we need to follow these steps:

1. Calculate the Total Energy Used by the BulbPermalink

First, let’s find out how much energy the bulb consumes in 1 hour. Since the bulb is rated at 10 watts, it consumes 10 joules of energy per second.

• Power (P) = 10 watts, and 1 watt = 1 joule/second.
• Therefore, the energy used in 1 hour (which is 3600 seconds) is:

$\text{Energy}=\text{Power}\times \text{Time}$ $\text{Energy}=10\text{W}\times 3600\text{s}=36,000\text{J}$

So, the total energy consumed by the bulb in 1 hour is 36,000 joules.

2. Calculate the Total ChargePermalink

To find the total charge, we use the relationship between energy, charge, and voltage:

Assuming the bulb operates at a typical household voltage of 120V (or 230V, depending on your region, but let’s use 120V for this example), we can rearrange the equation to solve for the total charge $(Q)$:

For 120V:

So, the total charge that has passed through the bulb is 300 Coulombs.

3. Calculate the Number of ElectronsPermalink

Now that we know the total charge, we can find the number of electrons that correspond to this charge. The charge of one electron is $(1.602\times {10}^{-19}\text{C})$.

The number of electrons $(N_e)$ is:

Summary:Permalink

To calculate the number of electrons that moved through the 10-watt bulb in 1 hour:

• Total energy used by the bulb in 1 hour = 36,000 joules.
• Total charge (Q) = 300 Coulombs.
• Number of electrons = $(1.87\times {10}^{21}\text{electrons})$.

So, approximately 1.87 × 10²¹ electrons flow through the bulb in 1 hour.

How to measure 1 joule?Permalink

To measure 1 joule of energy, we need to understand that a joule (J) is the standard unit of energy in the International System of Units (SI), and it represents the amount of energy transferred when one newton of force is applied to move an object one meter.

Here’s how you can measure 1 joule in different contexts:

1. Mechanical Work (Force and Distance)Permalink

One joule is the amount of energy used when applying a force of one newton to move an object one meter in the direction of the force. In simple terms, it can be expressed as:

• Force = 1 newton (N)
• Distance = 1 meter (m)

To measure 1 joule of energy using this method:

• Apply a force of 1 newton to an object.
• Move the object 1 meter in the direction of the force.
• The energy transferred will be 1 joule.

Example:Permalink

• Lifting an object weighing about 102 grams (0.102 kg) against gravity over a height of 1 meter will do 1 joule of work.

2. Electrical EnergyPermalink

In the context of electrical energy, 1 joule is the amount of energy consumed when 1 watt of power is used for 1 second.

• 1 joule = 1 watt × 1 second

So, if you have an electrical device that operates at 1 watt of power, and it runs for 1 second, the energy consumed is 1 joule.

Example:Permalink

• A 1-watt LED light bulb that stays on for 1 second will consume 1 joule of energy.

3. Heat EnergyPermalink

In terms of heat, 1 joule is also the amount of heat needed to raise the temperature of 0.239 grams of water by 1°C (or 1 Kelvin).

Example:Permalink

• Adding 1 joule of energy to 0.239 grams of water will increase its temperature by 1°C.

4. Using a CalorimeterPermalink

To measure 1 joule of energy in practical situations, you can use a calorimeter. A calorimeter measures the amount of heat energy absorbed or released by a substance. By using this method, you can calculate the energy involved in heating substances, and thus measure 1 joule in terms of heat.

5. In Terms of Electrical Circuit (Measuring Voltage and Current)Permalink

If you have a circuit with voltage (V) and current (I), you can calculate the energy in joules using the formula:

Where:

• $(E)$ = energy in joules (J),
• $(V)$ = voltage in volts (V),
• $(I)$ = current in amperes (A),
• $(t)$ = time in seconds (s).

For example, a circuit with:

• 1 volt across a resistor,
• 1 ampere of current flowing,
• and running for 1 second,

will use:

Summary:Permalink

To measure 1 joule, you can:

• Apply a force of 1 newton over 1 meter.
• Use 1 watt of electrical power for 1 second.
• Add energy to 0.239 grams of water to raise its temperature by 1°C.

By measuring the work done, power consumed, or heat transferred, you can effectively measure 1 joule of energy in various contexts.

There is confusion in 1 joule calculation.Permalink

To measure 1 joule, we can:

Apply a force of 1 newton over 1 meter.
Use 1 watt of electrical power for 1 second.
Add energy to 0.239 grams of water to raise its temperature by 1°C.

This is very amazing fact. How do we know that all 3 are same? any coversion principle between physical work, electricity and temprature?

You’re right to ask about how we know that these different methods—applying mechanical force, using electrical power, and heating water—are all equivalent to 1 joule. The key lies in the principle of energy conservation and the equivalence of different forms of energy.

The Conservation of EnergyPermalink

The conservation of energy is a fundamental principle in physics that states energy cannot be created or destroyed, only converted from one form to another. This means that when you perform mechanical work, apply electrical energy, or heat a substance, you are simply transferring energy from one form to another. The unit of energy remains the same in all contexts.

Conversion Between Forms of EnergyPermalink

1. Mechanical Work and Electrical Energy:
   • 1 joule of work (mechanical energy) is the energy required to move an object with a force of 1 newton over a distance of 1 meter.
   • 1 joule of electrical energy is the energy used when a current of 1 ampere flows through a resistance of 1 ohm for 1 second. You can also think of it as the energy consumed by a device with 1 watt of power for 1 second.

   These are both forms of mechanical energy and electrical energy, and we know they are equivalent because work and electricity both fall under the umbrella of energy transfer. A watt, for instance, is a unit of power that is defined as 1 joule per second. Thus, when electrical energy is used in a circuit for a given time, the conversion between electrical power and mechanical work can be traced back to the same unit of energy.

   • 1 joule = 1 watt × 1 second
   • The work done by a force (mechanical energy) or electric power can be described using the same unit because power and energy are interconnected, with energy being the integral of power over time.

2. Thermal Energy (Heating Water) and Mechanical/Electrical Energy: When you add thermal energy (heat) to a substance like water, you are transferring energy to increase its temperature. The equivalence between thermal energy and mechanical/electrical energy comes from the specific heat capacity of water and other materials, which allows us to calculate how much energy (in joules) is required to raise the temperature of a substance.

   The specific heat capacity of water is 4.186 joules per gram per degree Celsius. This means it takes 4.186 joules of energy to raise the temperature of 1 gram of water by 1°C. When you know the specific heat and the mass of the substance you’re heating, you can calculate how much energy (in joules) you need to supply.

   • For example: To raise the temperature of 0.239 grams of water by 1°C, you need exactly 1 joule of energy.

   This heat energy is essentially thermal energy, and it can be converted to mechanical energy or electrical energy. For instance, steam engines and thermoelectric generators are examples where heat energy is converted into mechanical or electrical energy.

The Key Concept: Energy is Conserved and Measured in JoulesPermalink

No matter how energy is transferred (as heat, work, or electricity), we use the same unit—joules—to measure it. This is because all forms of energy are equivalent under the principle of energy conservation, and they can be converted from one to another. Through experimental observation, we have established that:

• 1 joule of mechanical work = 1 joule of electrical energy = 1 joule of thermal energy (when measured over the same amount of time, distance, mass, or temperature change).

The conversion between physical work, electricity, and temperature change is established through empirical measurements, historical experiments, and the mathematical formulas we use to calculate work, heat, and electricity. These conversions are based on physical constants and the behavior of matter under different conditions.

Examples of Conversion:Permalink

1. Mechanical Work to Electrical Energy:
   • A generator converts mechanical energy (from moving parts) into electrical energy. This process relies on the fact that work done on a system can be converted into electrical energy in a consistent way.
2. Electrical Energy to Thermal Energy:
   • Resistors in electrical circuits convert electrical energy into heat energy due to Joule heating. The energy dissipated in a resistor (in joules) can be calculated by the formula: $\text{Energy (J)}=I^2\times R\times t$ Where I is the current, R is the resistance, and t is the time.
3. Thermal Energy to Mechanical Energy:
   • In heat engines, thermal energy (from burning fuel) is converted into mechanical work. The amount of work done is directly related to the thermal energy supplied to the engine, which is then measured in joules.

Summary:Permalink

The conversion principles between mechanical, electrical, and thermal energy are based on the law of conservation of energy. All energy forms can be converted into each other, and joules are the standard unit used to measure all types of energy. The equivalent energy values are determined through experiments and mathematical relationships (e.g., specific heat for heating, voltage and current for electrical work, force and distance for mechanical work). This is why 1 joule of energy is the same, whether it’s applied as mechanical work, electrical energy, or heat.

Conservation of energy is a fundamental principle in physicsPermalink

You said “The conservation of energy is a fundamental principle in physics that states energy cannot be created or destroyed, only converted from one form to another.” But when we destroy the matter then energy can be created. And we condense energy we can create matter, isn’t it?

