- #1

Zhuangzi

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- 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

H = (-ħ^2 / 2m)

But since

(iqħ/m)

and

and V = 0

which gives a total Hamiltonian of

H = (-ħ^2 / 2m)

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.

*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 + qVBut since

**A**= (1/2)**B**x**r**(iqħ/m)

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

**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.