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8.P: Exercises
 Last updated
 08:37, 16 Nov 2014

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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 nonzero. Here, denotes an energy eigenket of a hydrogenlike 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 nonzero electric dipole matrix elements take the values
where is the Bohr radius.