Yes, you’re absolutely right! You’ve touched upon some fascinating and deep concepts from modern physics. Let’s break down the ideas you’re raising:

1. Conservation of Energy and MatterPermalink

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. However, this principle holds within the known systems and classical physics. It applies well to most everyday situations.

However, Einstein’s theory of relativity (specifically the equation E = mc²) tells us that energy and matter are interchangeable. This is a revolutionary concept that goes beyond the classical understanding of energy conservation.

2. Matter and Energy Equivalence (E = mc²)Permalink

In the equation E = mc², E represents energy, m represents mass, and c represents the speed of light. This shows that mass and energy are two sides of the same coin. Essentially, mass can be converted into energy, and energy can be converted into mass.

Examples of Energy-Matter Interconversion:Permalink

• When matter is destroyed (like in particle-antiparticle annihilation), the mass of the particles is converted into pure energy in the form of gamma rays (high-energy photons). This is a real-world example of energy being “created” from matter, as the mass of the matter is transformed into energy.

• When energy condenses into matter, this is typically seen in high-energy processes like pair production. For example, when high-energy photons (gamma rays) interact with a strong electromagnetic field, they can produce a particle-antiparticle pair (such as an electron and a positron). This is how energy can be converted into matter, as the energy of the photons becomes the mass of the particles.

Thus, while energy cannot simply “disappear” or be “created from nothing,” matter can indeed be converted to energy and vice versa. This transformation process is fundamental to understanding the universe at both cosmological scales (like stars, supernovae, and black holes) and microscopic scales (like particle collisions).

3. “Destroying Matter” and Creating EnergyPermalink

When matter is destroyed or converted, as in nuclear reactions (fission or fusion), the energy released is far greater than what you would expect from chemical reactions. This is because the mass of the matter is not entirely destroyed; part of it is converted into energy, as explained by E = mc².

For instance, in the sun, hydrogen atoms undergo fusion to form helium, and in the process, a small amount of mass is lost and converted into energy (in the form of light and heat). This is a direct example of energy being released by converting mass into energy.

4. “Condensing Energy to Create Matter”Permalink

Yes, in the right conditions, energy can condense into matter. One of the most famous instances of this is the Big Bang, where immense amounts of energy condensed to form the matter of the universe. More commonly, pair production is a quantum process where energy (such as high-energy photons) converts into a particle-antiparticle pair.

How it Works:Permalink

• In pair production, a photon with enough energy (greater than 1.022 MeV, the combined mass-energy of an electron and a positron) can transform into an electron and a positron (which is the antimatter counterpart of the electron). The photon’s energy is effectively converted into mass in the form of two particles.

• This process occurs only under certain conditions (such as in the presence of a strong electromagnetic field), because the energy needs to be sufficient to create the rest mass of the particles.

5. The Role of Energy in Creating Matter:Permalink

In high-energy physics (like in particle accelerators), scientists can create new particles by colliding high-energy particles (such as protons) together. The energy of the collision is sometimes enough to create new particles, showing that energy can indeed be converted into matter.

For example, when two protons collide at very high speeds in a particle accelerator (like the Large Hadron Collider), the kinetic energy of the protons can convert into various new particles (like Higgs bosons or other exotic particles). This shows that energy is not only conserved but can create new matter under the right conditions.

Summary of Key Concepts:Permalink

• Energy conservation holds true in classical physics, where energy cannot be created or destroyed, only converted.
• Matter and energy are interchangeable—as expressed by E = mc²—and under certain conditions, energy can be converted into matter and vice versa.
• Destruction of matter (such as in particle-antiparticle annihilation) creates energy, while energy condensing into matter happens in processes like pair production or particle collisions.

How come massless gives rise to mass?Permalink

You said “This is how energy can be converted into matter, as the energy of the photons becomes the mass of the particles.”
It means massless photon give rise to a mass particle.

Yes, that’s correct! The key idea here is that photons, which are massless particles, can indeed give rise to massive particles under certain conditions. This is a fascinating concept in quantum field theory and high-energy physics, where energy and mass are intimately connected.

Here’s how it works:

Photon and Pair Production:Permalink

A photon, despite being massless, carries energy and momentum. When a photon with sufficiently high energy (greater than 1.022 MeV) interacts with a strong electromagnetic field (typically near a nucleus or in the presence of intense fields), it can produce a particle-antiparticle pair. For example, it can create an electron and a positron, which are massive particles.

The key to understanding this is that the energy of the photon (E) gets converted into mass (m) of the particles. This conversion follows the equation E = mc², where the energy of the photon is transformed into the rest mass of the newly created particles.

Example:Permalink

• A photon with enough energy (greater than 1.022 MeV, the combined rest mass energy of an electron and a positron) can produce an electron and a positron (which is the antimatter counterpart of the electron). Both of these particles have mass, even though the photon itself does not.

Energy to Mass Conversion:Permalink

This process doesn’t violate the principle that energy and mass are interchangeable (E = mc²). The energy of the photon is being converted into mass during pair production. Essentially, the photon’s energy “creates” mass when it’s able to produce a particle-antiparticle pair.

It’s important to note that this process requires certain conditions, such as:

• The photon must have high enough energy (at least 1.022 MeV).
• The process typically happens in the presence of a strong electromagnetic field, like that near a nucleus, because it helps conserve momentum during the interaction.

Why Doesn’t This Contradict the Idea of Massless Photons?Permalink

Photons are indeed massless in the sense that they have no rest mass. However, they still carry energy and momentum, and energy and momentum are related to mass through Einstein’s famous equation $(E=m{c}^2)$. When a photon interacts with a field, its energy can manifest as mass in the form of particles. This doesn’t contradict the fact that photons are massless; instead, it demonstrates how energy can create matter.

Key Takeaway:Permalink

So, even though photons are massless, their energy can lead to the creation of massive particles under the right conditions. This is a fundamental process in quantum mechanics and field theory and plays a role in phenomena like pair production.

How can a photon achieve high energy?Permalink

A photon can achieve high energy in several ways. The energy of a photon is directly related to its frequency or wavelength via the equation:

Where:

• E is the energy of the photon.
• h is Planck’s constant (approximately $(6.626\times {10}^{-34})$ joule-seconds).
• f is the frequency of the photon.

Alternatively, since the frequency and wavelength are inversely related $(f=\frac{c}{\lambda })$, where $(c)$ is the speed of light and $(\lambda )$ is the wavelength), the energy can also be expressed as:

Where:

• $(\lambda )$ is the wavelength of the photon.

How can a photon achieve high energy?Permalink

1. High Frequency (or Short Wavelength)Permalink

• Frequency and energy are directly proportional. Therefore, if a photon has a higher frequency, it will carry more energy.
• For example, gamma rays and X-rays are high-frequency photons and carry a lot of energy. In contrast, radio waves and microwaves have much lower frequencies and, hence, lower energy.

2. Emitting from High-Energy SourcesPermalink

Photons can be generated with high energy from sources that involve intense energy processes. Some examples include:

• Nuclear Reactions: In nuclear decay or reactions, high-energy photons (like gamma rays) are emitted. For example, during fission or fusion reactions, the energy released can produce high-energy photons.

• Particle Accelerators: High-energy photons can be generated by accelerating charged particles (like electrons) to very high speeds and then colliding them with matter. These collisions can produce photons with high energy, such as in Bremsstrahlung radiation or synchrotron radiation.

• Black Hole and Neutron Star Phenomena: In the intense gravitational fields near black holes or neutron stars, particles are accelerated to near light-speed, producing high-energy photons (like gamma rays) through processes such as inverse Compton scattering.

• Laser Systems: Lasers can also produce high-energy photons. For instance, high-powered lasers used in medical procedures, research, or military applications can emit photons with specific high energies depending on the material and the excitation method.

3. Relativistic Particles and Photon EmissionPermalink

• Relativistic particles (particles moving at speeds close to the speed of light) can emit high-energy photons when they are deflected or decelerated, such as in synchrotron radiation or Compton scattering.
• When a charged particle, such as an electron, is accelerated or decelerated rapidly (as in circular motion in a magnetic field), it emits photons with high energy. These are examples of electromagnetic radiation emitted at high energy.

4. Energy Transitions in Atoms and MoleculesPermalink

• In atoms and molecules, electrons can transition between different energy levels. When electrons fall from a higher energy level to a lower one, they emit photons. The energy of the emitted photon is equal to the difference between the two energy levels. In some cases, this can result in high-energy photons like UV or X-rays, depending on the energy levels involved.

5. Cosmic SourcesPermalink

• Gamma-ray bursts from distant galaxies and supernova explosions are cosmic events that release an enormous amount of energy in the form of high-energy photons (gamma rays).
• Cosmic rays (which are high-energy particles from outer space) can interact with the Earth’s atmosphere, producing high-energy photons through processes like Cherenkov radiation.

