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7.P: Exercises
 Last updated
 08:34, 16 Nov 2014

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 Calculate the energyshift in the ground state of the onedimensional harmonic oscillator when the perturbation
is added to
The properly normalized groundstate wavefunction is
 Calculate the energyshifts due to the firstorder Stark effect in the state of a hydrogen atom. You do not need to perform all of the integrals, but you should construct the correct linear combinations of states.
 The Hamiltonian of the valence electron in a hydrogenlike atom can be written
Here, the final term on the righthand side is the firstorder correction due to the electron's relativistic mass increase. Treating this term as a small perturbation, deduce that it causes an energyshift in the energy eigenstate characterized by the standard quantum numbers , , of
where is the unperturbed energy, and the fine structure constant.
 Consider an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers , , and . Show that if the energyshift due to spinorbit coupling (see Section 7.7) is added to that due to the electron's relativistic mass increase (see previous exercise) then the net fine structure energyshift can be written
Here, is the unperturbed energy, the fine structure constant, and the quantum number associated with the magnitude of the sum of the electron's orbital and spin angular momenta. You will need to use the following standard results for a hydrogen atom:
Here, is the Bohr radius. Assuming that the above formula for the energy shift is valid for (which it is), show that fine structure causes the energy of the states of a hydrogen atom to exceed those of the and states by .