Page 162 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.5 Magnetic moment 153

momentum is conserved and, since equation (5.81) applies, the magnetic
moment is constant in both magnitude and direction. Moreover, equation (5.81)
is also valid for orbital motion in quantum mechanics.

Interaction with a magnetic ®eld
The potential energy V of an atom with a magnetic moment M in a magnetic
®eld B is
:
V ˆÿM B ˆÿMB cos è (5:83)
where è is the angle between M and B. The force F acting on the atom due to
the magnetic ®eld is
F ˆÿ=V
or
@B @B
F x ˆÿM : ˆÿM cos è
@x @x
@B @B
F y ˆÿM : ˆÿM cos è (5:84)
@ y @ y
@B @B
F z ˆÿM : ˆÿM cos è
@z @z
If the magnetic ®eld is uniform, then the partial derivatives of B vanish and the
force on the atom is zero.
According to electrodynamics, the force F for a non-uniform magnetic ®eld
produces on the atom a torque T given by
ì B
T ˆ M 3 B ˆÿ L 3 B (5:85)
"
where equation (5.81) has been introduced as well. From the relation T ˆ
dL=dt in equation (5.6), we have
dL ì B
ˆÿ L 3 B (5:86)
dt "
Thus, the torque changes the direction of the angular momentum vector L and
the vector dL=dt is perpendicular to both L and B, as shown in Figure 5.5. As
a result of this torque, the vector L precesses around the direction of the
magnetic ®eld B with a constant angular velocity ù L . This motion is known as
Larmor precession and the angular velocity ù L is called the Larmor frequency.
Since the magnetic moment M is antiparallel to the angular moment L, it also
precesses about the magnetic ®eld vector B.
From equation (5.61), the Larmor angular frequency or velocity ù L is equal
to the velocity of the end of the vector L divided by the radius of the circular
path shown in Figure 5.5