The following description is bogus. That it, this is not ``really'' what intrinsic angular momentum is all about; but it is possible to understand it in ``common sense'' terms, so we can use it as a mnemonic technique. Many concepts are introduced this sort of ``cheating'' until students get comfortable enough with them to define them rigourously. (The truth about spin, like much of QM, can never be made to seem sensible; it can only be gotten used to!)
Imagine a big fuzzy ball of mass spinning about an axis.
While you're at it, imagine some electric charge sprinkled in,
a certain amount of charge for every little bit of mass.
(If you like, you can think of a cloud of particles,
each of which has the same charge-to-mass ratio,
all orbiting about a common axis.)
Each little mass element contributes a bit of angular momentum
and a proportional bit of magnetic moment, so that
(summed over all the mass elements) and, as for a single
particle,
(constant)
.
If the charge-to-mass ratio happens to be the same as for
an electron, then (constant)
,
the Bohr magneton.
Now imagine that, like a figure skater pulling in her/his arms
to spin faster, the little bits of charge and mass collapse
together, making r smaller everywhere. To conserve
angular momentum (which is always conserved!)
the momentum p has to get bigger - the bits must spin faster.
The relationship between L and
is such that
also remains constant as this happens.
Eventually the constituents can shrink down to a point spinning infinitely fast. Obviously we get into a bit of trouble here with both relativity and quantum mechanics; nevertheless, this is (sort of) how we think (privately) of an electron: although we have never been able to find any evidence for ``bits'' within an electron, we are able to rationalize its possession of an irreducible, intrinsic angular momentum (or ``spin'') in this way.
Such intrinsic angular momentum is a property
of the particle itself as well as a dynamical variable
that behaves just like orbital angular momentum.
It is given a special label (
instead of
)
just to emphasize its difference.
Like
, it is
quantized - i.e. it only
comes integer multiples of a fundamental quantum of
intrinsic angular momentum - but (here comes the weird part!)
that quantum can be either
,
as for
,
or
!
In the following, s is the ``spin quantum number''
analogous to the ``orbital quantum number''
such that the spin angular momentum
has a magnitude
\
and a z component
where
is the chosen spin quantization axis.
The magnetic quantum number for spin
has only two possible values, spin ``up''
(
) and spin ``down''
(
).
This is the explanation of the Stern-Gerlach result
for silver atoms: with no orbital angular momentum
at all, the Ag atoms have a single ``extra'' electron
whose spin determines their overall angular momentum
and magnetic moment.