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# 8.P: Exercises

1. Demonstrate that when , where is the momentum operator, and is a real function of the position operator, . Hence, show that the Hamiltonian (870) is Hermitian.

2. Find the selection rules for the matrix elements , , and to be non-zero. Here, denotes an energy eigenket of a hydrogen-like atom corresponding to the conventional quantum numbers, , , and .

3. Demonstrate that

where the average is taken over all directions of the incident radiation.

4. Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any 2p state to a 1s state of a hydrogen atom is

where is the fine structure constant. Hence, deduce that the natural line width of the associated spectral line is

The only non-zero electric dipole matrix elements take the values