# Homework Help: Spin Matrices for Spin 1

1. Nov 27, 2009

### mtmcavoy

1. The problem statement, all variables and given/known data
Construct the spin matrices (Sx,Sy,Sz) for a particle of spin 1. Determine the action of Sz, S+, and S- on each of these states.

2. Relevant equations
s=1 m=-1, 0, 1
Sz=hm |sm>
S+= h [2-m(m+1)]^1/2 |s m+1>
S-= h [2-m(m-1)]^1/2 |s m-1>
*"h" is actually h-bar

3. The attempt at a solution
I've been trying to follow the same method as for spin 1/2, where |1/2 1/2> is a vector (1 0) and |1/2 -1/2> is (0 1), but I don't understand how going between notation for vectors yields these results, and thus I don't know how to get the vector components for the spin 1 case.

2. Nov 27, 2009

### jdwood983

Use the following equations to construct the matrix:

$$\langle m'|S_x|m\rangle=(\delta_{m,m'+1}+\delta_{m+1,m'})\frac{\hbar}{2}\sqrt{s(s+1)-m'm}$$

$$\langle m'|S_y|m\rangle=(\delta_{m,m'+1}-\delta_{m+1,m'})\frac{\hbar}{2i}\sqrt{s(s+1)-m'm}$$

$$\langle m'|S_z|m\rangle=\delta_{mm'}m\hbar$$

with $s=1$, you know that $m=-1,0,1$.

3. Nov 27, 2009

### mtmcavoy

Eh, maybe I'm a little more confused than I thought. Can you be a little more...descriptive, maybe? I'm not seeing how those equations apply.

4. Nov 27, 2009

### jdwood983

The spin matrices--for spin 1--look like this:

$$\hat{S}_x=\left(\begin{array}{ccc} \langle 1|S_x|1\rangle & \langle 1|S_x|0\rangle & \langle 1|S_x|-1\rangle \\ \langle 0|S_x|1\rangle & \langle 0|S_x|0\rangle & \langle 0|S_x|-1\rangle \\ \langle -1|S_x|1\rangle & \langle -1|S_x|0\rangle & \langle -1|S_x|-1\rangle \end{array}\right)$$

so each 1,0 and -1 are the $m$ and $m'$ values. The delta's are the Kronecker delta:

$$\delta_{mn}= \left< \begin{array}{ll} 1 & m=n \\ 0 & m\neq n\end{array}\right.$$

It should just be matching the m's and delta's to get values for each component.

EDIT: For a quick example:

$$\langle 1|S_x|0\rangle=(1+0)\frac{\hbar}{2}\sqrt{1(1+1)-1\cdot0}=\sqrt{2}\frac{\hbar}{2}$$