- Consider a system of two spinless identical particles. Show that the orbital angular momentum
of their relative motion can only be even. (l= 0, 2 , 4 ,...) Show by direct calculation that, for
the triplet spin states of two spin^12 particles, ~σ 1 ·~σ 2 χ 1 m=χ 1 mfor all allowedm. Show that
for the singlet state~σ 1 ·~σ 2 χ 00 =− 3 χ 00.
- A deuteron has spin 1. What are the possible spin and total angular momentum states of two
deuterons. Include orbital angular momentum and assume the twoparticles are identical.
- The state of an electron is given byψ=R(r)[
√
1
3 Y^10 (θ,φ)χ++
√
2
3 Y^11 (θ,φ)χ−]. Find the
possible values and the probabilities of thezcomponent of the electron’s total angular mo-
mentum. Do the same for the total angular momentum squared. What is the probability
density for finding an electron with spin up atr,θ,φ? What is it for spin down? What is
the probability density independent of spin? (Do not leave your answer in terms of spherical
harmonics.)
- Then= 2 states of hydrogen have an 8-fold degeneracy due to the variouslandmstates
allowed and the two spin states of the electron. The spin orbit interaction partially breaks
the degeneracy by adding a term to the HamiltonianH 1 = Ae
2
2 m^2 c^2 r^3
L~·S~. Use first order
perturbation theory to find how the degeneracy is broken under the full Hamiltonian and
write the approximate energy eigenstates in terms ofRnl,Ylm, andχ±.
- The nucleus of a deuterium (A=2 isotope of H) atom is found to have spin 1. With a neutral
atom, we have three angular momenta to add, the nuclear spin, theelectron spin, and the
orbital angular momentum. DefineJ~=L~+S~in the usual way andF~=J~+~IwhereIdenotes
the nuclear spin operator. What are the possible quantum numbersjandffor an atom in
the ground state? What are the possible quantum numbers for an atom in the 2p state?
21.10Sample Test Problems
- Two identical spin^32 particles are bound together into a state with total angular momentum
l. a) What are the allowed states of total spin forl= 0 and forl= 1? b) List the allowed
states using spectroscopic notation forl= 0 and 1. (^2 s+1Lj)
- A hydrogen atom is in the stateψ=R 43 Y 30 χ+. A combined measurement of ofJ^2 and of
Jzis made. What are the possible outcomes of this combined measurement and what are the
probabilities of each? You may ignore nuclear spin in this problem.
- We want to find the eigenstates of totalS^2 andSzfor two spin 1 particles which have anS 1 ·S 2
interaction. (S=S 1 +S 2 )
(a) What are the allowed values ofs, the total spin quantum number.
(b) Write down the states of maximummsfor the maximumsstate. Use|sms〉notation and
|s 1 m 1 〉|s 2 m 2 〉for the product states.
(c) Now apply the lowering operator to get the othermsstates. You only need to go down
toms= 0 because of the obvious symmetry.
(d) Now find the states with the other values ofsin a similar way.
- Two (identical) electrons are bound in a Helium atom. What are the allowed states|jlsl 1 l 2 〉
if both electrons have principal quantum numbern= 1? What are the states if one hasn= 1
and the othern= 2?
