# B "Distribution" of a particle in different situations

1. Apr 7, 2017

### 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.

2. Apr 7, 2017

### Staff: Mentor

The bands are the limits of "infinite" spread.

Every system has an energy.
The overall speed of the electron does not matter. You can always change to a frame where its (expectation value of) momentum is zero.

3. Apr 7, 2017

### davidge

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.

4. Apr 7, 2017

### Staff: Mentor

No. At least not in this context.
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.

5. Apr 7, 2017

### 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?

6. Apr 7, 2017

### Staff: Mentor

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

7. Apr 7, 2017

### davidge

but certainly there are other important differences