Electron in Magnetic Field

An electron is in free space except for a constant magnetic field B in the z-direction.

a) Show that the magnetic field can be represented by the vector potential A = B(0, x, 0).

b) Use this vector potential to derive the exact eigenfunctions and eigenvalues for the electron.

SOLUTION

a) The relationship between the vector potential and magnetic field is   Using A = B (0, x, 0) does give B = Bẑ. So this vector potential produces the right field.

b) The vector potential enters the Hamiltonian in the form

(1)

One can show easily that p_y and p_z 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 x-dependence yet to be determined:

(2)

The Hamiltonian operating on ѱ gives

(3)

where E_z = h²k²_z/2m. We may write the energy E as

(4)

and find

(5)

(6)

(7)

The energy is given by the component E_z along the magnetic field and the energy E_x for motion in the (x, y) plane. The latter contribution is identical to the simple harmonic oscillator in the x-direction. The frequency is the cyclotron frequency ω_c, and the harmonic motion is centered at the point x₀ which depends upon k_y. The eigenvalues and eigenfunctions are

(8)

(9)

where ѱ_n(x) are the eigenfunctions for the one-dimensional harmonic oscillator.