Answer Happy

In this problem we consider the Zeeman effect - the behavior of (hydrogen-like) atoms in the presence of an external magnetic field. The total Hamiltonian is given in atomic units) by p2 H= +V(r) + HFS + Hz, 2 where His represents the fine-structure corrections while the Zeeman term is given by Hz = B(L2 +28;).

The coefficient a is the fine structure constant, B is the magnitude of the applied magnetic field (assumed to be aligned in the 2 direction), L, and S, are the z-components of the electron orbital and spin angular moment, respectively, and the relative factor of 2 represents the spin-factor of the electron (gs = 2). Note that the Bohr magneton is given by wb = a/2 in atomic units.

2 Next, consider the case where H, is much smaller, but still large enough that only part of the fine-structure contribution must be considered: the spin-orbit interaction. For this case, consider the unperturbed Hamiltonian to be Ho = a + V(r) + H2 + and consider HFS + Hso to be a perturbation. Use the uncoupled basis for Ho, and compute the energy correction AE arising from the spin-orbit term. To do this, you will need to write the operator product 7 in terms of Î4, St and Lz, Sz. You can use your results from problem 3.2 to evaluate the radial matrix element. Note that in this case, the correction vanishes for T = 0 levels. =