Question

An atom is in a state of total electron spin S, total orbital angular momentum L, and total angular momentum J. (The nuclear spin can be ignored for this problem). The z component of the atomic total angular momentum is J_z. By how much does the energy of this atomic state change if a weak magnetic field of strength B is applied in the z direction? Assume that the interaction with the field is small compared with the fine structure interaction.

The answer should be given as an explicit expression in terms of the quantum numbers J, L, S, J_z and natural constants.

Answer 1

The Hamiltonian and eigenfunctions before the introduction of magnetic field are as follows:

where the subscripts of r, 1,2,, . . ,n, represent the different electrons in the atom, and \(\phi_{SLJM_J}\) is the common eigenstate of (L², S², J², J_z), i.e,

\(\langle LM_LS, M_J - M_L|JM_J\rangle\) being Clebsch-Gordan coefficients. The corresponding unperturbed energy is E_nsLJ.

After switching on the weak magnetic field, the Hamiltonian becomes

As B is very small, we can still consider (L², S², J², J_z) as conserved quantities and take the wave function of the system as approximately \(\phi_{nLJM_J}\). The energy change caused by the term \(\frac{eB}{2mc}J_z \) is \(\Delta E_1 = M_jh\frac{eB}{2mc}\) as J_z has eigenvalue M_Jh. The matrix of \(\frac{eB}{2mc}S_z \) is diagonal in the subspace of the 2J + 1 state vectors for the energy \(E_{nLJ}\) and hence the energy change caused by it is