So we have completed the calculation of the coefficients. We will make use of these in the hydrogen
atom, particularly for the anomalous Zeeman effect.
Writing this in thenotation of matrix elements or Clebsch-Gordan coefficientsof the form,
〈jmjℓs|ℓmℓsms〉we get.
audio
〈(
ℓ+
1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓm1
2
1
2
〉
= α=√
ℓ+m+ 1
2 ℓ+ 1
〈(
ℓ+1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓ(m+ 1)1
2
− 1
2
〉
= β=√
ℓ−m
2 ℓ+ 1
〈(
ℓ+1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓm1
2
− 1
2
〉
= 0
〈(
ℓ+
1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓ(m+ 1)1
2
1
2
〉
= 0
Similarly
〈(
ℓ−1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓm1
2
1
2
〉
=
√
ℓ−m
2 ℓ+ 1
〈(
ℓ−1
2
)(
m+1
2
)
ℓ
1
2
∣
∣
∣
∣ℓ(m+ 1)1
2
− 1
2
〉
= −
√
ℓ+m+ 1
2 ℓ+ 121.8.5 Counting the States for|ℓ 1 −ℓ 2 |≤j≤ℓ 1 +ℓ
If we addℓ 1 toℓ 2 there are (2ℓ 1 + 1)(2ℓ 2 + 1) product states. Lets add up the number of states of
totalℓ. To keep things simple we assume we ordered things soℓ 1 ≥ℓ 2.
ℓ∑ 1 +ℓ 2ℓ=ℓ 1 −ℓ 2(2ℓ+ 1) =
∑^2 ℓ^2
n=0(2(ℓ 1 −ℓ 2 +n) + 1) = (2ℓ 2 + 1)(2ℓ 1 − 2 ℓ 2 + 1) + 2∑^2 ℓ^2
n=0n= (2ℓ 2 + 1)(2ℓ 1 − 2 ℓ 2 + 1) + (2ℓ 2 + 1)(2ℓ 2 ) = (2ℓ 2 + 1)(2ℓ 1 + 1)
This is what we expect.
21.9 Homework Problems
- Find the allowed total spin states of two spin 1 particles. Explicitly write out the 9 states
which are eigenfunctions ofS^2 andSz. - The Hamiltonian of a spin system is given byH=A+B
S~ 1 ·S~ 2
h ̄^2 +
C(S 1 z+S 2 z)
̄h. Find the eigenvalues
and eigenfunctions of the system of two particles (a) when both particles have spin^12 , (b) when
one particle has spin^12 and the other spin 1. What happens in (a) when the two particles are
identical?