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Spin
Michael Fowler 11/26/06
Introduction
The Stern
Gerlach experiment for the simplest possible atom, hydrogen in its ground
state, demonstrated unambiguously that the component of the magnetic moment of
the atom along the z-axis could only
have two values. It had been well
established by this time that the magnetic moment vector was along the same
axis as the angular momentum. This is
obviously true for the Bohr model of hydrogen, where the circulating electron
is equivalent to a ring current, generating a magnetic dipole. The problem is, though, that a magnetic
moment generated in this way by orbital
angular momentum will have a minimum
of three possible values of its z-component: the lowest nonzero orbital
angular momentum is 
with allowed values of
the z-component 
Recall, however, that in our derivation of allowed angular
momentum eigenvalues from very general properties of rotation operators, we
found that although for any system the allowed values of m form a ladder with spacing 
we could not rule out half-integral m values. The lowest such case, 
would in fact have
just two allowed m values: 
However, this cannot
be any kind of orbital angular
momentum because the z-component of
the orbital wave function 
has a factor 
and therefore picks up
a factor -1 on rotating through 
meaning is not single-valued,
which doesn’t make sense for a Schrödinger wave function.
Yet the experimental result is clear. Therefore, this must be a new kind of
non-orbital angular momentum. It is
called “spin”, the simple picture being that just as the Earth has orbital
angular momentum in its yearly circle around the sun, and also spin angular
momentum from its daily turning, the electron has an analogous spin. But the analogy has obvious limitations: the
Earth’s spin is after all made up of material orbiting around the axis through
the poles, the electron’s spin cannot similarly be imagined as arising from a
rotating body, since orbital angular
momenta always come in integral multiples of 
Fortunately, this lack of a simple quasi-mechanical picture
underlying electron spin doesn’t prevent us from using the general angular
momentum machinery previously developed, which followed just from analyzing the
effect of spatial rotation on a quantum mechanical system. Recall this led to the spacing 
of the ladder of eigenvalues, and to values of the matrix
elements of angular momentum components Ji
between the eigenkets 
enough information to
construct matrix representations of the rotation operators for a system of
given angular momentum. As an example,
for the orbital angular momentum 
state, we constructed
the 
matrix representation
of an arbitrary rotation operator 
in the space with
orthonormal basis 
(in the 
notation). The spin 
case can be handled in
exactly the same way.
Spinors, Spin Operators, Pauli Matrices
The Hilbert space of angular momentum states for spin
one-half is two dimensional. Various
notations are used: 
or even, more
graphically,
Any state of the spin can be written
and this two-dimensional ket is called a spinor.
Operators on spinors are necessarily
matrices. We shall
follow the usual practice of denoting the angular momentum components Ji by Si for spins.
From our definition of the spinor,
The general formulas for raising and lowering operators
become for 
simply
so
It follows immediately that an appropriate matrix representation for spin one-half is
These three matrices representing the (x, y, z) spin components are called the Pauli spin matrices. They are hermitian, traceless, and obey
This can be written 
The total spin 
Any matrix can be written
in the form
Exercise: prove the above statements, then use your results to show that
(a) 
(b) 
Relating the Spinor to the Spin Direction
But how do 
in 
relate to which way
the spin’s pointing? To find out, let’s
assume that it’s pointing up along
the unit vector 
that is, in the
direction 
In other words, it’s
in the eigenstate of the operator 
having eigenvalue
unity:
Evaluating, 
, using elementary trigonometric identities
where we have multiplied by an overall phase factor 
to make it look
nicer. Note that the spinor is also
correctly normalized.
The physically significant parameter for spin direction is
just the ratio 
Note that any complex number can be represented as
, with 
so for any possible
spinor, there’s a direction along which the spin points up with probability
one.
