Associated Legendre differential equation involved in solving spin function?

[bearcharge]

Amazed by the closeness of equations for orbital angular momentum L and spin angular momentum S, I can't help asking is associated Legendre differential equation involved in solving spin function? I only heard that spin naturally comes from treatment of quantum mechanics with relativity theory. The fact that the solution for spin is so similar to that for orbital angular momentum really intrigues me. I'm eager to be educated. Thanks.

 

[dextercioby]

The Wigner d functions appearing in spin theory are known to be related to the associated Legendre functions (which appear in theory of the orbital angular momentum) by the mathematical expressions from Edmond's 1957 book <Angular Momentum in Quantum Mechanics> which are found on page 59, formulas 4.1.24 and 4.1.25.

In an abstract fashion, the orbital ang. momentum and spin ang. momentum each have 3 generators obeying the same su(2) Lie algebra. The particularities which distinguish them completely are that they act on different Hilbert spaces due to their unrelated origins and the eigenvalues of L_z as opposed to S_z cannot be semiinteger.

[...] I only heard that spin naturally comes from treatment of quantum mechanics with relativity theory [...]

Correct. Either Galilean relativity or special relativity, it doesn't matter.

 

[bearcharge]

Great answer. Thanks! As someone who met group theory just one year ago, I think it would take some time for me to really understand what you said. But anyway, thanks for your answer. I'll keep your answer for later revisit.