Atomic Structure Theory

8 1 Angular Momentum
1.2 Spin Angular Momentum
The internal angular momentum of a particle in quantum mechanics is called
spin angular momentum and designated by S. Cartesian components of S
satisfy angular momentum commutation rules (1.4). The eigenvalue of S 2 is
¯h 2 s(s+1) and the 2s+1 eigenvalues of Sz are ¯hm with m = −s, −s+1, ··· ,s.
1.2.1 Spin 1/2 and Spinors
Let us consider the case s =1/2 which describes the spin of the electron.
We designate the eigenstates of S 2 and Sz by two-component vectors χµ,
µ = ±1/2:
χ 1/2 =
1
, χ−1/2 =
0
0
. (1.42)
1
These two-component spin eigenfunctions are called spinors. The spinors χµ
satisfy the orthonormality relations
The eigenvalue equations for S 2 and Sz are
χ † µχν = δµν. (1.43)
S 2 χµ = 3
4 ¯h2 χµ, Szχµ = µ¯hχµ.
We represent the operators S2 and Sz as 2 × 2 matrices acting in the space
spanned by χµ:
S 2 = 3
4 ¯h2
10
, Sz =
01
1
2 ¯h
1 0
.
0 −1
One can use (1.18,1.19) to work out the elements of the matrices representing
the spin raising and lowering operators S±:
01
00
S+ =¯h , S− =¯h .
00
10
These matrices can be combined to give matrices representing Sx =(S+ +
S−)/2 andSy =(S+ − S−)/2i. The matrices representing the components
of S are commonly written in terms of the Pauli matrices σ =(σx,σy,σz),
whicharegivenby
σx =
01
10
through the relation
, σy =
0 −i
1 0
, σz = , (1.44)
i 0
0 −1
S = 1
¯hσ . (1.45)
2