# Eigenfunctions of spin operator

1. Nov 25, 2012

### function22

What are the eigenfunctions of the spin operators? I know the spin operators are given by Pauli matricies (https://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin), and I know what the eigenvalues are (and the eigenvectors), but I have no idea what the eigenfunctions of the spin operator are. I searched google but I could not find a derivation. Does anyone know how to find the eigenfunctions of the spin operator?

2. Nov 25, 2012

### Bill_K

Do you mean spin spherical harmonics? These are the functions that represent finite rotations in spin space, half-integer analogs to the Ylm's. They're briefly mentioned in Wikipedia, covered more in this article, and especially in books on angular momentum, like the monograph by Edmonds.

3. Nov 25, 2012

### function22

Like the Pauli matricies Sx, Sy, Sz, I know the eigenvalues/eigenvectors, but what are the eigenfunctions? I have no idea what they could be.

4. Nov 25, 2012

### dextercioby

The spin space is an abstract finite dimensional (essentially C^(2s+1)) vector space. There are no <wavefunctions>, just normal quadratic matrices and matrices with one column.

5. Nov 25, 2012

### Bill_K

No, NOT like the Pauli matrices. The Pauli matrices are the generators. I'm referring to the matrix functions that represent finite rotations. They are functions in the sense that they are functions of the three Euler angles. And they are eigenfunctions of S and Sz.