Examples of High-Energy Photons:Permalink

1. Gamma Rays: These photons have extremely high energy (greater than 100 keV) and are produced in nuclear reactions, particle accelerators, or certain astrophysical phenomena (e.g., black holes, supernovae).

2. X-rays: These photons have high energy, typically ranging from a few keV to a few hundred keV. They are produced in high-energy processes like X-ray machines and cosmic events.

3. UV Light: Ultraviolet photons are less energetic than X-rays but still high-energy compared to visible light. These photons are emitted by the Sun and some artificial sources like UV lamps.

To Summarize:Permalink

A photon can achieve high energy by:

• Having a high frequency or short wavelength.
• Being emitted by a source with intense energy processes, like nuclear reactions, particle accelerators, or astrophysical phenomena.
• Interacting with high-energy particles or systems.

How photon can have momentum when mass is zero?Permalink

A photon, despite having zero mass, can still carry momentum due to its energy. This is a key concept in relativity. The key to understanding this is the relationship between energy, momentum, and mass in special relativity.

Energy and Momentum of a Photon:Permalink

Even though a photon has zero rest mass, it still carries energy and momentum. This is described by the equation:

Where:

• E is the total energy of the particle (photon in this case).
• p is the momentum of the photon.
• c is the speed of light.
• m₀ is the rest mass of the photon (which is zero).

Since the rest mass of a photon is zero $(m_0=0)$, this simplifies to:

This shows that the energy of a photon is related to its momentum through the speed of light, $(c)$.

• The energy of a photon is given by $(E=h\cdot f)$, where $(h)$ is Planck’s constant and $(f)$ is the frequency of the photon.
• Using this in the equation, we get:

This leads to the photon momentum being:

or equivalently:

Key Points:Permalink

• Massless photons have non-zero momentum.
• Their momentum is directly proportional to their energy and inversely proportional to the speed of light.

Why Does This Matter?Permalink

Even though photons are massless, they still interact with matter and can transfer momentum. This property is what allows phenomena such as:

• The pressure of light: When photons strike a surface, they exert pressure (known as radiation pressure). This is why solar sails can be propelled by light from the Sun, even though the photons have no rest mass.
• Compton scattering: In this phenomenon, a photon collides with an electron, transferring momentum to the electron and causing a shift in the photon’s wavelength.

Example of Photon Momentum in Action:Permalink

Consider light striking a mirror:

• The light carries momentum. When it strikes the mirror, the momentum of the photons is transferred to the mirror, causing it to move slightly.
• If the mirror absorbs the light, the photons impart all their momentum to the mirror. If the mirror reflects the light, it transfers twice the momentum because the direction of the photon’s momentum changes.

To Summarize:Permalink

A photon can carry momentum despite having zero rest mass because of its energy. The energy and momentum of a photon are related through the speed of light, and this allows the photon to interact with matter just like a particle with mass, even though it doesn’t have rest mass itself.

What is the size of photon and electron that they collide and effect each other?Permalink

you said “Compton scattering: In this phenomenon, a photon collides with an electron, transferring momentum to the electron and causing a shift in the photon’s wavelength.” What is there size to collide with each other?

The size of a photon and an electron are fundamentally different, and in quantum mechanics, particles like photons and electrons are not thought of as having a “size” in the classical sense. Instead, their properties are governed by wavefunctions and probability distributions.

However, let’s break down how Compton scattering works in terms of their interaction:

Photon Size:Permalink

• A photon is a quantum of electromagnetic radiation and is considered a point particle in terms of classical physics.
• It doesn’t have a well-defined size because it doesn’t have a physical structure like a macroscopic object. The “size” of a photon is often described in terms of its wavelength. For instance, a photon with a wavelength of 500 nm (in the visible light spectrum) corresponds to a “size” of around that value, but this is more of a spatial property of the wave, not the photon itself.
• The photon’s interaction with matter is governed by its energy (related to its frequency) and momentum, not its “size.”

Electron Size:Permalink

• An electron is often described as a point-like particle with no known internal structure. It has a very small size, but in quantum mechanics, it’s considered a fundamental particle with a probability distribution.
• In terms of classical physics, we could describe the classical radius of an electron, which is based on how its electromagnetic field interacts with other charges. This value is extremely small — about 2.8179 × 10⁻¹⁵ meters, known as the classical electron radius. However, this is not a real “size” in a strict sense, since electrons don’t have a definite size or shape like macroscopic objects. It’s more of a derived quantity based on classical field theory.

How Compton Scattering Works:Permalink

Compton scattering occurs when a photon (with energy and momentum) interacts with an electron, typically in a material.

• At the quantum level, this interaction is not about the physical size of the electron and photon but about their wavefunctions and probability distributions.
• When a photon collides with an electron, it transfers energy and momentum to the electron. The electron, in turn, gains kinetic energy (a shift in momentum), and this causes the photon to lose some of its energy, resulting in a longer wavelength (redshift).

Interaction of Photon and Electron:Permalink

• The interaction happens when the photon’s electromagnetic field interacts with the electron’s charge. The photon doesn’t need a specific “size” to interact with the electron — it is more about the energy and the probability of interaction between their respective quantum states.
• The Compton wavelength of an electron (which is different from the photon’s wavelength) is a key factor in determining the dynamics of the scattering event. For an electron, the Compton wavelength is given by:

Where:

• $({\lambda }_C)$ is the Compton wavelength,
• $(h)$ is Planck’s constant,
• $(m_e)$ is the mass of the electron,
• $(c)$ is the speed of light.

For the electron, this is a characteristic scale related to its quantum properties, not its physical size.

Why Do They Interact?Permalink

• In Compton scattering, the momentum and energy transfer between the photon and the electron happens due to their wave-like properties and the electromagnetic interaction between the photon and the charged electron.
• The “collision” doesn’t require the electron and photon to have specific sizes but instead depends on the probability of interaction at the quantum level.

Summary:Permalink

• The photon doesn’t have a defined size in the traditional sense; its properties are described by its wavelength and energy.
• The electron is considered point-like in modern physics, but its properties are described by a wavefunction that governs the probability of where it might be found.
• In Compton scattering, the interaction between the photon and electron occurs due to the exchange of energy and momentum, not because of physical sizes or direct collisions in the classical sense. It’s a quantum mechanical interaction based on the probability of the photon and electron overlapping in terms of their wavefunctions.

We don’t know the size of a light particle of 500 nm wavelength!Permalink

You are saying a light particle of 500 nm wavelength we don’t know the size of it?

Yes, that’s correct. When we talk about a photon (a light particle), we don’t define its size in the way we do for macroscopic objects or even other particles like electrons. Instead, the photon is described by its wave properties, particularly its wavelength and frequency, which are related to its energy.

Why don’t we define a photon’s size in traditional terms?Permalink

1. Photons are Quantum Objects: A photon is a quantum of electromagnetic radiation. It doesn’t have a physical size like a classical object because it doesn’t have a spatially extended structure. It is described by its wavefunction, which gives the probability of where the photon is likely to be found.

2. Wave-Particle Duality: The photon behaves both as a particle (with energy and momentum) and as a wave (with wavelength and frequency). The wave nature means that instead of thinking of it as a particle with a physical size, we consider it in terms of its wavelength. For example, a photon with a wavelength of 500 nm (nanometers) is in the visible light spectrum and has certain properties that describe its energy and momentum. The wavelength of the photon is related to the spatial extent of its wave-like behavior.

3. Photon’s “Size” as Wavelength: In some ways, you can think of the wavelength as a “spatial” characteristic. For a photon with a wavelength of 500 nm, this is the spatial extent over which its wave oscillates. However, this does not mean the photon has a “size” like a ball or a particle. It simply means that the probability of the photon being found in a specific place follows a wave distribution that can be described by its wavelength.

4. Uncertainty Principle: The Heisenberg Uncertainty Principle tells us that for quantum objects like photons, we can’t precisely know both their position and momentum at the same time. This is another reason why we don’t think about photons in terms of fixed, localized sizes. Instead, they are described by their wavefunction, which gives the probability of detecting the photon in different regions of space.

The Wavelength and SizePermalink

For a photon with a wavelength of 500 nm (which is in the visible light range), the wavelength refers to the distance between two consecutive crests of the electromagnetic wave. This gives us an idea of the spatial extent of the photon’s wave-like behavior, but not its “size” in a classical sense.

In summary, we don’t define a photon’s size in traditional terms because it doesn’t have a well-defined spatial extent like classical objects. Instead, its properties are described by its wavelength (like 500 nm for visible light), energy, and momentum. These properties determine how the photon interacts with matter and how its behavior is observed.

When electron don’t have definite size then how do we measure their mass?Permalink

You said “electrons don’t have a definite size or shape like macroscopic objects.” then how do we measure the mass of this? Secondly, then what is that for which we are saying that “probability of finding electron”

Let’s break this down step by step:

1. Measuring the Weight of Electrons:Permalink

Electrons are subatomic particles, so they don’t have a “size” in the classical sense (like a baseball or a rock). However, they do have measurable mass and energy.

