Page 211 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

202 Spin
1
According to the Biot and Savart law of electromagnetic theory, the
magnetic ®eld B at the `®xed' electron due to the revolving positively charged
nucleus is given in SI units to ®rst order in v=c by
1
B ˆ (E 3 v n ) (7:26)
c 2
where E is the electric ®eld due to the revolving nucleus, v n is the velocity of
the nucleus relative to the electron, and c is the speed of light. The electric
force F is related to E and the potential energy V(r) of interaction between the
nucleus and the electron by
F ˆÿeE ˆÿ=V
Thus, the electric ®eld at the electron is
r n dV(r)
E ˆ (7:27)
er dr
where r n is the vector distance of the nucleus from the electron. The vector r
from nucleus to electron is ÿr n and the velocity v of the electron relative to the
nucleus is ÿv n . Accordingly, the angular momentum L of the electron is
L ˆ r 3 p ˆ m e (r 3 v) ˆ m e (r n 3 v n ) (7:28)
Combining equations (7.26), (7.27), and (7.28), we have
1 dV(r)
B ˆ L (7:29)
2
em e c r dr
:
The spin±orbit energy ÿM s B may be related to the spin and orbital
angular momenta through equations (7.1) and (7.29)
1 dV(r)
: :
ÿM s B ˆ L S
2 2
m c r dr
e
This expression is not quite correct, however, because of a relativistic effect in
changing from the perspective of the electron to the perspective of the nucleus.
2
The correction, known as the Thomas precession, introduces the factor 1 on
2
the right-hand side to give
1 dV(r)
: :
ÿM s B ˆ L S
2 2
2m c r dr
e
^
The corresponding spin±orbit Hamiltonian operator H so is, then,
1 dV(r)
^ ^ : ^
H so ˆ L S (7:30)
2 2
2m c r dr
e
1 R. P. Feyman, R. B. Leighton, and M. Sands (1964) The Feynman Lectures on Physics, Vol. II (Addison-
Wesley, Reading, MA) section 14-7.
2 J. D. Jackson (1975) Classical Electrodynamics, 2nd edition (John Wiley & Sons, New York) pp. 541±2.