Question

Two electrons at fixed positions on the z-axis have a magnetic dipole-dipole interaction (energy) E = A(s₁. s₂ - 3s_1z, s_2z), where Si = $i, Ci being Pauli抯 spin matrix, A = constant.

(a) Express E/A in terms of the total spin operator S = s₁ + s₂.

(b) State the eigenvalues of E/A and their degeneracy (statistical weights).

Answer 1

(a) As

(b) For the common eigenstate |S, M) of S and S_z, we have

Note that for states with M \(\ne\) 0, the energy levels are two-fold degenerate, i.e., D = 2.