Physics 5403 Homework 1 solved

Description

1 Addition of angular momentum

Consider two particles with spin 3/2 and spin 1/2.
a) Compute the total angular momentum states |j1, j2; J, Mi in terms of the single particle
product states |j1, m1i|j2, m2i.

b) Suppose the Hamiltonian of the two particles has the form
H = αS1 · S2,
with α a constant. If the system is initially (t = 0) in the following eigenstate of S
2
1
, S
2
2
, S1z, S2z,
|j1j2; m1m2i =

3
2
1
2
;
1
2
1
2

,
what is the probability of finding the system in state

3
2
1
2
;
3
2 −
1
2
E
at time t > 0?

2 Clebsh-Gordan Coefficients

Using recursion relations, verify the special case of the Clebsh-Gordan coefficient
hj1; j0|j1; jji =
s
j
j + 1
.

Hint: write down the relevant Clebsh-Gordan completeness relation that includes this coefficient. That will give you one equation with two unknown coefficients. Find then a convenient
recursion relation to obtain the second equation for those two same coefficients and solve them.

3 Wigner-Eckart theorem

Consider a system formed formed by two spinless particles with angular momentum j1 = 1
and j2 = 1.

a) Assuming that the system is subjected to a spherically symmetric potential, using the
Wigner-Eckart theorem, find the selection rules for the matrix elements of the momentum
operator components Px, Py and Pz
hα, j′
, m′
|Pi
|αjmi,
1
where α is a quantum number which is independent of the of the magnetic quantum numbers m
and m′
.

Compute the matrix elements explicitly for j
′ = 2 and j = 1 (use the Clebsh-Gordan
table for that).

b) Using the definition for the product of spherical tensors,
T
(k)
q =
X
q1q2
T
(k1)
q1
T
(k2)
q2
hk1k2; q1q2|k1k2; kqi,
compute the generic form for T
(2)
q in terms of the operator components of P.