• The mass of an electron is approximately 9.11 × 10^−31 kilograms. This mass is one of the fundamental properties of the electron, and it is not related to the electron’s “size” in a physical sense but is an intrinsic property.

• The weight of an electron depends on the gravitational field it is in, just like any other object. However, because the mass is so small, the weight of an electron in Earth’s gravitational field is minuscule and difficult to measure directly.

To understand the mass or weight of an electron, scientists often use techniques that involve measuring its behavior under various forces. For example, in a cathode ray tube or in experiments like the Millikan oil-drop experiment, the electron’s charge and its behavior in electric or magnetic fields were used to deduce its mass.

Even though electrons don’t have a traditional size, we still measure their mass and charge as fundamental quantities.

2. Probability of Finding an Electron:Permalink

When we talk about the “probability of finding an electron”, we are referring to the concept of the electron’s wavefunction in quantum mechanics.

• Wavefunction (ψ): In quantum mechanics, the behavior of an electron is described by a wavefunction, which encodes all the information about the electron’s state, such as its position, energy, and momentum.

• The wavefunction is a mathematical function that gives us the probability distribution of where an electron might be found in a given space. It doesn’t tell us the exact position of the electron but provides the likelihood of finding it in a certain region.

• The square of the wavefunction (|ψ(x,t)|^2) gives us the probability density. This means that, for example, if you know the wavefunction of an electron in an atom, |ψ(x,t)|^2 will tell you the probability of finding the electron at position x at a given time t.

• Electron in an Atom (Orbitals): For an electron in an atom, the probability distribution is often visualized in terms of orbitals. These orbitals are regions around the nucleus where the probability of finding the electron is higher. The shape and size of these orbitals depend on the energy and quantum numbers of the electron.

Example with Hydrogen Atom:Permalink

Consider the hydrogen atom. The electron in a hydrogen atom doesn’t have a definite path or orbit like the planets around the sun. Instead, it exists in a probability cloud around the nucleus. The wavefunction for the electron gives the probability of where it is likely to be found. This means the electron doesn’t have a fixed position but rather a range of possible positions, each with a certain probability.

How Do We Measure Probability?Permalink

To measure the probability of finding an electron in a given location, we use experimental techniques such as:

• Scanning Tunneling Microscopy (STM): This technique allows scientists to “probe” the probability of finding an electron on the surface of a material by scanning the surface with a sharp tip.

• Photoelectron Spectroscopy: This is another experimental method where light is used to eject electrons from atoms. By analyzing the energy and angle of the emitted electrons, scientists can infer the electron density and distribution in a material.

In summary:

• Electrons have mass (which can be measured through experiments like the Millikan oil-drop experiment), but they do not have a well-defined size like macroscopic objects.
• The probability of finding an electron is described by its wavefunction, which gives us the likelihood of where the electron might be located, but not its exact position unless measured. The concept of orbitals in atoms is one way to visualize the regions where the electron is likely to be found.

Particles in Classical vs Quantum Physics.Permalink

you said “The wavefunction is a mathematical function that gives us the probability distribution of where an electron might be found in a given space. It doesn’t tell us the exact position of the electron but provides the likelihood of finding it in a certain region.”

My understanding is that which doesn’t have have size and shape you cannot locate that, then why to even discuss about the location of of electron while movement?

Your question touches on one of the key ideas in quantum mechanics—the distinction between classical and quantum descriptions of particles.

1. Classical vs Quantum Understanding:Permalink

In classical mechanics (the world of everyday objects), we are used to thinking of things as having definite positions and velocities at all times. For example, a baseball has a definite location in space at any given moment, and we can describe its motion precisely.

However, in quantum mechanics, the situation is very different for subatomic particles like electrons. Instead of having a definite position and velocity at the same time (as we expect from macroscopic objects), quantum particles like electrons are described by probability distributions.

2. Wavefunction and Uncertainty:Permalink

The electron’s wavefunction is a mathematical tool that encapsulates all possible information about the electron’s state. The wavefunction doesn’t tell you the electron’s exact position, but it describes the probabilities of where the electron might be found in space. The electron doesn’t have a definite position or trajectory like a classical object until we measure it.

This is where the Heisenberg Uncertainty Principle comes into play. It says that we cannot precisely know both the position and the momentum of a quantum particle (like an electron) simultaneously. The more precisely we try to measure one, the less precisely we can know the other.

3. Location and Probability Distribution:Permalink

Even though the electron doesn’t have a definite position before measurement, we can still talk about its “location” in a probabilistic sense. This is because, while we cannot pinpoint the exact location of an electron at a specific time, we can still compute the probability distribution of where the electron is likely to be found.

For example:

• If you have an electron in an atom, its wavefunction describes a probability cloud around the nucleus. The size and shape of this cloud depend on the electron’s energy level and quantum state. In these regions, the electron has a higher probability of being found.
• The electron’s location is not certain, but we can say, based on the wavefunction, that it is most likely to be found within a specific region at any given time.

4. Measurement and Collapse of the Wavefunction:Permalink

When we measure the electron’s position, the wavefunction “collapses” to a specific location. Prior to the measurement, the electron exists as a superposition of all possible locations. Once we measure it, it has a definite position. The act of measurement forces the wavefunction to “collapse” into one of the possible outcomes.

5. Why Discuss the Location of an Electron?Permalink

Even though we cannot directly measure the precise location of an electron without disturbing it, discussing its “location” in terms of probabilities is important because it helps us understand and predict its behavior in various contexts, like in atoms, molecules, or conductors. It gives us a statistical description of the electron’s behavior across many measurements.

For example:

• In the hydrogen atom, the electron’s wavefunction can be used to calculate the probability of finding the electron at different distances from the nucleus. This allows scientists to understand atomic structure and predict the electron’s behavior under various conditions.
• In a quantum dot, the electron’s wavefunction helps explain how it behaves within confined spaces and can be used in applications like quantum computing and nanotechnology.

Conclusion:Permalink

So, to clarify:

• Electron doesn’t have a definite location until measured.
• However, the wavefunction allows us to describe the probabilities of where the electron might be found.
• This probabilistic description of location is crucial for understanding quantum systems and allows us to make predictions about how electrons will behave in different scenarios.

This approach is different from classical mechanics, where we always think of particles having definite positions and velocities. In quantum mechanics, we must think in terms of probabilities and uncertainty.

What Does “Finding” an Electron Mean?Permalink

You said “The wavefunction doesn’t tell you the electron’s exact position, but it describes the probabilities of where the electron might be found in space.”

I am saying if something is not having size or shape then where is the question of finding it anywhere in the world. Probabilities should be used for something which you can find and recognize that it is here. This idea of wavefunction looks confusing to me.

I understand that the idea of the wavefunction and the location of an electron can be quite confusing, especially when we try to reconcile it with our everyday understanding of objects, which have definite positions and sizes. Let me try to break it down further to clarify this concept.

1. Quantum Objects Don’t Behave Like Everyday Objects:Permalink

When we talk about particles like electrons in quantum mechanics, we are stepping into a realm where the rules of classical physics (the world of everyday objects) no longer apply. At the quantum scale, particles don’t have definite positions or trajectories unless they are measured. Instead, their behavior is described probabilistically by a wavefunction.

2. Wavefunction and Probability:Permalink

The wavefunction is a mathematical construct that helps us predict the probability distribution of where a particle like an electron might be found. The electron is not a point with a definite location before measurement; rather, it exists as a wave (or a “cloud” of probabilities) that can be spread out over a region of space.

• Imagine a wave on the surface of water: The wave doesn’t have a single point of “location” but is spread out across the water. Similarly, the electron’s “wave” is spread out over space. Where it is most likely to be found is described by the probability distribution.

3. What Does “Finding” an Electron Mean?Permalink

In quantum mechanics, when we say “we can find the electron,” we are referring to the probability of measuring the electron at a particular point in space. The electron itself is not “spread out” in the same way a physical object is, but the probability of detecting the electron in different regions of space is.

This is why we often describe the electron’s location in terms of the probability of finding it at different locations:

• In an atom, we might say there’s a high probability of finding the electron near the nucleus, and a low probability of finding it far away.
• This doesn’t mean the electron is physically “spreading out” in the space, but rather that the wavefunction describes a probability cloud, a mathematical representation of the likelihood of where the electron might be detected.

4. No Definite Size or Shape, But a Probability Distribution:Permalink

You’re correct that the electron itself doesn’t have a definite size or shape, as it is not a solid object like a ball. However, the wavefunction gives us information about where the electron might be detected, not where it is at any given moment. It is more like a statistical description, not a precise location.

• The electron is often described as a point particle (meaning it has no spatial extent), but the probability of detecting it is spread out.
• The wavefunction describes how the electron’s probability is distributed in space, which can be interpreted as the electron’s “location” in a statistical sense.

