Page 208 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

7.3 Spin one-half 199

j÷iˆ c á jái‡ c â jâi (7:11)
where c á and c â are complex constants. If the ket j÷i is normalized, then
equation (7.10) gives
2
2
jc á j ‡jc â j ˆ 1
The ket j÷i may also be expressed as a column matrix, known as a spinor

c á 1 0
j÷iˆ ˆ c á ‡ c â (7:12)
c â 0 1
where the eigenfunctions jái and jâi in spinor notation are

1 0
jáiˆ , jâiˆ (7:13)
0 1
1
Equations (7.4), (7.5), and (7.8) for the s ˆ case are
2
^ 2 3 2 ^ 2 3 2 (7:14)
S jáiˆ " jái,
S jâiˆ " jâi
4 4
^ 1 ^ 1 (7:15)
S z jáiˆ "jái,
2 S z jâiˆÿ "jâi
2
^ ^ (7:16a)
S ‡ jáiˆ 0,
S ÿ jâiˆ 0
^ ^ (7:16b)
S ‡ jâiˆ "jái,
S ÿ jáiˆ "jâi
^
^
Equations (7.16) illustrate the behavior of S ‡ and S ÿ as ladder operators. The
^
operator S ‡ `raises' the state jâi to state jái, but cannot raise jái any further,
^
while S ÿ `lowers' jái to jâi, but cannot lower jâi. From equations (7.7) and
(7.16), we obtain the additional relations
^ 1 ^ 1 (7:17a)
S x jáiˆ "jâi,
S x jâiˆ "jái
2 2
^ i ^ i (7:17b)
S y jáiˆ "jâi,
S y jâiˆÿ "jái
2 2
We next introduce three operators ó x , ó y , ó z which satisfy the relations
^ 1 S y ˆ "ó y , ^ 1 (7:18)
^
1
S x ˆ "ó x ,
2
2 2 S z ˆ "ó z
From equations (7.15) and (7.17), we ®nd that the only eigenvalue for each of
2
2
2
the operators ó , ó , ó is 1. Thus, each squared operator is just the identity
x
y
z
operator
2
2
2
ó ˆ ó ˆ ó ˆ 1 (7:19)
x y z
According to equations (7.2) and (7.18), the commutation rules for ó x , ó y , ó z
are
[ó x , ó y ] ˆ 2ió z , [ó y , ó z ] ˆ 2ió x , [ó z , ó x ] ˆ 2ió y (7:20)
The set of operators ó x , ó y , ó z anticommute, a property which we demon-
strate for the pair ó x , ó y as follows