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I was reading through Feynman's Lectures Vol. III when I stumbled upon Eqs. 5.38 which describe the transformation amplitudes for a Stern-Gerlach experiment rotated about the y-Axis for an arbitrary angle $\alpha$.

I already know the basics of angular momentum theory, so I know that one can represent the generators (commonly the $L_i$ operators) and then through exponentiating one finds a representation for a rotation about the $i$-axis. That gives us exactly the matrix with the transition amplitudes given in Eq. 5.38.

Of course this all requires some tedious algebra but it doesn't contain any particularly difficult computations, maybe computing the exponential for higher spin representation is harder but certainly doable (I guess..?).

My question is the following:

Is there another completely different method to find these amplitudes, not involving the generators of rotations? Maybe something analoguous to what Feynman shows in Chapter 6 can be adapted for spin-1 (and higher) systems as well?

(For my references see http://www.feynmanlectures.caltech.edu/III_05.html)

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  • $\begingroup$ Yes, you want to learn about Wigner's rotation matrices which Feynman is illustrating with his "seat-of-the-pants" language. They, the Ds and ds generalize to all representations beautifully. Your idea of exponentiating spin matrices for large representations to find finite rotations is not terribly practical, but it is doable. Still, the mainstream solution to the general problem, to be found in all non-introductory QM texts, is the d matrices. $\endgroup$
    – Cosmas Zachos
    Jan 16, 2018 at 15:33
  • $\begingroup$ I think I saw them also in Sakurai's Modern Quantum Mechanics, I will surely look into them! Many thanks for the comment. $\endgroup$
    – Bonsay
    Jan 16, 2018 at 16:04

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