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

In summary, the wave function provided does not specify which Ylm each spin X± corresponds to. However, in one example, X- is paired with a higher m value (m=1), while in another example it is paired with a lower m value (m=-1). This is due to the fact that spin and orbital angular momentum are not related, allowing for freedom in choosing the appropriate combination in the tensor product space. In simpler terms, spin up and down can occupy any position without restriction.
  • #1
QuarkDecay
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TL;DR Summary
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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  • #2
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.
 

1. What is the wavefunction and why is it important in determining spin?

The wavefunction is a mathematical function that describes the quantum state of a particle. It is important in determining spin because it contains information about the particle's energy, position, and momentum, including its spin quantum number.

2. How is spin represented in the wavefunction?

Spin is represented in the wavefunction as a quantum number, either +1/2 or -1/2, which corresponds to the two possible spin states of a particle. This quantum number can be used to calculate the probability of the particle having a particular spin orientation.

3. How can we determine where the up and down spin go in the wavefunction?

The up and down spin states are represented by different mathematical functions within the wavefunction. By solving the Schrödinger equation, we can determine the probability amplitudes for each spin state at a particular position in space.

4. Can the up and down spin states change in the wavefunction?

Yes, the up and down spin states can change in the wavefunction due to interactions with other particles or external forces. This is known as spin-flipping and can result in a change in the particle's spin quantum number.

5. How does the Pauli exclusion principle relate to determining spin in the wavefunction?

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. This means that in a multi-particle system, each particle must have a unique combination of quantum numbers, including spin, in the wavefunction.

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