Eigenvectors, spinors, states, values

[SoggyBottoms]

For spin-1/2, the eigenvalues of $S_x, S_y$ and $S_z$ are always $\pm \frac{\hbar}{2}$ for spin-up and spin-down, correct?

What is the difference between eigenvectors, eigenstates and eigenspinors? I believe eigenstates = eigenspinors and eigenvectors are something else? I'm just getting confused, because my teachers use the letter $\chi$ for everything.

 

[tiny-tim]

Hi SoggyBottoms!

I think "eigenvector" is the correct technical term for integral spin, and "eigenspinor" for half-integral spin.

But i wouldn't worry about it. ​

 

[SoggyBottoms]

So all three terms are actually the same? At least as far as my introductory QM is concerned? Eigenvector = eigenspinor = eigenstate?

 

[tiny-tim]

well, not exactly the same, since a spinor isn't a vector, and a vector isn't a spinor

but your introductory QM probably doesn't go into the difference between vectors and spinors in detail anyway

 

[SoggyBottoms]

It doesn't indeed, but they use all the terms interchangeably it seems, so it's confusing. Thanks.

 

[kith]

State is short for state vector, so eigenstate and eigenvector are the same. These terms are general and apply to every quantum system.

Spinors are a specific way to express spin state vectors. For spin 1/2 particles, they have two or four components (Pauli spinor vs. Dirac spinor).

State vectors are written as |ψ>. If you want to write them in a specific base, you use column vector notation (<a₁|ψ> <a₂|ψ> ... )^T. The same notation is often used for spinors, although rectangular brackets are arguably better to make it clear you are talking about spinors: χ=[c₁ c₂]^T.