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Spin Precession
According to classical physics, a small current loop possesses a magnetic moment of magnitude , where
is the current circulating around the loop, and the area of the loop.
The direction of the magnetic moment is conventionally taken to be
normal to the plane of the loop, in the sense given by a standard
righthand circulation rule. Consider a small current loop consisting of an electron in uniform circular motion. It is
easily demonstrated that the electron's orbital angular momentum
is related to the magnetic moment
of the loop via

(758) 
where is the magnitude of the electron charge, and the electron mass.
The above expression suggests that there may be a similar relationship between
magnetic moment and spin angular momentum. We can write

(759) 
where is called the gyromagnetic ratio. Classically, we would
expect . In fact,

(760) 
where
is the socalled
finestructure constant. The fact that the gyromagnetic ratio is
(almost) twice that expected from classical physics is only explicable using relativistic
quantum mechanics. Furthermore, the small corrections to the relativistic result come from quantum field theory.
The energy of a classical magnetic moment
in a uniform magnetic field is

(761) 
Assuming that the above expression also holds good in quantum mechanics,
the Hamiltonian of an electron in a directed magnetic field of magnitude
takes the form

(762) 
where

(763) 
Here, for the sake of simplicity, we are neglecting the electron's translational degrees of freedom.
Schrödinger's equation can be written
[see Eq. (199)]

(764) 
where the spin state of the electron is characterized by the spinor .
Adopting the Pauli representation, we obtain

(765) 
where
. Here, is the probability of observing the
spinup state, and the probability of observing the spindown
state. It follows from Eqs. (748), (755), (762),
(764), and (765) that

(766) 
where
.
Hence,

(767) 
Let
The significance of the angle will become apparent presently.
Solving Eq. (767), subject to the initial conditions
(768) and (769), we obtain
We can most easily visualize the effect of the time dependence in the above
expressions for by calculating the expectation
values of the three Cartesian components of the electron's spin angular momentum. By analogy
with Eq. (192), the expectation value of a general spin operator
is simply

(772) 
Hence, the expectation value of is

(773) 
which reduces to

(774) 
with the help of Eqs. (770) and (771). Likewise,
the expectation value of is

(775) 
which reduces to

(776) 
Finally, the expectation value of is

(777) 
According to Eqs. (774), (776), and (777),
the expectation value of the spin angular momentum vector subtends
a constant angle with the axis, and precesses about
this axis at the frequency

(778) 
This behaviour is actually equivalent to that predicted by classical physics.
Note, however, that a measurement of , , or will always
yield either or . It is the relative probabilities
of obtaining these two results which varies as the expectation value
of a given component of the spin varies.
Subsections
Next: Exercises
Up: Spin Angular Momentum
Previous: Pauli Representation
Richard Fitzpatrick
20100720