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Magnetic Moments
Consider a particle of electric charge
and speed
performing a circular orbit of radius
in the

plane. The charge is equivalent to a current loop of radius
, lying
in the

plane, and carrying a current
. The magnetic moment
of
the loop is of magnitude
and is directed along the
axis (the direction is given
by a righthand rule with the fingers of the righthand circulating in the same direction as the current) [49].
Thus, we can write

(5.42) 
where
and
are the vector position and velocity of the particle,
respectively. However, we know that
, where
is the particle's vector momentum, and
its mass. We also know that
, where
is the orbital angular momentum.
It follows that

(5.43) 
Using the standard analogy between classical and quantum mechanics, we
expect the previous relation to also hold between the quantum mechanical operators,
and
, which represent magnetic moment and orbital angular momentum,
respectively.
This is indeed found to the the case experimentally.
Spin angular momentum also gives rise to a contribution to the magnetic
moment of a charged particle. In fact, relativistic quantum
mechanics predicts that a charged particle possessing spin must also
possess a corresponding magnetic moment [30,32,9]. We can write

(5.44) 
where the parameter
is called the
factor. For an electron this factor
is found to be

(5.45) 
where

(5.46) 
is the fine structure constant.
The factor
is correctly predicted by Dirac's famous relativistic theory of the electron [30,32]. (See Chapter 11.)
The small correction
is due to
quantum field effects [101]. We shall ignore this correction in the following,
so

(5.47) 
for an electron. (Here,
is the magnitude of the electron charge, and
the electron mass.)
Next: Spin Precession
Up: Spin Angular Momentum
Previous: Rotation Operators in Spin
Richard Fitzpatrick
20160122