Physics 325
Winter 2022
Homework Assignment # 6
Due Friday Feb 25, in Canvas by 11:59PM
Turn in all problems and clearly note all constants and assumptions you use.
1. Perturbation
A 0
Consider a quantum system whose unperturbed Hamiltonian is given by
. A small
0 B
a b
perturbation is added in the form of H 0 =
.
b c
(a) Find first-order and second-order correction to energy eigenvalues.
(b) Compare new energy eigenvalues with the exact eigenvalues for the perturbed system.
(c) Find the eigenstates of the perturbed system up to first order.
2. The interval rule
The spin-orbit interaction splits the unperturbed atomic energy levels with the first-order energy
shift in perturbation theory given by
Eso (j) =
βso
(j(j + 1) − l(l + 1) − s(s + 1)),
2
(1)
where βso is independent of j.
(a) What are the values of j corresponding to n = 2, l = 1 in hydrogen?
(b) What is the numerical value of βso for n = 2, l = 1 in hydrogen?
(c) Show that Eso (j) − Eso (j − 1) = βso j (note l and s are fixed). This is called the interval rule.
(d) It is found spectroscopically that a certain energy level in a particular multi-electron atom
(i.e, s need not be 1/2) gets split by the spin-orbit interaction according to Eq. (1), generating
three fine structure levels with energy separations (between adjacent j states) in the ratio 5:3.
Find the values of l and s in this case.
3. Hyperfine splitting in hydrogen-like systems
Calculate the hyperfine splitting of the ground state of the following systems. Pay attention to
the mass - you will have to use the appropriate reduced mass to find the corresponding Bohr’s
radius.
(a) Muonic hydrogen (which is a hydrogen atom in which the electron is replaced by a muon,
a particle whose charge, spin and g-factor are the same as those of an electron, but the mass is
about 200 times larger).
(b) Positronium (which is a hydrogen atom in which the proton is replaced by a positron, whose
mass, spin and g-factor are the same those of an electron, but the charge is +e).
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