12.2 The fun­da­men­tal com­mu­ta­tion re­la­tions

An­a­lyz­ing nonor­bital an­gu­lar mo­men­tum is a chal­lenge. How can you say any­thing sen­si­ble about an­gu­lar mo­men­tum, the dy­namic mo­tion of masses around a given point, with­out a mass mov­ing around a point? For, while a par­ti­cle like an elec­tron has spin an­gu­lar mo­men­tum, try­ing to ex­plain it as an­gu­lar mo­tion of the elec­tron about some in­ter­nal axis leads to gross con­tra­dic­tions such as the elec­tron ex­ceed­ing the speed of light [25, p. 172]. Spin is def­i­nitely part of the law of con­ser­va­tion of an­gu­lar mo­men­tum, but it does not seem to be as­so­ci­ated with any fa­mil­iar idea of some mass mov­ing around some axis as far as is known.

There goes the New­ton­ian anal­ogy, then. Some­thing else than clas­si­cal physics is needed to an­a­lyze spin.

Now, the com­plex dis­cov­er­ies of math­e­mat­ics are rou­tinely de­duced from ap­par­ently self-ev­i­dent sim­ple ax­ioms, such as that a straight line will cross each of a pair of par­al­lel lines un­der the same an­gle. Ac­tu­ally, such ax­ioms are not as ob­vi­ous as they seem, and math­e­mati­cians have de­duced very dif­fer­ent an­swers from chang­ing the ax­ioms into dif­fer­ent ones. Such an­swers may be just as good or bet­ter than oth­ers de­pend­ing on cir­cum­stances, and you can in­vent imag­i­nary uni­verses in which they are the norm.

Physics has no such lat­i­tude to in­vent its own uni­verses; its mis­sion is to de­scribe ours as well as it can. But the idea of math­e­mat­ics is still a good one: try to guess the sim­plest pos­si­ble ba­sic law that na­ture re­ally seems to obey, and then re­con­struct as much of the com­plex­ity of na­ture from it as you can. The more you can de­duce from the law, the more ways you have to check it against a va­ri­ety of facts, and the more con­fi­dent you can be­come in it.

Physi­cist have found that the needed equa­tions for an­gu­lar mo­men­tum are given by the fol­low­ing “fun­da­men­tal com­mu­ta­tion re­la­tions:”

$$[{\widehat J}_x,{\widehat J}_y]= {\rm i}\hbar{\widehat J}_z \... ... [{\widehat J}_z,{\widehat J}_x] = {\rm i}\hbar{\widehat J}_y$$ (12.1)

They can be de­rived for or­bital an­gu­lar mo­men­tum (see chap­ter 4.5.4), but must be pos­tu­lated to also ap­ply to spin an­gu­lar mo­men­tum {N.26}.

At first glance, these com­mu­ta­tion re­la­tions do not look like a promis­ing start­ing point for much analy­sis. All they say on their face is that the an­gu­lar mo­men­tum op­er­a­tors ${\widehat J}_x$, ${\widehat J}_y$, and ${\widehat J}_z$ do not com­mute, so that they can­not have a full set of eigen­states in com­mon. That is hardly im­pres­sive.

But if you read the fol­low­ing sec­tions, you will be as­ton­ished by what knowl­edge can be teased out of them. For starters, one thing that im­me­di­ately fol­lows is that the only eigen­states that ${\widehat J}_x$, ${\widehat J}_y$, and ${\widehat J}_z$ have in com­mon are states ${\left\vert\:0\right\rangle}$ of no an­gu­lar mo­men­tum at all {D.63}. No other com­mon eigen­states ex­ist.

One as­sump­tion will be im­plicit in the use of the fun­da­men­tal com­mu­ta­tion re­la­tions, namely that they can be taken at face value. It is cer­tainly pos­si­ble to imag­ine that say ${\widehat J}_x$ would turn an eigen­func­tion of say ${\widehat J}_z$ into some sin­gu­lar ob­ject for which an­gu­lar mo­men­tum would be ill-de­fined. That would of course make ap­pli­ca­tion of the fun­da­men­tal com­mu­ta­tion re­la­tions im­proper. It will be as­sumed that the op­er­a­tors are free of such patho­log­i­cal nas­ti­ness.