Introduction to Quantum Mechanics · Problem 4.43

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · landau_levels

Problem

Suppose

\mathbf{A} = \frac{B_0}{2}\left(x\hat{j} - y\hat{i}\right)

, and

\varphi = Kz^2

, where

B_0

and

K

are constants. (a) Find the fields

\mathbf{E}

and

\mathbf{B}

. (b) Find the allowed energies, for a particle of mass

m

and charge

q

, in these fields. Answer:

E(n_1, n_2) = \left(n_1 + \frac{1}{2}\right)\hbar\omega_1 + \left(n_2 + \frac{1}{2}\right)\hbar\omega_2

(n_1, n_2 = 0, 1, 2, \ldots)

, (4.195) where

\omega_1 \equiv qB_0/m

and

\omega_2 \equiv \sqrt{2qK/m}

. Comment: In two dimensions (

x

and

y

, with

K = 0

) this is the quantum analog to cyclotron motion;

\omega_1

is the classical cyclotron frequency, and

\omega_2

is zero. The allowed energies,

\left(n_1 + \frac{1}{2}\right)\hbar\omega_1

, are called Landau Levels.