Understanding Potential Energy in QM

[leinadle]

Hi all,

I'm having conceptual troubles understanding the significance of the potential energy term in Schrödinger's equation. More specifically, the physical meaning of the potential "wells" is not clicking with me; my textbook is not making this very clear.

For clarity, consider an electron in some wire that has a potential energy function that is not constant throughout the length and non-zero. How exactly can I interpret this? Does this curve characterize the potential energy of the electron at any point in the wire? What could cause this potential energy to vary with position, and how does that physically affect the electron?

An overall conceptual understanding of the potential energy for an electron in a well would be much appreciated, as I am quite confused on how to understand these curves intuitively.

 

[Nugatory]

For clarity, consider an electron in some wire that has a potential energy function that is not constant throughout the length and non-zero. How exactly can I interpret this? Does this curve characterize the potential energy of the electron at any point in the wire? What could cause this potential energy to vary with position, and how does that physically affect the electron?

Do you already have a clear understanding of classical electricity and magnetism, and especially Maxwell's equations? This part of QM is just like classical E&M - the potential will vary from one point to another if there is an electric field.

 

[leinadle]

Do you already have a clear understanding of classical electricity and magnetism, and especially Maxwell's equations? This part of QM is just like classical E&M - the potential will vary from one point to another if there is an electric field.

On an introductory level, yes. Are you saying the potential energy profile is solely due to the superposition of electric fields from surrounding charges? I guess my confusion with that is the sudden jumps that I'm seeing in the potential energy for finite square wells and the tunneling barriers. When an electron is in the barrier, for example, what is that constant potential energy value relative to? For a cut wire, would that just be the work function for whatever the wire is made of?

We're only talking about the 1D non-rigid box model in my class (electron in a wire), so I think my confusion stems from not knowing how we can have different potential setups for this scenario. Specifically, why exactly would the rounded potential well be a more accurate depiction of an electron in a wire? How would a higher potential near the ends be more appropriate?

These questions are probably very obvious to you, but I just want to make sure that I have a solid foundation.

 

[K^2]

These are approximations. Real potentials are very different. But you can learn a lot about the physics of conductors and lattices by studying such approximate potentials.

 

[leinadle]

These are approximations. Real potentials are very different. But you can learn a lot about the physics of conductors and lattices by studying such approximate potentials.

So what makes the rounded square well a better approximation? Why is it sensible to have the potential start to vary near the edges rather than stay zero?

 

[K^2]

It all depends on your models. Typically, potential wells are a decent model for a lattice. You have ions located at specific points on the lattice, which means the potential is going to be low there and high everywhere else. Because the most critical feature is the periodicity of the potential, square well for each ion makes sense.