INTRODUCTION TO QUANTUM MECHANICS IPROBLEM SET 11

due December 12th, before class

I. WIGNER-ECKART AGAIN

a) Show that the three operators x, y and z form a k = 1 tensor (that is, vector) operator.b) Suppose 〈1/2, 1/2, α |z|1/2, 1/2, β〉 = A. Find 〈1/2, 1/2, α |x|1/2,−1/2, β〉. You can read off the necesary Cleb-

sches out of a table.

II. SPIN PROBLEM FROM SAKURAI

The spin-dependent hamiltonia of an electron-positron system in the presence of a magnetic field in the z-directioncan be written as

H = AS(−).S(+) +eB

mcS(−)z − S(+)

z , (1)

were the (+) and (−) refer ot the positron and electron. Supposed the spn pat of the wave function of the system isgiven by χ(−)χ(+), namely, it factorizes into a product of a electron wave function and a positron wavefunction.

i) Is this an eigenfunction of H in the limit A→ 0, eB 6= 0? If it is, what is the eigenvalue?ii) Same problem when eB/mc→ 0, A 6= 0.

III. ELEMENTARY FACTS ABOUT TIME REVERSAL

1. φ(p) is the momentum space wave function for the state |φ〉. What is the momentum space wave function for

the time reversed state T |φ〉.

2. Assuming that the hamiltonian of a spinless particle is time-reversal invariant show that the position space wavefunction os a non-degenerate state can be chosen to be real.

3. The wave function for a plane wave state is ei~p.~r/~. Is this in contradiction to the item above.

4. Suppose a spinless particle is bound to a fixed center by a potential V (~r) so assymetrical that no energy level

is degenerate. Using time reversal prove that 〈~L〉=0. This is known as quenching of the angular momentum.