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- 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.

- 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 .

- Demonstrate that

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

- 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

where is the Bohr radius.