Why is the spin quantum number of an electron +1\2 or -1\2 in an atom? Is this an experimental value or theoretical value?
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1$\begingroup$ Actually, only the projection of the spin of an electron has a sign and is given in unints of $\hbar$. This fact was initially put forward by experiments but could later be explained using relativistic quantum mechanics. $\endgroup$– PaulNov 18, 2016 at 9:44
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$\begingroup$ An electron has two quantum numbers the spin which is $ 1/2 $ and the projection (also called magnetic or azimuthal) quantum number onto an axis which has values $\pm 1/2$. $\endgroup$– porphyrinNov 18, 2016 at 12:44
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$\begingroup$ Here is the corresponding question (with answer!) on Physics.SE. $\endgroup$– a-cyclohexane-moleculeSep 20, 2018 at 0:10
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1 Answer
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It is both experimental and theoretical. Electrons have angular momentum, and sometimes we want to discuss whether it lines up with another direction (ie typically the direction an electromagnetic field, since electrons are charged their angular momentum makes them magnetism and that interacts with electromagnetic field).
How should we talk about the direction of the angular momentum? We can say "up" and "down" or "forwards" and "backwards", and sometimes we do that. But if we want to use it in calculations, it is easier to give them numbers. So what numbers should we give? They are opposite directions so really it makes sense to use opposite numbers. So what opposite numbers should we use? We can use any, so why not use a gap of "1" between them? So what opposite numbers have a gap of 1 between them? 1/2 and -1/2.
There are some other detailed reasons why 1/2 is a convenient number, but it is in my view fair to say we just want to represent two opposite directions in calculations, and the numbers 1/2 and -1/2 suit this well.
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$\begingroup$ -1. The spin of a particle is fundamental and certainly cannot be reduced to an arbitrary choice of opposite numbers. $\endgroup$ Sep 19, 2018 at 13:07
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1$\begingroup$ @a-cyclohexane-molecule yes, I agree that if we want the absolute value of the spin quantum number to match particle's spin property it is not an arbitrary choice. This is what I was alluding to with my mention of "other detailed reasons". However to understand what this means requires a large amount of theory. So I was trying to focus on what the spin quantum number practically represents: fundamentally it is just a direction. I accept my answer is not ideal and welcome any edits to it or specific suggestions. I think/hope it currenly carries more benefit than no answer. $\endgroup$ Sep 19, 2018 at 21:28
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$\begingroup$ @a-cyclohexane-molecule thank you though for explaining your downvote. $\endgroup$ Sep 19, 2018 at 21:38
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$\begingroup$ @a-cyclohexane-molecule Although I am not certain, I believe the historical arrival at the 1/2 labelling for electron spin followed the reasoning I used in my answer, and the labelling of the particle property was chosen to match this, and not the other way round, as I believe you are implying. It is not explicitly stated (or a perfect source) but I believe this is suggested in en.wikipedia.org/wiki/Spin_quantum_number $\endgroup$ Sep 19, 2018 at 21:52