B "Distribution" of a particle in different situations

[davidge]

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.

 

[mfb]

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.

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.

 

[davidge]

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.

 

[mfb]

So that means that the very existence of a particle is dependent on the frame?

No. At least not in this context.

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.

 

[davidge]

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?

 

[Nugatory]

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.

 

[davidge]

You can conclude that any differences between the two are frame-dependent.

but certainly there are other important differences