Page 212 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

7.4 Spin±orbit interaction 203

For a hydrogen atom, the potential energy V(r) is given by equation (6.13)
^
with Z ˆ 1 and H so becomes
^ : ^
^
H so ˆ î(r)L S (7:31)
where
e 2
î(r) ˆ (7:32)
2 2 3
8ðå 0 m c r
e
^
Thus, the total Hamiltonian operator H for a hydrogen atom including spin±
orbit coupling is
^
^
^
^
^ : ^
H ˆ H 0 ‡ H so ˆ H 0 ‡ î(r)L S (7:33)
^
where H 0 is the Hamiltonian operator for the hydrogen atom without the
inclusion of spin, as given in equation (6.14).
The effect of the spin±orbit interaction term on the total energy is easily
shown to be small. The angular momenta jLj and jSj are each on the order of "
and the distance r is of the order of the radius a 0 of the ®rst Bohr orbit. If we
also neglect the small difference between the electronic mass m e and the
reduced mass ì, the spin±orbit energy is of the order of
2 2
e "
2
ˆ á jE 1 j
2 2 3
8ðå 0 m c a
e 0
where jE 1 j is the ground-state energy for the hydrogen atom with Hamiltonian
^
operator H 0 as given by equation (6.57) and á is the ®ne structure constant,
de®ned by
e 2 " 1
á ˆ ˆ
4ðå 0 "c m e ca 0 137:036
Thus, the spin±orbit interaction energy is about 5 3 10 ÿ5 times smaller than
jE 1 j.
^
While the Hamiltonian operator H 0 for the hydrogen atom in the absence of
^
^
the spin±orbit coupling term commutes with L and with S, the total Hamilto-
^
^
^
nian operator H in equation (7.33) does not commute with either L or S
^ : ^
because of the presence of the scalar product L S. To illustrate this feature,
^ ^ : ^
^ : ^
^
we consider the commutators [L z , L S] and [S z , L S],
^ ^ : ^
^ ^
^
^
^ ^
^ ^
^ ^
^
^ ^
[L z , L S] ˆ [L z ,(L x S x ‡ L y S y ‡ L z S z )] ˆ [L z , L x ]S x ‡ [L z , L y ]S y ‡ 0
^ ^
^ ^
ˆ i"(L y S x ÿ L x S y ) 6ˆ 0 (7:34)
^ ^
^ ^ : ^
^ ^ ^
^ ^ ^
^ ^
[S z , L S] ˆ [S z , S x ]L x ‡ [S z , S y ]L y ˆ i"(L x S y ÿ L y S x ) 6ˆ 0 (7:35)
where equations (5.10) and (7.2) have been used. Similar expressions apply to
^
^
the other components of L and S. Thus, the vectors L and S are no longer