Page 209 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

200 Spin
2i(ó x ó y ‡ ó y ó x ) ˆ (2ió x )ó y ‡ ó y (2ió x )

ˆ (ó y ó z ÿ ó z ó y )ó y ‡ ó y (ó y ó z ÿ ó z ó y )
2 2
ˆÿó z ó ‡ ó ó z
y y
ˆ 0
where the second of equations (7.20) and equation (7.19) have been used. The
same procedure may be applied to the pairs ó y , ó z and ó x , ó z, giving
(ó x ó y ‡ ó y ó x ) ˆ (ó y ó z ‡ ó z ó y ) ˆ (ó z ó x ‡ ó x ó z ) ˆ 0 (7:21)
Combining equations (7.20) and (7.21), we also have
ó x ó y ˆ ió z , ó y ó z ˆ ió x , ó z ó x ˆ ió y (7:22)

Pauli spin matrices
An explicit set of operators ó x , ó y , ó z with the foregoing properties can be
formed using 2 3 2 matrices. The properties of matrices are discussed in
Appendix I. In matrix notation, equation (7.19) is

2 2 2 1 0
ó ˆ ó ˆ ó ˆ 0 1 (7:23)
y
z
x
We let ó z be represented by the simplest 2 3 2 matrix with eigenvalues 1 and
ÿ1

1 0
ó z ˆ (7:24)
0 ÿ1
To ®nd ó x and ó y , we note that

a b 1 0 a ÿb
ˆ
c d 0 ÿ1 c ÿd
and

1 0 a b a b
ˆ
0 ÿ1 c d ÿc ÿd
Since ó x and ó y anticommute with ó z as represented in (7.24), we must have

a ÿb ÿa ÿb
ˆ
c ÿd c d
so that a ˆ d ˆ 0 and both ó x and ó y have the form

0 b
c 0
Further, we have from (7.23)

0 b 0 b bc 0 1 0
2 2
ó ˆ ó ˆ ˆ ˆ
x y c 0 c 0 0 bc 0 1