Thermodynamics, Statistical Physics, and Quantum Mechanics

328 SOLUTIONS

5.74 Electron in Magnetic Field (Stony Brook,

Moscow Phys-Tech)

a) The relationship between the vector potential and magnetic field is

Using does give So this vector

potential produces the right field.

b)The vector potential enters the Hamiltonian in the form

One can show easily that and each commute with the Hamiltonian

and are constants of motion. Thus, we can write the eigenfunction as

plane waves for these two variables, with only the yet to be

determined:

The Hamiltonian operating on gives

where We may write the energy E as

and find

The energy is given by the component along the magnetic field and the
energy for motion in the plane. The latter contribution is identical
to the simple harmonic oscillator in the The frequency is the