# Eigenvectors, spinors, states, values

1. Mar 8, 2012

### 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.

2. Mar 9, 2012

### 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.

3. Mar 9, 2012

### SoggyBottoms

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

4. Mar 9, 2012

### 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

5. Mar 9, 2012

### SoggyBottoms

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

6. Mar 9, 2012

### 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.