Eigenstates of a free electron in a uniform magnetic field

[Zhuangzi]

Homework Statement

Consider the (non-relativistic) Hamiltonian of a particle of charge -e in the presence of an external magnetic field B=B_0*ẑ, in the symmetric gauge A=(1/2)B x r.

a) Explicitly write the Hamiltonian described and show that p_z is a constant of motion.
b) Using your reasoning from (a), show that the problem admits a separation of variables with eigenfunctions of the form ψ(r)=exp(i*k_z*z)φ(x,y).

Relevant Equations

H = (1/2m)(p-qA)^2 + qV
L_z = -iħ(x∂_y - y∂_x)
B=B_0*ẑ
A=(1/2)B x r
ψ(r)=exp(i*k_z*z)φ(x,y)

I started with the first of the relevant equations, replacing the p with the operator -iħ∇ and expanding the squared term to yield:

H = (-ħ^2 / 2m)∇^2 + (iqħ/m)A·∇ + (q^2 / 2m)A^2 + qV

But since A = (1/2)B x r

(iqħ/m)A·∇ = (iqħ / 2m)(r x ∇)·B = -(q / 2m)L·B = -(qB_0 / 2m)L_z

and A^2 = (1/4)(r^2*B^2 - (r·B)^2) = (B_0^2 / 4)(x^2 + y^2)

and V = 0

which gives a total Hamiltonian of

H = (-ħ^2 / 2m)∇^2 + (eB_0 / 2m) L_z + (e^2*B_0^2 / 8m)(x^2 + y^2).

At this point, however, I get stuck. I tried plugging in the wavefunction suggested in the problem, but I couldn't get an eigenvalue to pop out (I've attached a picture of my work). I want to know if I've made a mistake in calculating the Hamiltonian or in applying it to the wavefunction.

 

[vela]

The idea is to assume a solution of the form $\psi(x,y,z) = \phi(x,y)Z(z)$, substitute it into the Schrödinger equation, and show it separates into a differential equation for $\phi$ and one for $Z$.