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

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Determining the allocation of spin states in a wavefunction can be ambiguous, as the wavefunction does not specify which spherical harmonic corresponds to each spin state. In the first example, the up spin X+ is associated with Y10, while the down spin X- corresponds to Y11, indicating a trend where the down spin aligns with a higher m value. Conversely, in the second example, the down spin X- is linked to Y10, and the up spin X+ is associated with Y1-1, illustrating that the relationship between spin and orbital angular momentum is flexible. This flexibility allows for various combinations of tensor product spaces, meaning spins can be assigned to different states without strict rules. Ultimately, the choice of spin and orbital combinations is largely a matter of convenience and interpretation.
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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?

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+ )
 
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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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