In the quantum mechanics book I have, they first cover the mechanics of a generic particle (say an electron), without ever considering the spin. They encode the state of the electron as a function F in L2, where the F^*F is the probability density for the location of the electron. Later, they discuss spin and immediately start talking about the state of an electron as a 2 dimensional vector (a linear combination of the "spin up" and "spin down" vectors about some axis). They never mention it, but obviously this is not really the state of the particle as two numbers cannot encode all the information stored in F. Conversely, F doesn't seem to encode any information about the spin.(adsbygoogle = window.adsbygoogle || []).push({});

So I guess the state of a particle is some combination of the above two pieces of information, along with possibly some additional information? If S1 is the space in which F lives and S2 is the 2D spin space, how do we combine them to form a larger space representing the entire state. Tensor product, Cartesian product, ...? Is there any other information we need to include in the state (along with the spin and position density)? If not, how do we know there's nothing else to include? Some rationale that says the behavior of a particle is determined only by these pieces of information?

Thanks

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# State of an electron, including spin

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