Principles of Quantum Mechanics · Problem 12.3.8

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · charged-particle-magnetic-field

Problem

Consider a particle of charge

q

in a vector potential

\mathbf{A} = \frac{B}{2}(-y\mathbf{i} + x\mathbf{j})

(1) Show that the magnetic field is

\mathbf{B} = B\mathbf{k}

. (2) Show that a classical particle in this potential will move in circles at an angular frequency

\omega_0 = qB/\mu c

. (3) Consider the Hamiltonian for the corresponding quantum problem:

H = \frac{[P_x + qYB/2c]^2}{2\mu} + \frac{[P_y - qXB/2c]^2}{2\mu}

Show that

Q = (cP_x + qYB/2)/qB

and

P = (P_y - qXB/2c)

are canonical. Write

H

in terms of

P

and

Q

and show that allowed levels are

E = (n+1/2)\hbar\omega_0

. (4) Expand

H

out in terms of the original variables and show

H = H\left(\frac{\omega_0}{2}, \mu\right) - \frac{\omega_0}{2} L_z

where

H(\omega_0/2, \mu)

is the Hamiltonian for an isotropic two-dimensional harmonic oscillator of mass

\mu

and frequency

\omega_0/2

. Argue that the same basis that diagonalized

H(\omega_0/2, \mu)

will diagonalize

H

. By thinking in terms of this basis, show that the allowed levels for

H

are

E = (k + \frac{1}{2}|m| - \frac{1}{2}m + \frac{1}{2})\hbar\omega_0

, where

k

is any integer and

m

is the angular momentum. Convince yourself that you get the same levels as from the earlier formula

[E = (n+1/2)\hbar\omega_0]

. We shall return to this problem in Chapter 21.