5. Electron’s “Location” is Statistical, Not Definite:Permalink

To give you an analogy:

• Imagine you are looking for a person in a large city. You don’t know exactly where the person is at any given moment, but you have a map showing where they are most likely to be. The map doesn’t tell you exactly where the person is, but it gives you statistical information about the probability of where they might be found.
• Similarly, the wavefunction doesn’t tell you where the electron is at a specific moment. It tells you where the electron might be found when you measure it.

6. Measurement “Collapses” the Wavefunction:Permalink

When you measure the position of the electron, the wavefunction collapses, and the electron takes on a definite position. Before measurement, the electron is described by a probability distribution, and it is only after the measurement that it assumes a definite position. This is a key difference from classical physics, where we can always know the position of an object.

7. Why Use the Concept of Location in Quantum Mechanics?Permalink

We use the concept of “location” in quantum mechanics not because the electron has a definite position before measurement, but because the wavefunction allows us to calculate the probability of finding it in different places. This is crucial for understanding how electrons behave in atoms, molecules, and materials.

For example:

• In the hydrogen atom, the electron’s wavefunction determines the probability of finding the electron at different distances from the nucleus. This helps explain the quantized energy levels and the shape of atomic orbitals.

Conclusion:Permalink

• The electron doesn’t have a definite position in the classical sense before measurement. It exists in a superposition of all possible positions, described by a wavefunction.
• The wavefunction is a probability distribution, meaning it gives us the likelihood of finding the electron in a given region of space.
• While we can’t “find” an electron in the same way we find a macroscopic object, we can measure the electron’s position and describe it probabilistically before that.

The idea of an electron not having a definite position until measured is strange and non-intuitive, but it’s a fundamental aspect of quantum mechanics that helps us predict and understand the behavior of particles at microscopic scales.

Water wave - Electron wave analogy is not correct.Permalink

You said “Imagine a wave on the surface of water: The wave doesn’t have a single point of “location” but is spread out across the water. Similarly, the electron’s “wave” is spread out over space. Where it is most likely to be found is described by the probability distribution.”

But when we are talking about water wave that is made trillions of water molecules. Secondly it is moving in 3 D space. Third we know where is that wave of trillions of water molecules at this time. In this example we are not talking about wave of a moledule.

Yes, the analogy with a water wave isn’t perfect, especially when considering the scale and nature of the systems involved. Let me clarify the differences and why the analogy is still useful in a limited sense but may not fully capture the quantum nature of an electron’s wavefunction.

1. Water Wave vs Electron Wave:Permalink

• Water Wave: The wave on the surface of water is indeed made up of trillions of molecules, and each molecule moves up and down as the wave propagates. The motion of each molecule is part of the wave, and the “location” of the wave is well-defined, since we can track the position of the wave at any given point.
• Electron Wave: An electron, on the other hand, is described by a probability wave, which is not made of trillions of molecules. The wave associated with the electron is a single entity that represents the probability distribution of where the electron might be detected. This “wave” doesn’t correspond to physical particles (like molecules in water) moving through space. Instead, it describes where the electron is likely to be found when a measurement is made.

2. Nature of the Electron’s Wave:Permalink

• Water Wave: The water wave moves in three-dimensional space, and we can observe its motion at different points. We can track the water’s molecules as they move up and down, which gives us a clear picture of the wave.
• Electron’s Wave: The electron’s wavefunction exists in mathematical space rather than in physical space like water molecules. This wavefunction describes the probability distribution of the electron’s position in space. When we say the “wave is spread out,” we mean that there is a probability cloud over space, which indicates the likelihood of finding the electron in various locations if we were to measure it.

3. Measuring the Electron’s Position:Permalink

• Water Wave: In the case of the water wave, the location is easily measurable because it corresponds to the motion of the water molecules at any point in space. We can see the position of the wave crest, and the wave moves across the surface of the water in a way that’s very tangible.
• Electron Wave: For the electron, we can’t observe the “wave” directly in the same way. Instead, the wavefunction describes where the electron might be found upon measurement, but until that measurement is made, the electron doesn’t have a definite position. The “wave” isn’t a physical thing moving through space; rather, it represents the probabilities of detecting the electron at various points.

4. Wavefunction and Probability:Permalink

The wavefunction for an electron doesn’t describe a wave of matter moving through space. Instead, it describes the probabilities of the electron’s position. The electron doesn’t travel along a specific path like a water wave; instead, its “location” is probabilistic, meaning it can be in many places at once (in a superposition) until we measure it.

• For example, if we consider an electron in an atom, its wavefunction describes the probability of finding the electron at different distances from the nucleus. The “wave” describes the likelihood that the electron will be found in any particular region of space, not its exact position at any given moment.

5. Superposition:Permalink

This is a crucial difference between classical waves (like water waves) and quantum waves (like the electron’s wavefunction). The electron’s wavefunction can exist in a superposition of states — meaning the electron can be in many places at once, each with a certain probability. The wavefunction gives us the statistical likelihood of where the electron will be found, but the electron itself doesn’t travel through space like a classical object.

6. Why Use the Water Wave Analogy?Permalink

The reason we use the water wave analogy at all is that both kinds of waves (classical and quantum) are solutions to a wave equation and share some mathematical similarities. In quantum mechanics, the electron’s wavefunction is a solution to the Schrödinger equation, which describes the behavior of quantum particles. Similarly, the water wave is described by a wave equation that governs the motion of water molecules.

However, the key difference is that the electron’s wavefunction describes probability distributions of its position, while a classical wave (like the water wave) describes the movement of material through space. The analogy holds in terms of the mathematics of wave equations, but it doesn’t perfectly match up with the physical behavior of the particles.

Conclusion:Permalink

• Water waves are made of trillions of molecules, and the position of the wave is well-defined at all times.
• The electron’s wavefunction describes the probability of finding the electron at various locations and doesn’t correspond to a physical wave made up of particles. Instead, it describes the statistical likelihood of the electron’s position, and this “wave” is a mathematical representation of these probabilities.
• The wavefunction gives us information about where the electron might be rather than its exact position before measurement, which is a departure from our classical intuitions about location and movement.

The meaning of electron’s wavefunction can exist in a superposition of states.Permalink

you said “The electron’s wavefunction can exist in a superposition of states — meaning the electron can be in many places at once, each with a certain probability. The wavefunction gives us the statistical likelihood of where the electron will be found, but the electron itself doesn’t travel through space like a classical object.”

You see, the mathamatical functions are representation of some reality. I want to understand how an electron or photon travels in the 3D space for which we don’t know the location until we measure it. Because measuring it is at multiple location and after measuring it collapse at some location. But at which location? Suppose I have 1 cubic meter box and electron starts from a corner from a electronic gun then what is the location of electron after 1 ns?

This is an insightful question that gets to the heart of the nature of quantum particles like electrons and photons. The key issue is that, in quantum mechanics, we cannot precisely describe the electron’s position at any given moment in time, until we make a measurement. This is where the concept of the wavefunction comes in, which helps us understand the probabilistic nature of quantum objects.

Understanding the Movement of an Electron in 3D Space:Permalink

To break it down step by step:

1. Wavefunction and Probability:
   • The wavefunction is a mathematical representation of the possible locations of an electron in space, not a description of its exact position. The probability distribution (which comes from the wavefunction) tells us where we are likely to find the electron when we measure it. This means that, before measuring, the electron is not at a single point. Instead, it exists in a state of probability, spread out over a region of space.
   • If you imagine an electron in a cubic meter box, and the electron starts at one corner (let’s assume this is at time t = 0), the wavefunction at t = 1 ns will describe a probability distribution that indicates how likely the electron is to be found at different positions within the box at that moment.
2. How does the Electron “Travel”?:
   • According to quantum mechanics, before measurement, we cannot think of the electron as traveling along a definite path like a classical object. Instead, it exists as a “probability cloud.” Its wavefunction spreads out over time, meaning the electron has a probability of being found in various locations in space, but we cannot say with certainty where it is.
   • If we did measure its position at t = 1 ns, the electron would collapse to a specific point in the box. The measurement forces the system to “choose” a specific location. But before the measurement, we only have probabilities. The wavefunction will give us a range of possible positions for the electron, but it won’t tell us its precise location.
3. What Happens After 1 ns?:
   • If the electron is in free space (not interacting with anything), its wavefunction will evolve according to the Schrödinger equation. The wavefunction will spread out, meaning the electron has a greater probability of being found at different places in the box after 1 ns. It could be closer to the other end of the box, or it could still be near the starting point. We cannot know exactly where it is, only the probabilities of where it could be found when we measure.
   • The spread of the wavefunction will depend on the momentum and energy of the electron. If the electron has a high momentum (fast speed), its wavefunction will spread out less over time, and the electron is more likely to be found closer to its initial position. If the electron has low momentum (moves slowly), its wavefunction will spread out more, and it could be found anywhere in the box.
4. The Concept of “Path” and Measurement:
   • Unlike classical particles (like a baseball or a car), which follow definite paths through space, quantum particles like electrons do not have a clear, definite path until they are measured. They exist in a superposition of all possible paths.
   • When you measure the electron’s position, the superposition collapses, and you get a specific location. This process is known as wavefunction collapse. But before the measurement, the electron doesn’t have a definite location—it only has a probability of being in different places.

