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 [24, 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 $\big\vert\:0\big\rangle$ of no angular momentum at all {D.64}. 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.