Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 329

cyclotronfrequency and theharmonic motion is centered at thepoint

which depends upon Theeigenvalues andeigenfunctions are

where are the eigenfunctions for the one-dimensional harmonic oscil-
lator.

5.75 Electric and Magnetic Fields (Princeton)

a) Many vectorpotentialsA(r) can be chosen so that For

the present problem the most convenient choice is Thus the

Hamiltonian is

The abovechoice isconvenientsince only fails tocommutewithH,so

and are constants ofmotion. Bothpotentials have been made to depend

on

b) Since and are constants ofmotion, we canwrite theeigenstates

and energies as

The lastequation determines theeigenvalue andeigenfunctions
The potential isa combination oflinear andquadratic terms in So the
motion behaves as asimpleharmonicoscillator,where the termslinear in
determine thecenter ofvibration. Aftersome algebra we canwrite the