Intrinsic Angular Momentum

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.