How to Prove the Commutator Relation for Quantum Spin Operators?

[jdstokes]

Homework Statement

Using the orthonormality of $|+\rangle$ and $|-\rangle$, prove

$[S_i,S_j]= i \varepsilon_{ijk}S_k$

where
$S_x = \frac{\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + |$
$S_y = -\frac{i\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + |$
$S_z = \frac{\hbar}{2}|+\rangle \langle + | - | - \rangle \langle - |$

The Attempt at a Solution

Since S_x and S_y commute, their commutator should be zero which contradicts $[S_x,S_y]= i S_z$. What am I missing here?

 

[malawi_glenn]

S_x and S_y does not commute, check again.

If you are unsure, please write the procedure you did to get that S_x and S_y commutes.

 

[jdstokes]

Umm, is this not right?

$[S_x,S_y] = \frac{\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + |\left(-\frac{i\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + | \right) -\left(-\frac{i\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + | \right)\frac{\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + | = 0$?

 

[jdstokes]

Is this a typo or what?

 

[George Jones]

Is this a typo or what?

Must be; the way it's written,

$$S_y = -iS_x.$$

 

[malawi_glenn]

jdstokes You must use paranthesis more carefully!

According to my copy of Sakurai:

$S_y = \frac{i\hbar}{2}(-|+\rangle \langle - | + | - \rangle \langle + |)$

 

[jdstokes]

OMG why do I miss these obvious things!

Thanks for your patience malawi_glenn and George.

 

[jdstokes]

If you guys have a spare moment, would you please have a look at my new post in the quantum physics forum?