"Distribution" of a particle in different situations

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Discussion Overview

The discussion revolves around the "distribution" of a particle, specifically focusing on the wave function of electrons in various contexts, such as bound electrons in atoms, conduction electrons in wires, and free electrons. Participants explore the implications of these distributions on quantum mechanics and frame dependence.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that an electron bound in an atom has a broad wave function and questions the broadness of conduction electrons in a wire and free electrons.
  • Another participant suggests that the wave function's broadness for conduction electrons is limited by energy bands, implying a connection to their confinement in a material.
  • It is proposed that the speed of a free electron influences the broadness of its wave function, although one participant argues that the overall speed does not matter due to the possibility of changing frames.
  • A question is raised about whether the existence of a particle is frame-dependent, leading to a clarification that while wave functions may appear different to different observers, this does not imply that a particle's existence is dependent on the frame.
  • Participants discuss the implications of frame dependence on the differences between wave functions for electrons in a current versus free electrons, with one suggesting that any differences are frame-dependent, while another hints at the existence of other significant differences.

Areas of Agreement / Disagreement

Participants express differing views on the implications of frame dependence for the wave functions of electrons, with some suggesting that differences are purely frame-dependent while others indicate that there are additional important differences that need to be considered. The discussion remains unresolved regarding the specific nature of these differences.

Contextual Notes

Participants acknowledge the vagueness in defining proximity in terms of interactions affecting the wave function and the implications of changing frames on the interpretation of wave functions.

davidge
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I was thinking about the "distribution" of a particle in space in different situations. An electron bound in a atom have a wave function that is broad.

What about the broadness of the wave function of conduction electrons in a wire? Or doesn't it even make sense to quantum-mechanicaly speak of conduction electrons in a wire?

What about the broadness of the wave function of a free electron? By free I mean an electron that is not interacting with any nearby particles or "classical" fields. (I know that just saying nearby is vague (How nearby?), but I hope you understand what I mean.)

So I was trying to answer myself the two questions above, and I started by thinking this way:

The time evolution of a wave function is governed by the hamiltonian of the system and I guess the hamiltonian is non-zero only if the system has energy. So in both cases above, I would expect the wave function to be changing in time.
Also, the broader the wave function, the narrower the momentum function -which measures the distribution of momentum through space-. In the case of the electrons in a wire, I guess they are constantly being atracted by the other charges that form the atoms and by the other electrons as well, but I don't have an idea of how fast they move (in average, at least). For a free electron, it would be easy to conclude that how fast the electron is moving will dictate how broader is its wave function.

I used the term wave function to mean position function, actually.
 
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davidge said:
What about the broadness of the wave function of conduction electrons in a wire?
The bands are the limits of "infinite" spread.

Every system has an energy.
davidge said:
For a free electron, it would be easy to conclude that how fast the electron is moving will dictate how broader is its wave function.
The overall speed of the electron does not matter. You can always change to a frame where its (expectation value of) momentum is zero.
 
mfb said:
You can always change to a frame where its (expectation value of) momentum is zero.
So that means that the very existence of a particle is dependent on the frame? Because we can change to a frame where its wave function has a different value at a same position in space.
 
davidge said:
So that means that the very existence of a particle is dependent on the frame?
No. At least not in this context.
davidge said:
Because we can change to a frame where its wave function has a different value at a same position in space.
The wave function will look different for different observers, of course. That is not a result of quantum mechanics, you already have different properties in classical mechanics.
 
Ok. So, returning to the point of the thread, what can we conclude about the wave function for electrons in a current and for a free electron?
 
davidge said:
Ok. So, returning to the point of the thread, what can we conclude about the wave function for electrons in a current and for a free electron?
You can conclude that any differences between the two are frame-dependent.
 
Nugatory said:
You can conclude that any differences between the two are frame-dependent.
but certainly there are other important differences
 

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