The Spin Rotation Operator
The rotation operator for rotation through an angle
about an axis in the direction of the unit vector 
is, using 
(Warning: we’re
following standard notation here, but don’t confuse this --angle turned through—with the in writing 
in terms of 
)
Expanding the exponential,
and using 
Writing this in the same D-notation we used for orbital angular momentum earlier (the superscript refers to the j-value)
The rotation operator
is a matrix operating on
the ket space
Explicitly, it is
Notice that this matrix has the form
with
The inverse of this rotation operator is clearly given by
replacing with 
that is,
These matrices have determinant
and so are unitary.
They clearly form a group, since they represent operations of rotation
on a spin. This group is called SU(2), the 2 refers to the
dimensionality, the U to their being
unitary, and the S signifying determinant
+1.
Note that for rotation about the z-axis, 
it is more natural to replace 
and the rotation
operator becomes
In particular, the wave function is multiplied by -1 for a
rotation of 
Since this is true
for any initial wave function, it is clearly also true for rotation through
about any axis.
Exercise: write down the infinitesimal version of
the rotation operator 
for spin ½ , and prove
that 
that is, 
is rotated in the same
way as an ordinary three-vector—note particularly that the change depends on
the angle rotated through, as opposed to the half-angle, so, reassuringly,
there is no -1 for a complete rotation (as there cannot be—the direction of the
spin is a physical observable, and cannot be changed on rotating the measuring
frame through ).
Spin Precession in a Magnetic Field
As a warm up exercise, consider a magnetized classical object spinning about its
center of mass, with angular momentum 
and parallel magnetic
moment
. The constant 
is called the gyromagnetic ratio. Now add a magnetic field 
, say in the z-direction.
This will exert a torque 
, easily solved to find the angular momentum vector precessing about the
magnetic field direction with angular velocity of precession 
.
(Proof: from 
, take 
Of course, dLz/dt
= 0, since is perpendicular to , which is in the z-direction.)
The exact same result
comes from the quantum mechanics of
an electron spin in a magnetic field.
The electron has magnetic dipole moment 
, where 
and g (known as the Landé g-factor) is very close to 2. (This g-factor
terminology is used more widely: the magnetic moment of an atom is written
is the Bohr magneton,
and g depends on the total orbital
angular momentum and total spin of the particular atom.)
The Hamiltonian for the interaction of the electron’s dipole
moment with the magnetic field is 
, hence the time development is
with the propagator
but this is exactly the rotation
operator (as shown earlier) through an angle
about !
For an arbitrary initial spin orientation
the propagator for a magnetic field in the z-direction
so the time-dependent spinor is
The angle between the spin and the
field stays constant, the azimuthal angle around the field increases as 
exactly as in the classical case.
Exercise: for a spin initially pointing along the x-axis, prove that 
Paramagnetic Resonance
We have shown that the spin precession frequency is
independent of the angle of the spin to the field. Consider how all this looks in a frame of
reference which is itself rotating with angular velocity 
about the z-axis.
Let’s call the magnetic field 
, because we’ll soon be adding another one.
In the rotating frame, the observed precession frequency is 
, so there is a different effective field 
in the rotating frame.
Obviously, if the frame rotates exactly at the precession frequency, 
spins pointing in any
direction will remain at rest in that frame—there’s no effective field
at all.
Suppose
now we add a small rotating magnetic field with angular frequency in the x,y
plane, so the total magnetic field
The
effective magnetic field in the frame rotating with the same frequency as the small added
field is
Now, if we tune the angular frequency of the small rotating
field so that it exactly matches the precession frequency in the original
static magnetic field, all the magnetic
moment will see in the rotating frame is the small field in the x-direction! It will therefore precess about the x-direction
at the slow angular speed 
This matching of the
small field rotation frequency with the large field spin precession frequency
is the “resonance”.
If the spins are lined up preferentially in the z-direction
by the static field, and the small resonant oscillating field is switched on
for a time such that 
the spins will be
preferentially in the y-direction in the rotating frame, so in the lab
they will be rotating in the x,y plane, and a coil will pick up
an ac signal from the induced emf.