How Do You Construct and Analyze Spin Matrices for a Spin 1 Particle?

[mtmcavoy]

Homework Statement

Construct the spin matrices (S_x,S_y,S_z) for a particle of spin 1. Determine the action of S_z, S₊, and S_- on each of these states.

Homework Equations

s=1 m=-1, 0, 1
S_z=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

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.

 

[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 $m=-1,0,1$.

 

[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.

 

[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:

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