How Does a Magnetic Field Influence a 2D Harmonic Oscillator?

[iamnoy]

Homework Statement

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

Homework Equations

L = r x p[/B]

Hamiltonian for HO

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?

 

[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!