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

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 Ylm each spin X± goes. So how do I know?


(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+ )


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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.

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