2D Harmonica Oscillator For Holes In Magnetic Field

[MartinCort]

Hello

I am bemused by a sign convention for Holes, My questions are as follow:

For an electron inside the 2D Circular Quantum Well. We can write our Hamiltonian as

H = 1/2m * ( p - q A)^2 + 1/2 m w^2 r^2 (Should we use minus in the momentum term? I think for Holes, it is)

If we expand this Hamiltonian under gauge A = -B/2 (y,x,0), we will obtain a effective 2D Harmonica Oscillator with a extra term :

H = 2D-Harmonica Oscillator - w_c/2 l_z (This sign is confusing)

where l_z is the angular momentum operators in z direction.
My question is, are the two minus signs make sense in my equations?

I am not too sure if it is correct because I have seen some literature get a total different sign, And this will affect the spectrum of the hole.

Thanks!

 

[eljose79]

Hello there,

I understand your confusion with the sign convention for holes in the context of the 2D Circular Quantum Well. Let me try to address your questions:

1. For the Hamiltonian, the sign convention for the momentum term depends on the convention used for the vector potential A. In your case, you have used the convention A = -B/2 (y,x,0), which results in a minus sign in the momentum term. This is a valid convention and is often used in literature.

2. The extra term in the expanded Hamiltonian, H = 2D-Harmonica Oscillator - w_c/2 l_z, also has a minus sign. This is because the angular momentum operator l_z has a minus sign in its definition, which is l_z = -i(hbar)(d/dz). Therefore, the overall sign in this term is also correct.

3. It is important to note that different literature may use different sign conventions, so it is always a good idea to check the conventions used in a particular study before comparing results.

I hope this helps clarify your questions. If you have any further doubts or concerns, please don't hesitate to ask. Thank you for your question and keep up the good work in your research!