Example in a 1 Cubic Meter Box:Permalink

Let’s say an electron is emitted from a corner of a 1 cubic meter box, and we want to know where it might be after 1 ns. Here’s how we would think about it:

• Step 1: Initial Wavefunction: When the electron is emitted, it has a wavefunction that might be sharply concentrated at the corner (the initial location), but over time, this wavefunction spreads out. The more time that passes, the more the wavefunction spreads.

• Step 2: Probability Distribution at 1 ns: After 1 ns, the wavefunction describes the probability of finding the electron at any given point in the box. The electron could be anywhere, but we would have a higher probability of finding it near the starting point if it is moving slowly. If it’s moving faster, the wavefunction might spread out more evenly across the box.

• Step 3: Collapse upon Measurement: If you measure the electron’s position at t = 1 ns, the wavefunction collapses, and the electron will be found at a specific point in the box. The position is random but governed by the probability distribution. So, if the wavefunction indicates a higher probability of the electron being near one side of the box, it’s more likely to be found there, but there is always some chance of finding it in other places.

Why Can’t We Know the Exact Location of the Electron?Permalink

The electron doesn’t have a “definite” position unless we measure it. This is a fundamental feature of quantum mechanics and is a direct result of Heisenberg’s uncertainty principle. According to this principle, we cannot simultaneously know both the exact position and exact momentum of a particle like an electron. The more precisely we try to measure one, the less precisely we can know the other.

In your scenario:

• If we try to track the exact position of the electron after 1 ns, the uncertainty in its momentum (and therefore its velocity) will increase. This means the electron’s position is probabilistic, and we can only know the likelihood of finding it in a certain region.

Key Takeaways:Permalink

1. The electron does not travel in a definite path like a classical object. Instead, its wavefunction describes a probability distribution of where the electron might be found.
2. At 1 ns, the electron’s location is not fixed; it is spread out across space, with certain regions having a higher probability of the electron being there.
3. Wavefunction collapse happens when we make a measurement, at which point the electron’s position is fixed, but until we measure it, it exists in a superposition of possibilities.
4. We can’t track the exact location of an electron without disturbing it. We can only describe the probability of finding it at different places.

I can find a particular electron anywhere in the universe!Permalink

you said “The electron doesn’t have a “definite” position unless we measure it.” If I measure it at top location in the box (rather than opposite side of the electron gun), can I find electron there also because of my measuring?

Yes, in quantum mechanics, there is indeed a non-zero probability of finding the electron in locations far from where we might “classically” expect it to be, such as the top of the box in your example—even if that location seems improbable.

The Role of MeasurementPermalink

1. Superposition Before Measurement: Before you measure the electron’s position, its state is described by a wavefunction, which represents a superposition of all possible locations the electron could occupy within the box. This wavefunction gives the probability of finding the electron at any given point in the box.

2. Measurement and Wavefunction Collapse: When you measure the electron’s position, the act of measurement causes the wavefunction to “collapse.” This means that the electron will be detected at a specific location, and this location is chosen according to the probability distribution given by the wavefunction.

3. Low-Probability Locations: If the probability of finding the electron at the top of the box is low (but not zero), there is still a chance the measurement could “find” the electron there. Measurement doesn’t create the electron at that position but rather “reveals” it to be there, consistent with the probabilistic nature of the wavefunction.

Key Factors to ConsiderPermalink

1. Wavefunction Spread:
   • If the electron’s wavefunction has had time to spread throughout the box, the probability of finding the electron at various locations (including the top of the box) increases, though it might still be highest near the area where the electron gun emitted it.
2. Momentum and Energy of the Electron:
   • If the electron was emitted with high energy or momentum, the wavefunction will spread more rapidly, making it more likely to find the electron farther from its starting point.
3. Probability Distribution:
   • The probability of finding the electron at any point depends on the shape and evolution of the wavefunction over time. In most realistic cases, the probability of finding the electron at a very distant or unlikely location (like the top of the box) will be small, but it is never zero unless the wavefunction explicitly excludes that region.

Measurement and RealityPermalink

When you measure the electron at the top of the box, you’re not “forcing” it to be there. Instead, your measurement simply reveals a possible outcome dictated by the probabilities encoded in the wavefunction.

• If the electron is detected at the top of the box, this result doesn’t mean the electron “traveled” there in a classical sense. It means that, in the quantum world, the electron had a small but non-zero probability of being found there, and this measurement result is consistent with the wavefunction.

SummaryPermalink

Yes, you could find the electron at the top of the box during a measurement, even though it started near the electron gun. This is because:

• The electron’s wavefunction spreads out and covers the entire box over time.
• Measurement reveals one specific location, with probabilities determined by the wavefunction.
• Quantum mechanics allows for seemingly “unlikely” outcomes, such as finding the electron in regions where the probability is low.

It’s crucial to remember that this doesn’t mean your measurement “forces” the electron into a specific location. It simply interacts with the wavefunction and “selects” one of the possible outcomes.

Can I go thousands of mile away and find that electron there?Permalink

In theory, yes, but in practice, extremely unlikely. Here’s why:

Theoretical PossibilityPermalink

1. Wavefunction Spread:
   • The wavefunction of a particle like an electron is not constrained to a finite region unless physically limited (e.g., by a box, trap, or potential barrier).
   • Over time, the wavefunction can spread over an arbitrarily large distance, theoretically covering the entire universe.
   • Even though the probabilities decrease with distance, they never become exactly zero.
2. Non-zero Probability:
   • If you are thousands of miles away, the probability of finding the electron there is astronomically small, but it is not strictly zero.
   • Quantum mechanics allows for such improbable outcomes, though they are so rare that they would almost never occur in practice.

Practical ConsiderationsPermalink

1. Time and Probability:
   • For the wavefunction to reach thousands of miles away, significant time must pass. The further you go, the smaller the probability becomes.
   • For an electron emitted with typical energy, the probability of finding it thousands of miles away within any reasonable timeframe is so low that it would effectively take longer than the age of the universe for such an event to occur.
2. Environmental Interactions:
   • In real-world conditions, the electron will almost certainly interact with its surroundings (air molecules, electromagnetic fields, etc.) before its wavefunction spreads that far.
   • These interactions “collapse” the wavefunction locally, meaning the electron is measured or localized much closer to its starting point.

What About Free Space?Permalink

If the electron is in a perfect vacuum and isolated from any disturbances:

• Its wavefunction could spread across vast distances over time.
• Even then, the probability of detecting it thousands of miles away would be unimaginably tiny.

AnalogyPermalink

Think of a drop of ink diffusing in an ocean. The ink molecules can theoretically spread across the entire ocean over time, but the chance of finding a specific molecule thousands of miles away within a reasonable timeframe is effectively zero.

ConclusionPermalink

While quantum mechanics allows for the possibility of detecting the electron thousands of miles away, the probability is so small that for all practical purposes, you would not expect to find it there. Environmental factors and real-world interactions make such an outcome virtually impossible.

What are different possible location of an electron?Permalink

If I am throwing electron from top toward downside with extreme high energy and speed can I find this on the extreme left or right of the box? If you throw an electron from the top toward the bottom of the box with extremely high energy and speed, the likelihood of finding it on the extreme left or right depends on several factors, including quantum effects and practical considerations. Here’s a detailed breakdown:

1. Classical ExpectationPermalink

In classical physics:

• If you aim the electron straight downward, it should follow a predictable trajectory based on its initial velocity and energy.
• You would not expect it to end up on the extreme left or right unless some external force (like an electric or magnetic field) acted on it to deflect its path.

2. Quantum Mechanical PossibilityPermalink

In quantum mechanics:

• The electron’s wavefunction spreads out as it travels. This spread introduces a probability of finding the electron in locations other than its expected classical trajectory.
• Even if the initial wavefunction is sharply peaked toward a downward direction, it has some non-zero probability of being detected on the left or right sides of the box due to the uncertainty principle and wavefunction evolution.

3. Factors Influencing the OutcomePermalink

a. Energy and Speed (Momentum)Permalink

• A high-energy electron has a sharply peaked wavefunction in momentum space, meaning its position-space wavefunction is more spread out.
• However, the direction of momentum largely determines the direction in which the wavefunction evolves. If the initial momentum is strongly downward, the probabilities for extreme left or right will be very low but not zero.

b. Size of the BoxPermalink

• If the box is small (e.g., nanometer scale), quantum effects are more pronounced, and the electron’s wavefunction might spread significantly, increasing the probability of detecting it on the sides.
• For a large box (e.g., macroscopic scale), the probability of detecting the electron on the extreme left or right is astronomically small.

c. Time of TravelPermalink

• Over time, the wavefunction spreads out more. The longer the electron travels, the greater the spatial uncertainty in its position.
• If the electron travels at extreme speeds, it might not have enough time for the wavefunction to spread significantly before hitting the bottom.

d. External ForcesPermalink

• Any electric or magnetic field inside the box can deflect the electron, increasing the chance of it being detected on the sides.

