PHYS 402 Homework---Due March 11
1. In this problem we consider two spin ½ particles interacting via the Hamiltonian 21 ˆˆ
ˆssaH
r
r
⋅= where a is a
constant. The computation of the energies using total spin is straightforward. Show that there is a nondegenerate
state ( “a singlet”) with energy
4
3ah
− and triply degenerate state (“a triplet”) with energy
4
ah .
2. In the next two problems we will work through the system discussed in problem 1 the dumb way: using the
2121 mmss basis. The states in this basis can be denoted ↑↑ ↑↓ ↓↑ ↓↓
a. Show that
↓↓=↓↓↓↑−=↓↑↑↓−=↑↓↑↑=↑↑
4
ˆˆ
4
ˆˆ
4
ˆˆ
4
ˆˆ 2
21
2
21
2
21
2
21
hhhh
zzzzzzzz ssssssss
b. Show that
↑↑=↓↓↑↓=↓↑↓↑=↑↓↓↓=↑↑
4
ˆˆ
4
ˆˆ
4
ˆˆ
4
ˆˆ 2
21
2
21
2
21
2
21
hhhh
xxxxxxxx ssssssss
c. Show that
↑↑−=↓↓↑↓=↓↑↓↑=↑↓↓↓−=↑↑
4
ˆˆ
4
ˆˆ
4
ˆˆ
4
ˆˆ 2
21
2
21
2
21
2
21
hhhh
yyyyyyyy ssssssss
3. In this problem we construct and diagonalize the Hamiltonian matrix:
a. Using the results of problem 2. show that Hamiltonian in problem one can be expresses as the following
matrix in the basis ↑↑ ↑↓ ↓↑ ↓↓ :
−
−
=
1000
0120
0210
0001
4
2
ha
H
b. Find the eignenvalues of the preceding Hamiltonian and verify that there there are three degenerate
eigenvalue of
4
ah and one of
4
3ah
− as seen in problem 1.
4. This problem a concerns a system of of two particles particle one with spin ½ and particle two with spin 1.
Suppose that the Hamiltonian for the system 21 ˆˆ
ˆssaH
r
r
⋅= where a is a constant. The first problem concerns
some general properties of the system
a. Show on general grounds that there are six states in the Hilbert space.
b. What are the possible total spins?
c. Explain why one expects only two distinct energies. One level which is four-fold degenerate and another
which is two fold degenerate.
d. Use properties of the total spin to show that the four fold degenerate states have an energy of
2
ah while
the doubly degenerate state has an energy of -a
h
.
e. Write the state 1/2m 3/2S| == in the 2121 mmss basis.