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In a homework exercise i am asked why is the spin wave function (for the H-Atom) expressed as a product between the wave function and the spin function (spin function has the spin vector operator as a "variable" ).

So i used probability in order to explain it :

There is a probability to find an atom at a certain place in space, and there is also a probability that this atom has a total spin (+1/2 or -1/2) so the combined probability is a multiplication of both of these probabilities (statistics). Probability of finding the atom at certain place in space is related to the wave function. Probability of having one of the 2 values of spin is related to the Spin wave function. So the combined (multiplied ) probability must be related to the combined (multiplied) wave function. Which we can call the spin wave function.

Is there a better explanation as to why? One that has to do with the respective Hilbert spaces?

Thank you

Dari
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  • I think this answer is relevant here. You should note that the 'product' in your question is really a tensor product. (I think your explanation is okay and the answer I'm linking to formalises this somewhat.) – jacob1729 Jun 16 '20 at 18:17
  • I also thought that the product is a tensor product. The thing is that even tho i am in the 4th semester no one has explained to us tensors or product of tensors, their meaning etc etc. The exercise says explain why its a product. Anyways thank you for the link – Dari Jun 16 '20 at 18:23

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