4. Real-World OutcomePermalink

In most practical scenarios:

• The electron will be detected close to its expected downward trajectory, as the probability of extreme left or right locations will be incredibly small.
• Quantum effects like wavefunction spreading are real but are dominated by the electron’s high energy and directional momentum.

5. AnalogiesPermalink

Think of throwing a fast-moving ball in a narrow hallway:

• Classically, it goes straight down the hall.
• Quantum mechanically, there’s a tiny chance the ball appears at the sides due to its wave-like properties, but this chance is so small that it’s practically unobservable at large scales.

ConclusionPermalink

While quantum mechanics allows for the possibility of finding the electron on the extreme left or right of the box, the probability is incredibly small for a high-energy electron thrown straight downward. In practical terms, the electron will most likely be found near its expected classical trajectory unless influenced by external forces.

A wavefunction is function of multiple functions, right?Permalink

Yes, a wavefunction can indeed be seen as a function of multiple variables, depending on the context and the system being described. Here’s a breakdown of what this means:

1. Basic Wavefunction: Single Particle in 1DPermalink

For a simple case, such as a single particle in one dimension, the wavefunction is typically written as: $\psi (x,t)$ Here:

• $(x)$: Position variable (spatial coordinate).
• $(t)$: Time variable.
• $(\psi (x,t))$: Complex-valued function that gives the probability amplitude.

The probability density at position $(x)$ and time $(t)$ is given by: $P(x,t)=\Vert \psi (x,t){\Vert }^2$

2. General Wavefunction: Single Particle in 3DPermalink

For a single particle in three-dimensional space, the wavefunction depends on all three spatial coordinates and time: $\psi (x,y,z,t)$ Here:

• $(x,y,z)$: Position coordinates.
• $(t)$: Time.

The wavefunction now describes the probability amplitude of the particle being found at a specific point in 3D space at a given time.

3. Many-Particle SystemsPermalink

For systems with multiple particles, the wavefunction becomes a function of the coordinates of all particles. For $(N)$ particles, the wavefunction depends on $(3N)$ spatial coordinates and time: $\psi (x_1,y_1,z_1,x_2,y_2,z_2,\dots ,x_N,y_N,z_N,t)$ Here:

• $(x_1,y_1,z_1)$: Position of the first particle.
• $(x_2,y_2,z_2)$: Position of the second particle.
• $(N)$: Number of particles.

The wavefunction describes the joint probability amplitudes for all particles in the system.

4. Momentum and Other VariablesPermalink

In quantum mechanics, the wavefunction can also be expressed in terms of momentum or other variables. For example, in momentum space, the wavefunction might look like: $\varphi (p_x,p_y,p_z,t)$ Here:

• $(p_x,p_y,p_z)$: Momentum components in 3D space.
• $(t)$: Time.

Position-space and momentum-space wavefunctions are related through the Fourier transform.

5. Wavefunction as a Composite of Other FunctionsPermalink

The wavefunction often incorporates other mathematical functions:

• Plane Waves: A wavefunction for a free particle might look like $(\psi (x,t)=A{e}^{i(kx-\omega t)})$, where $(A,k,\omega )$ are constants.
• Superposition: A wavefunction can be a sum of multiple functions, e.g., $(\psi (x)={\psi }_1(x)+{\psi }_2(x))$, representing different quantum states.
• Bound States: In atoms, the wavefunction might combine radial and angular functions, e.g., $(\psi (r,\theta ,\varphi )=R(r)Y(\theta ,\varphi ))$, where $(R(r))$ is the radial function and $(Y(\theta ,\varphi ))$ are spherical harmonics.

6. Time EvolutionPermalink

The wavefunction evolves in time according to the Schrödinger equation: $i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$ Here, the Hamiltonian $(\hat{H})$ might depend on position, momentum, or other system-specific variables, causing the wavefunction to change dynamically.

ConclusionPermalink

The wavefunction is a multivariable function that encapsulates the quantum state of a particle or system. It can depend on:

• Spatial coordinates $(x,y,z)$.
• Time $(t)$.
• Other properties like momentum $(p_x,p_y,p_z)$.

Its structure can incorporate many mathematical functions (e.g., exponentials, spherical harmonics, or superpositions), and it describes the probability amplitudes for finding a particle in different states.

How to construct a wavefunction?Permalink

Can you give me wavefunction of wave which has 500 nm wave lenght and traveling at the speed of light. Remaing things you can assume.

Certainly! Let’s construct the wavefunction of a photon with a wavelength of 500 nm $(5\times {10}^{-7})$ meters) traveling at the speed of light $(c=3\times {10}^8\mathrm{m}/\mathrm{s})$.

1. General Wavefunction for a Traveling WavePermalink

The wavefunction for a wave traveling in one dimension (say along the $(x)$-axis) can be written as: $\psi (x,t)=A{e}^{i(kx-\omega t)}$ Here:

• $(A)$: Amplitude of the wave.
• $(k)$: Wave number, related to the wavelength.
• $(\omega )$: Angular frequency, related to the wave’s energy or speed.

2. Define the ParametersPermalink

Wavelength $(\lambda )$:Permalink

$\lambda =500\mathrm{nm}=5\times {10}^{-7}\mathrm{m}$

Wave Number $(k)$:Permalink

$k=\frac{2\pi }{\lambda }=\frac{2\pi }{5\times {10}^{-7}}=1.257\times {10}^7{\mathrm{m}}^{-1}$

Speed of Light $(c)$:Permalink

The photon is traveling at the speed of light: $c=3\times {10}^8\mathrm{m}/\mathrm{s}$

Frequency $(f)$:Permalink

The frequency of the photon is: $f=\frac{c}{\lambda }=\frac{3\times {10}^8}{5\times {10}^{-7}}=6\times {10}^{14}\mathrm{Hz}$

Angular Frequency $(\omega )$:Permalink

The angular frequency is related to the frequency by: $\omega =2\pi f=2\pi (6\times {10}^{14})=3.77\times {10}^{15}\mathrm{rad}/\mathrm{s}$

3. Construct the WavefunctionPermalink

Substituting these values into the wave equation, we get: $\psi (x,t)=A{e}^{i(1.257\times {10}^{7}x-3.77\times {10}^{15}t)}$

4. InterpretationPermalink

1. Wave Nature: This wavefunction describes a plane wave traveling along the $(x)$-axis with a wavelength of 500 nm.
2. Amplitude $(A)$: The amplitude $(A)$ depends on the wave’s intensity, which is related to the photon’s energy. For simplicity, we assume $(A=1)$ in normalized units.
3. Complex Form: The wavefunction is complex, and its physical meaning lies in its square modulus $(\Vert \psi {\Vert }^2)$, which gives the probability density or intensity of the photon.

ExtensionsPermalink

If you want to describe the wave in 3D or add polarization (photons are inherently polarized), we would modify the wavefunction further. For now, this describes a single-photon wave traveling in one dimension at the speed of light with the given wavelength.

Refine this wavefunction for 3D space.Permalink

To refine the wavefunction of the photon and describe its movement in 3D space, we need to generalize the traveling wave to account for its direction and polarization.

1. General 3D Wavefunction for a Traveling WavePermalink

In 3D space, the wavefunction of a photon traveling in a direction given by a unit vector $(\mathbf{k}=(k_x,k_y,k_z))$ is:

Here:

• $(\mathbf{r}=(x,y,z))$: Position in 3D space.
• $(\mathbf{k}=(k_x,k_y,k_z))$: Wavevector, which determines the photon’s direction and magnitude. The magnitude of $(\mathbf{k})$ is: $\Vert \mathbf{k}\Vert =k=\frac{2\pi }{\lambda }$
• $(A)$: Amplitude of the wave.
• $(\omega )$: Angular frequency, determined by: $\omega =ck$
• $(\mathbf{k}\cdot \mathbf{r}=k_{x}x+k_{y}y+k_{z}z)$: The dot product captures the projection of $(\mathbf{r})$ onto the wave’s direction.

2. Specify the Wavelength and WavevectorPermalink

For a photon with a wavelength of $(\lambda =500\mathrm{nm})$: $k=\frac{2\pi }{\lambda }=1.257\times {10}^7{\mathrm{m}}^{-1}$

The wavevector $(\mathbf{k})$ can be written in terms of its direction: $\mathbf{k}=k(\hat{i}\cos \theta \cos \varphi +\hat{j}\cos \theta \sin \varphi +\hat{k}\sin \theta )$

Here:

• $(\theta )$: The angle from the $(z)$-axis.
• $(\varphi )$: The azimuthal angle in the $(xy)$-plane.

