Page 160 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.5 Magnetic moment 151

To ®nd the degeneracy of the eigenvalue E J , we note that for a given value
of J, the quantum number m has values m ˆ 0, 1, 2, ... , J. Accordingly,
there are (2J ‡ 1) spherical harmonics for each value of J and the energy level
E J is (2J ‡ 1)-fold degenerate. The ground-state energy level E 0 is non-
degenerate.

5.5 Magnetic moment

Atoms are observed to have magnetic moments. To understand how an electron
circulating about a nuclear core can give rise to a magnetic moment, we may
apply classical theory. We consider an electron of mass m e and charge ÿe
bound to a ®xed nucleus of charge Ze by a central coulombic force F(r) with
potential V(r)
dV(r) ÿZe 2
F(r) ˆÿ ˆ (5:75)
dr 4ðå 0 r 2
ÿZe 2
V(r) ˆ (5:76)
4ðå 0 r
Equation (5.75) is Coulomb's law for the force between two charged particles
separated by a distance r. In SI units, the charge e is expressed in coulombs
(C), while å 0 is the permittivity of free space with the value
2
å 0 ˆ 8:854 19 3 10 ÿ12 J ÿ1 C m ÿ1
According to classical mechanics, a stable circular orbit of radius r and angular
velocity ù is established for the electron if the centrifugal force m e rù 2
balances the attractive coulombic force
Ze 2
2
m e rù ˆ
4ðå 0 r 2
This assumption is the basis of the Bohr model for the hydrogen-like atom.
When solved for ù, this balancing equation is
2 1=2
Ze
ù ˆ (5:77)
4ðå 0 m e r 3
An electron in a circular orbit with an angular velocity ù passes each point
in the orbit ù=2ð times per second. This electronic motion constitutes an
electric current I, de®ned as the amount of charge passing a given point per
second, so that
eù
I ˆ (5:78)
2ð
From the de®nition of the magnetic moment in electrodynamics, a circulat-