Electric and Magnetic Fields

Consider a particle of mass m and charge e which is in perpendicular electric and magnetic fields: E = |E|ẑ, B = |B|ŷ.

a) Write the Hamiltonian, using a convenient gauge for the vector potential.

b) Find the eigenfunctions and eigenvalues.

c) Find the average velocity in the x-direction for any eigenstate.

SOLUTION

a) Many vector potentials A(r) can be chosen so that For the present problem the most convenient choice is  Thus the Hamiltonian is

(1)

The above choice is convenient since only p_z fails to commute with H, so p_x and p_y are constants of motion. Both potentials have been made to depend on z.

b) Since p_x and p_y are constants of motion, we can write the eigenstates and energies as

(2)

(3)

(4)

(5)

The last equation determines the eigenvalue E_z and eigenfunctions ѱ(z). The potential is a combination of linear and quadratic terms in z. So the motion behaves as a simple harmonic oscillator, where the terms linear in z determine the center of vibration. After some algebra we can write the above expression as

(6)

So, we obtain

(8)

The total energy is E_z plus the kinetic energy along the y-direction. The z-part of the eigenfunction is a harmonic oscillator ѱ_n(z – z₀).

c) In order to find the average velocity, we take a derivative with respect to the wave vector k_x:

(9)

This is the drift velocity in the x-direction. It agrees with the classical answer.