Franz Schwabl Quantum Mechanics 4th Edit · Problem 7.3

Operators, Matrices, State Vectors

Physics — Quantum Mechanics · charged-particle-crossed-fields

Problem

Consider a particle with mass

m

and charge

e

in a homogeneous electromagnetic field

\boldsymbol{B} = (0,0,B)

\boldsymbol{E} = (E,0,0)

with

|E| < |B|

. Take the gauge

\boldsymbol{A} = (0, Bx, 0)

. Determine the eigenfunctions and eigenvalues for the Hamiltonian

H = \frac{1}{2m}\left(\boldsymbol{p} - \frac{e}{c}\boldsymbol{A}\right)^2 - eEx \;.

In the case

E = 0

, discuss also the degeneracy of the energy levels.