Question

The Stark effect. The energy levels of the n = 2 states of atomic hydrogen are illustrated in Fig.

The S_1/2 and P_1/2 levels are degenerate at an energy \(\varepsilon_0\) and the P_3/2 level is degenerate at an energy \(\varepsilon_0\) + A.

A uniform static electric field E applied to the atom shifts the states to energies \(\varepsilon_1\), \(\varepsilon_2\) and \(\varepsilon_3\). Assuming that all states other than these three are far enough away to be neglected, determine the energies \(\varepsilon_1\), \(\varepsilon_2\) and \(\varepsilon_3\) to second order in the electric field E.

Answer 1

Suppose the matrix elements of the perturbation Hamiltonian H’ = -eE . r are

since \(\langle l'|H'|l\rangle\) = 0 for I', I states of the same parity. Then for energy level P_3/2, we have

For energy levels P_1/2 and S_1/2, we diagonalize the Hamiltonian in the corresponding subspace, i.e, solve

The roots are \(\lambda\) = \(\pm|a|\), which give the new wave functions