How to know where the up and down spin go in the wavefunction?

[QuarkDecay]

TL;DR

Wavefunction with spin, and how to know in which part up and down Spin go

We are given the wave function with spin, but it doesn't say in which Y_l^m each spin X_± goes. So how do I know?

Examples;

(1) Ψ = 1/√3 R₂₁(r) ( Y₁⁰
√2Y₁¹ )

Here we have the up Spin X₊ to Y₁⁰ and the X_- to Y₁¹

I notice the X_- went to the higher m (m=1). However, in this other example it's the opposite;

(2) Ψ = R₂₁ ( √(1/6) Y₁⁰ X_- - √(5/6) Y₁^-1X₊ )

 

[dextercioby]

Spin and orbital angular momentum are not related, so there is quite a freedom to choose which combination of the tensor product space makes sense to use. In your not rigorously chosen words, spin up and down are free to go everywhere.