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- Calculate the energy-shift in the ground state of the one-dimensional harmonic oscillator when the perturbation

is added to

The properly normalized ground-state wavefunction is

- Calculate the energy-shifts due to the first-order 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 hydrogen-like atom can be written

Here, the final term on the right-hand side is the first-order correction due to the electron's relativistic mass increase. Treating this term as a small perturbation, deduce that it causes an energy-shift 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 energy-shift due to spin-orbit 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 energy-shift 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 .