D.69 More awk­ward­ness about spin

How about that? A note on a note.

The pre­vi­ous note brought up the ques­tion: why can you only change the spin states you find in a given di­rec­tion by a fac­tor $\vphantom{0}\raisebox{1.5pt}{$-$}$1 by ro­tat­ing your point of view? Why not by ${\rm i}$, say?

With a bit of knowl­edge of lin­ear al­ge­bra and some thought, you can see that this ques­tion is re­ally: how can you change the spin states if you per­form an ar­bi­trary num­ber of co­or­di­nate sys­tem ro­ta­tions that end up in the same ori­en­ta­tion as they started?

One way to an­swer this is to show that the ef­fect of any two ro­ta­tions of the co­or­di­nate sys­tem can be achieved by a sin­gle ro­ta­tion over a suit­ably cho­sen net an­gle around a suit­ably cho­sen net axis. (Math­e­mati­cians call this show­ing the “group” na­ture of the ro­ta­tions.) Ap­plied re­peat­edly, any set of ro­ta­tions of the start­ing axis sys­tem back to where it was be­comes a sin­gle ro­ta­tion around a sin­gle axis, and then it is easy to check that at most a change of sign is pos­si­ble.

(To show that any two ro­ta­tions are equiv­a­lent to one, just crunch out the mul­ti­pli­ca­tion of two ro­ta­tions, which shows that it takes the al­ge­braic form of a sin­gle ro­ta­tion, though with a unit vec­tor ${\vec n}$ not im­me­di­ately ev­i­dent to be of length one. By not­ing that the de­ter­mi­nant of the ro­ta­tion ma­trix must be one, it fol­lows that the length is in fact one.)