When you have an ordinary (read: spin zero) particle with a quantum state |ψ>, the corresponding position-space wave function <x|ψ> is an ordinary scalar function of position. How do things work when the particle has spin? In that case its quantum state will live in the tensor product of two Hilbert spaces: an infinite-dimensional (rigged) Hilbert space for position, and presumably a finite dimensional Hilbert space for its spin states. So its state will be of the form |ψ>=|ρ>|χ>, where |ρ> is the positional state and |χ> is the spin state, and the amplitude to find it in a particular position and spin is given by <x|<s||ρ>|χ>. So then the position space wave function should be written as ψ(x)=ρ(x)χ(s), where ρ(x)=<x|ρ> and χ(s)=<s|χ> are scalar-valued wave functions representing the positional and spin states.(adsbygoogle = window.adsbygoogle || []).push({});

But instead I think I've seen people write ψ(x)=ρ(x)χ(x), where χ(x)=<x|χ> is called a "spinor-valued wavefunction" representing the spin state. What's going on here? Is this correct, and what does this mean? I don't know too much about spinors, other than the fact that they can be represented in terms of matrices. Are spinors supposed to be the irreducible representations of the group SU(2) in the positional Hilbert space?

Any help would be greatly appreciated.

Thank You in Advance.

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# How does spin work in the position representation?

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