Question

An electron of mass m and charge -e moves in a region where a uniform magnetic field B = V x A exists in z direction.

(a) Set up the Schrodinger equation in rectangular coordinates.

(b) Solve the equation for all energy levels.

(c) Discuss the motion of the electron.

Answer 1

(a) The Hamiltonian is

we can take A_z = A_x = 0, A_x = B_x, i.e. A = Bx\(\hat y\), and write the Schrodinger equation as

(b) As \([\hat P_y, \hat H] = [\hat P_z, \hat H] =0\), P_y and P_z are conserved. Choose H, P_y, P_z as a complete set of mechanical variables and write the SchrGdinger equation as

The above shows that \(\hat H - \frac{\hat P_x^2}{2m}\) is the Hamiltonian of a one-dimensional harmonic oscillator of angular frequency w = \(\frac{eB}{mc}\). Hence the energy levels of the system are

E = (n + 1/2)h\(\omega\) + P_z²/2m, n = 0, 1, 2, . . .

Because the expression of E does not contain P_y explicitly, the degeneracies of the energy levels are infinite.

(c) In the coordinate frame chosen, the energy eigenstates correspond to free motion in the z direction and circular motion in the x - y plane, i.e. a helical motion. In the z direction, the mechanical momentum mw_z = P_z is conserved, describing a uniform linear motion. In the x direction there is a simple harmonic oscillation round the equilibrium point x = -cP_y/eB. In the y direction, the mechanical momentum is mv_y = P_y + eBx/c = eB\(\xi\) /c = m\(\omega\xi\) and so there is a simple harmonic oscillation with the same amplitude and frequency.