State of an electron, including spin

[msumm21]

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.

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

 

[olgranpappy]

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.

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

Basically you need two functions $F_{\rm down}$ and $F_{\rm up}$ instead of just one function $F$ for a spinless particle.

Often one can speak of the spin and the space parts of the wavefunction separately. This is why we bother studying the spin part "by itself". But, of course, as you say, there is always the space part of the wavefunction to deal with as well. For pedagogical reasons some texts treat the spin-only problem first.

But press on in your book (or switch to a different book--I like Messiah's text) and I'm sure you will see how to include the spin and the spatial parts of the wavefunctions at once and learn what sort of space this wavefunction lives in. Cheers.

 

[tiny-tim]

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, ...?

Hi msumm21!

the wave-function for an scalar spin-0 particle is Ae^iψ(t,x), where A is an ordinary number;

the wave-function for an electron is Se^iψ(t,x) where S is a spinor;

(in other words, the amplitude of an electron is really a spinor, but elementary books tend not to tell you that )

it's just a direct (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?

No … and we know that from experiments … if experiments showed there was something else, we'd add it in (and we probably wouldn't call it an electron).

Alternative answer … yes … there's things like lepton number.