# Hamiltonian of charge in 2D

1. Dec 1, 2014

### iamnoy

1. The problem statement, all variables and given/known data
charge e is within 2 dimensions in presence of magnetic field.

H = 1/2m (p - e/c A)^2

A = 1/2* B x r

p and r have two components

Show: H in terms of B along z axis resembles 2D HO (with some extra term)

express H in terms of x, y, p_x, L_y

2. Relevant equations

L = r x p

Hamiltonian for HO

3. The attempt at a solution

I plugged in B into H equation above so that gives:

H = 1/2m [p- e/c (1/2*Bxr)]

but i do not know what do do next. How do I express H in terms of B along z axis?

2. Dec 1, 2014

### ShayanJ

Well, if you use the vector potential $\vec A= B x \hat y$, then you can find the complete calculation here. I'm not sure using other vector potentials, gives the same effect or not!