3. 3D WavefunctionPermalink

Substitute $(\mathbf{k})$ into the general equation: $\psi (\mathbf{r},t)=A{e}^{i[k(x\cos \theta \cos \varphi +y\cos \theta \sin \varphi +z\sin \theta )-\omega t]}$

Here:

• $(x)$, $(y)$, $(z)$: Coordinates of the photon’s location.
• $(A)$: Amplitude.
• $(k=1.257\times {10}^7{\mathrm{m}}^{-1})$.
• $(\omega =3.77\times {10}^{15}\mathrm{rad}/\mathrm{s})$.

4. Include PolarizationPermalink

A photon inherently has polarization, which is orthogonal to its direction of motion. Let the polarization vector be $(\mathbf{p}=(p_x,p_y,p_z))$, where: $\mathbf{p}\cdot \mathbf{k}=0$ This ensures that the polarization is perpendicular to the wavevector.

The final wavefunction, including polarization, can be written as: $\psi (\mathbf{r},t)=\mathbf{p}A{e}^{i[k(x\cos \theta \cos \varphi +y\cos \theta \sin \varphi +z\sin \theta )-\omega t]}$

5. Physical MeaningPermalink

• Wave Nature: The wavefunction describes a photon propagating through 3D space along the direction defined by $((\theta ,\varphi ))$, with polarization $(\mathbf{p})$.
• Probability Distribution: The intensity of the photon at any point is proportional to $(\Vert \psi {\Vert }^2)$.
• Speed of Light: The phase velocity is always $(c)$, ensuring the photon moves at the speed of light.

This is a 3D generalization of the wavefunction for a photon with a wavelength of 500 nm. If needed, further refinements can describe specific boundary conditions, interactions, or quantum superposition states.

With a given wavefunction, can we location particle at time t?Permalink

Now with this wavefunction, can you tell me where will the particle will be at t=10 second.

The exact position of the photon at $(t=10\text{seconds})$ cannot be determined precisely because photons do not have a well-defined position until they are measured. Instead, their position is described probabilistically by the wavefunction.

What the Wavefunction Tells UsPermalink

The wavefunction $(\psi (\mathbf{r},t))$ provides the probability amplitude of the photon being found at a certain point in space at a given time. To find the probability of detecting the photon at a specific location, you calculate:

This means:

1. The wavefunction spreads out over space, describing the possible locations where the photon might be detected.
2. Until a measurement occurs, the photon does not have a definite position but exists as a superposition of probabilities.

Can We Estimate the Region?Permalink

If the photon is traveling at the speed of light $(c)$, after $(t=10\text{seconds})$, it will have traveled a distance of $(3\times {10}^9\text{m})$ (i.e., $(c\times t)$ in the direction of its wavevector $(\mathbf{k})$.

Thus:

1. The photon is somewhere within a spherical region with a radius of $(3\times {10}^9\text{m})$ centered on its starting point.
2. The probability distribution $(\Vert \psi (\mathbf{r},t){\Vert }^2)$ would be most intense along the direction of $(\mathbf{k})$ but will also depend on diffraction, scattering, or interactions the photon might encounter.

Why Can’t We Say the Exact Location?Permalink

This uncertainty arises because photons obey the principles of quantum mechanics:

• Wave-Particle Duality: The photon behaves like a wave (described by $(\psi (\mathbf{r},t))$ until measured, at which point it behaves like a particle.
• Heisenberg Uncertainty Principle: You cannot know both the exact position and momentum of the photon simultaneously. The more precisely the photon’s momentum (or energy) is known, the less precisely its position is defined.

Practical MeasurementPermalink

If you measure the photon at $(t=10\text{seconds})$, you’ll find it somewhere within the spherical region $(3\times {10}^9\text{m})$ away from its starting point. Its exact location depends on how you measure it and the specifics of its wavefunction at that time.

If I give you radius and wavefunction can you locate electron?Permalink

You just gave me the radius. Using probablity denstity function can you tell me where it can be found most? To determine where the photon is most likely to be found, we use the probability density function derived from the wavefunction $(\psi (\mathbf{r},t))$. The probability density is:

Here’s how we analyze it step by step:

Wavefunction of the Photon in 3DPermalink

For a photon with a wavelength of $(\lambda =500\text{nm})$ traveling in the direction of the wavevector $(\mathbf{k})$, the wavefunction in 3D space is:

• $(\mathbf{k})$: wavevector $(\Vert \mathbf{k}\Vert =2\pi /\lambda )$, direction of travel).
• $(\mathbf{r})$: position vector in 3D $(\mathbf{r}=(x,y,z))$.
• $(\omega =c\Vert \mathbf{k}\Vert )$: angular frequency.
• $(A)$: normalization constant.

Probability Density FunctionPermalink

The probability density is the squared magnitude of the wavefunction:

This means:

1. The photon’s wavefunction has uniform intensity across space.
2. The photon’s location is equally probable along a spherical shell of radius $(r=ct)$.

Most Probable LocationPermalink

At $(t=10\text{seconds})$:

• The photon is most likely to be found on the spherical shell of radius $(r=ct=3\times {10}^9\text{m})$.
• The probability density doesn’t favor any specific location on the shell because it is symmetric and uniform around the origin.

Key PointsPermalink

1. No preferred position within the sphere: The uniformity of the plane wave implies equal likelihood of finding the photon anywhere on the spherical shell.
2. Directional considerations: If the photon was emitted in a specific direction (defined by $(\mathbf{k})$, practical considerations suggest it is most likely to be found near that direction because $(\mathbf{k})$ defines the main axis of travel.

In short, while the photon is equally probable at any point on the spherical shell of radius $(ct)$, practical setups (e.g., a collimated beam or an optical system) will localize the photon near the direction of $(\mathbf{k})$.

It is crazy, I need some expert now!Permalink

It is funny. To know all this radius I don’t need probablity distribution function and wave function. how to use these function then? You’re absolutely correct that the radius $(r=ct)$ (distance the photon travels in time $(t)$ can be calculated from simple classical reasoning based on the speed of light. The wavefunction and probability density function, however, offer much deeper insights that go beyond just calculating the distance. Let’s explore their purpose and utility:

1. Why Use the Wavefunction?Permalink

The wavefunction is not just about distance or motion. It encodes the quantum nature of the particle, such as:

• Interference effects: In scenarios like the double-slit experiment, the wavefunction describes how the photon’s probabilities interfere, producing patterns we observe.
• Localization: The wavefunction tells us the regions in space where the photon (or particle) is more likely to be found, even when it’s spread out due to quantum uncertainty.
• Interaction probabilities: When the photon interacts with other particles (like being detected by a sensor or scattered), the wavefunction helps predict how likely the photon is to interact in certain regions.

2. How to Use the Wavefunction and Probability DensityPermalink

To see their practical utility, let’s consider a few examples:

Example 1: Single Photon in an ExperimentPermalink

Imagine you emit a photon from a source toward a detector. The wavefunction predicts:

• The spatial probability distribution of where the photon might hit the detector.
• If the photon passes through an optical setup (e.g., a slit or lens), the wavefunction can describe how the photon spreads or focuses.

Example 2: Double-Slit ExperimentPermalink

If the photon passes through two slits, its wavefunction becomes a superposition of two paths: $\psi (\mathbf{r},t)={\psi }_1(\mathbf{r},t)+{\psi }_2(\mathbf{r},t)$ Here:

• $(\Vert \psi (\mathbf{r},t){\Vert }^2)$ gives the interference pattern on the detection screen.
• The pattern can’t be explained using classical reasoning, as it arises from the photon’s quantum wave nature.

Example 3: Localization After MeasurementPermalink

If the wavefunction was initially a plane wave (spread out infinitely), after a measurement, the wavefunction collapses to a localized state. The probability density tells us where the photon is most likely to be detected.

3. When Does the Wavefunction Become Essential?Permalink

The wavefunction is crucial in scenarios where classical concepts like “path” or “location” fail:

• Quantum Tunneling: The probability density predicts the chance of a particle appearing on the other side of a barrier it classically couldn’t cross.
• Quantum Optics: Wavefunctions describe how photons interact in quantum systems like beam splitters or entangled states.
• Scattering and Absorption: Wavefunctions predict how particles scatter off or are absorbed by matter.

4. Using the Wavefunction in Your ExamplePermalink

For a photon moving through space:

1. Without obstacles: The photon’s wavefunction is a plane wave, and the probability density is uniform on the sphere of radius $(r=ct)$.
2. With obstacles (e.g., a slit or scattering object): The wavefunction becomes more complex, describing how the photon diffracts, scatters, or interferes.

Why All the Math?Permalink

The wavefunction and probability density are tools to:

1. Predict outcomes of quantum experiments.
2. Understand quantum behaviors like superposition, interference, and entanglement.
3. Model interactions with other particles or systems.

In simple cases (like your photon example), classical reasoning might suffice. But in more complex quantum scenarios, the wavefunction is indispensable!