# Sping Matrices and Commutation Relations

1. Dec 1, 2007

### Rahmuss

1. The problem statement, all variables and given/known data
Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.

2. Relevant equations
Eq. 4.147a --> $$S_{x} = \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$

Eq. 4.147b --> $$S_{y} = \frac{\hbar}{2}\begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}$$

Eq. 4.145 --> $$S_{z} = \frac{\hbar}{2}\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$

$$\left[ S_{x}, S_{y}\right] = i\hbar S_{z}$$

$$\left[ S_{y}, S_{z}\right] = i\hbar S_{x}$$

$$\left[ S_{z}, S_{x}\right] = i\hbar S_{y}$$

3. The attempt at a solution

Well, from Eq. 4.147 I have $$S_{x} = \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$ and also $$S_{y} = \frac{\hbar}{2}\begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}$$

So $$\left[ S_{x}, S_{y} \right] = i\hbar S_{z} \Rightarrow$$

$$\left[ \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}, \frac{\hbar}{2}\begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \right] \Rightarrow$$

$$\frac{\hbar^{2}}{4}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} - \frac{\hbar^{2}}{4}\begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \Rightarrow$$

$$\frac{\hbar^{2}}{4}\begin{pmatrix}i & 0 \\ 0 & -i \end{pmatrix} - \frac{\hbar^{2}}{4}\begin{pmatrix}-i & 0 \\ 0 & i \end{pmatrix} = \frac{\hbar^{2}}{4}\begin{pmatrix}2i & 0 \\ 0 & -2i \end{pmatrix}$$

It doesn't quite seem right to me because the answer shows:

$$\left[ S_{x}, S_{y}\right] = i\hbar S_{z}$$

But I can't see how I'd get that from where I'm going.

Last edited: Dec 1, 2007
2. Dec 1, 2007