7.P: Exercises

1. 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
2. Calculate the energy-shifts due to the first-order Stark effect in the $n=3$ 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.
3. 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 $n$ , $l$ , $m$ of where $E_n$ is the unperturbed energy, and $\alpha$ the fine structure constant.
4. Consider an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers $n$ , $l$ , and $m$ . 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, $E_n$ is the unperturbed energy, $\alpha$ the fine structure constant, and $j=l\pm 1/2$ 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:
$\left\langle \frac{a_0}{r}\right\rangle$ $= \frac{1}{n^2},$ $\left\langle \frac{a_0^{\,2}}{r^2}\right\rangle$ $= \frac{1}{(l+1/2)\,n^3},$ $\left\langle \frac{a_0^{\,3}}{r^3}\right\rangle$ $= \frac{1}{l\,(l+1/2)\,(l+1)\,n^3}.$ Here, $a_0$ is the Bohr radius. Assuming that the above formula for the energy shift is valid for $l=0$ (which it is), show that fine structure causes the energy of the $(2p)_{3/2}$ states of a hydrogen atom to exceed those of the $(2p)_{1/2}$ and $(2s)_{1/2}$ states by $4.5\times 10^{-5}\,{\rm eV}$ .

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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