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
1,
l
=
with allowed values of the
z
-component
,1
,
0
,
mm
=−
=
1
.
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
,
=
1/2,
l
=
1/2, 1/2.
m
ψ
has a factor
,
i
e
ϕ
±
and therefore picks up a factor -1 on rotating through
2,
π
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
J
i
between the eigenkets
,:
j m
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
1
jl
=
=
state, we constructed
the
matrix representation of an arbitrary rotation operator
33
×
iJ
e
θ
⋅
−
G G
=
in the space with
orthonormal basis
1,1 , 1, 0
1
−
, 1,
(in the
,
lm
notation).
The spin
1/2
js
=
=
case can be
handled in exactly the same way.