N.26 Physics of the fundamental commutators

The fundamental commutation relations look much like a mathematical axiom. Surely, there should be some other reasons for physicists to believe that they apply to nature, beyond that they seem to produce the right answers?

Addendum {A.17} explained that the angular momentum operators correspond to small rotations of the axis system through space. So, the commutator $[{\widehat J}_x,{\widehat J}_y]$ really corresponds to the difference between a small rotation around the $y$-axis followed by a small rotation around the $x$ axis, versus a small rotation around the $x$-axis followed by a small rotation around the $y$ axis. As shown below, in our normal world this difference is equivalent to the effect of a small rotation about the $z$-axis.

So, the fundamental commutator relations do have physical meaning; they say that this basic relationship between rotations around different axes continues to apply in the presence of spin.

This idea can be written out more precisely by using the symbols $R_{x,\alpha}$, $R_{y,\beta}$, and $R_{z,\gamma}$ for, respectively, a rotation around the $x$-axis over an angle $\alpha$, around the $y$-axis over an angle $\beta$, and the $z$-axis over an angle $\gamma$. Then following {A.17}, the angular momentum around the $z$ axis is by definition:

$${\widehat J}_z \approx \frac{\hbar}{{\rm i}} \frac{R_{z,\gamma}-I}{\gamma}$$

(To get this true exactly, you have to take the limit $\gamma\to0$. But to keep things more physical, taking the mathematical limit will be delayed to the end. The above expression can be made arbitrarily accurate by just taking $\gamma$ small enough.)

Of course, the $x$ and $y$ components of angular momentum can be written similarly. So their commutator can be written as:

$$[{\widehat J}_x,{\widehat J}_y]\equiv {\widehat J}_x {\wideha... ...R_{y,\beta}-I}{\beta} \frac{R_{x,\alpha}-I}{\alpha} \right)$$

or multiplying out

$$[{\widehat J}_x,{\widehat J}_y]\approx \frac{\hbar^2}{{\rm i}... ...,\alpha} R_{y,\beta} - R_{y,\beta} R_{x,\alpha}}{\alpha\beta}$$

The final expression is what was referred to above. Suppose you do a rotation of your axis system around the $y$ axis over a small angle $\beta$ followed by a rotation around the $x$-axis around a small angle $\alpha$. Then you will change the position coordinates of every point slightly. And so you will if you do the same two rotations in the opposite order. Now if you look at the difference between these two results, it is described by the numerator in the final ratio above.

All those small rotations are of course a complicated business. It turns out that in our normal world you can get the same differences in position in a much simpler way: simply rotate the axis system around a small angle $\gamma=-\alpha\beta$ around the $z$-axis. The change produced by that is the numerator in the expression for the angular momentum in the $z$-direction given above. If the two numerators are the same for small $\alpha$ and $\beta$, then the fundamental commutation relation follows. At least in our normal world. So if physicists extend the fundamental commutation relations to spin, they are merely generalizing a normal property of rotations.

To show that the two numerators are the indeed the same for small angles requires a little linear algebra. You may want to take the remainder of this section for granted if you never had a course in it.

First, in linear algebra, the effects of rotations on position coordinates are described by matrices. In particular,

$$R_{x,\alpha} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ ... ... & 0 \\ \sin\beta & 0 & \cos\beta \\ \end{array}\right)$$

By multiplying out, the commutator is found as

$$[{\widehat J}_x,{\widehat J}_y]\approx \frac{\hbar^2}{{\rm ... ...lpha) & -\sin\alpha(1-\cos\beta) & 0 \\ \end{array}\right)$$

Similarly, the angular momentum around the $z$-axis is

$${\widehat J}_z \approx \frac{\hbar}{{\rm i}\gamma} \left( ... ... & \cos\gamma -1 & 0 \\ 0 & 0 & 0 \\ \end{array}\right)$$

If you take the limit that the angles become zero in both expressions, using either l’Hôpital or Taylor series expansions, you get the fundamental commutation relationship.

And of course, it does not make a difference which of your three axes you take to be the $z$-axis. So you get a total of three of these relationships.