Description
1 Parity operator
A wavefunction is written in the momentum representation ψ(p) ≡ hp|ψi.
a) Using the Wigner definition of parity, πψ(p) = hp|π|ψi, calculate the inverted wavefunction in momentum space.
b) A charged particle with charge q sits in a 1D quantum well with eigenstates |ni, n =
1, 2, . . . . A small potential U(x) = −qE0x
m due to a weak electric field is then introduced,
where m is a positive integer and x = 0 at the center of the well.
Using the properties of the
parity operator, derive the parity selection rule for the matrix element
hn
′
|U(x)|ni.
2 Time reversal symmetry I
a) If |n, ˆ −i is a two component eigenstate of the spin projection S · nˆ = 1
2
~σ · nˆ, with eigenstate
−~/2, show that application of the time reversal operator on this state, namely −iσyK|n, ˆ −i,
results in a state with the spin reversed.
b) A spin 1 particle has the Hamiltonian
H = αS2
z + β(S
2
x − S
2
y
).
Is the Hamiltonian invariant under time reversal symmetry? Prove your answer using the
properties of the time reversal symmetry operator T .
c) Calculate the exact eigenstates of the Hamiltonian in part b, and show that those states
obbey the same symmetry you found for the Hamiltonian under time reversal symmetry.
3 Time reversal symmetry II
Consider the time reversal symmetry operator T = UK acting in angular momentum states
|j, mi, where U is a unitary operator and K the conjugation.
a) Using the properties of the time reversal symmetry operator T and of the conjugation
operator K, calculate:
i) UJzU
ii) UJ±U
1
iii) UJ
2U
Express your answer in angular momentum operators only. Find whether each of those angular
momentum operators commute or anticommute with U.
b) Using your result in a), calculate the selection rule for the matrix elements hj, m′
|U|j, mi.
c) Now show that
hj, m′
|U|j, −m′
i
hj, m|U|j, −mi
= i
2(m′−m)
.
Choosing hj, m|U|j, −mi = (i)
2m, then find that
T |j, mi = (−1)m|j, −mi.
d) Find the time reversed state of the rotated ket D(R)|jmi.
e) Starting from the ket D(R)T |j, mi, show that
D
(j)∗
m′
,m(R) = (−1)m−m′
D
(j)
−m′
,−m(R).
Hint: Use the commutator [D(R), T ] in your derivation.
f) If a system is time reversal invariant and has no degeneracy in the energy spectrum
H|Ei = E|Ei, show that
hE|L|Ei = 0.
