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.19} 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 ${\cal R}_{x,\alpha}$, ${\cal R}_{y,\beta}$, and ${\cal R}_{z,\gamma}$ 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.19}, the an­gu­lar mo­men­tum around the $z$-​axis is by de­f­i­n­i­tion:

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

(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:

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

or mul­ti­ply­ing out

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

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 $\gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\alpha\beta$ 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,

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

By mul­ti­ply­ing out, the com­mu­ta­tor is found as

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

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

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

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.