Page 210 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

7.4 Spin±orbit interaction 201

giving the relation bc ˆ 1. If we select b ˆ c ˆ 1 for ó x , then we have

0 1
ó x ˆ
1 0
The third of equations (7.22) determines that ó y must be

0 ÿi
ó y ˆ
i 0
In summary, the three matrices are

0 1 0 ÿi 1 0
ó x ˆ , ó y ˆ , ó z ˆ (7:25)
1 0 i 0 0 ÿ1
and are known as the Pauli spin matrices.
The traces of the Pauli spin matrices vanish
Tr ó x ˆ Tr ó y ˆ Tr ó z ˆ 0
and their determinants equal ÿ1
det ó x ˆ det ó y ˆ det ó z ˆÿ1
The unit matrix I

1 0
I ˆ
0 1
and the three Pauli spin matrices in equation (7.25) form a complete set of
2 3 2 matrices. Any arbitrary 2 3 2 matrix M can always be expressed as the
linear combination
M ˆ c 1 I ‡ c 2 ó x ‡ c 3 ó y ‡ c 4 ó z
where c 1 , c 2 , c 3 , c 4 are complex constants.

7.4 Spin±orbit interaction
The spin magnetic moment M s of an electron interacts with its orbital magnetic
moment to produce an additional term in the Hamiltonian operator and,
therefore, in the energy. In this section, we derive the mathematical expression
for this spin±orbit interaction and apply it to the hydrogen atom.
With respect to a coordinate system with the nucleus as the origin, the
electron revolves about the ®xed nucleus with angular momentum L. However,
with respect to a coordinate system with the electron as the origin, the nucleus
revolves around the ®xed electron. Since the revolving nucleus has an electric
charge, it produces at the position of the electron a magnetic ®eld B parallel to
L. The interaction of the spin magnetic moment M s of the electron with this
:
magnetic ®eld B gives rise to the spin±orbit coupling with energy ÿM s B.