 Problem Statement

Consider the (nonrelativistic) 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)(pqA)^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.
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.
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