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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.

First, consider the case where Hz > HFs, so that we can ignore fine-structure completely. Com- pute the corrections AE to the energy-levels, using the uncoupled basis states (némems). Draw a diagram showing how the 6 degenerate states associated with the 2p energy level are shifted by the application of the external magnetic field.