# Show that Isospin operators satisfy the SU(2) algebra

1. Dec 2, 2008

### diegzumillo

Hi there! Doesn't seem like a hard problem..

1. The problem statement, all variables and given/known data

Show that the 3 isospin operators, defined by $$T_{+}\left\vert p\right\rangle =0$$, $$T_{-}\left\vert n\right\rangle =0$$, $$T_{+}\left\vert n\right\rangle =\left\vert p\right\rangle$$, $$T_{-}\left\vert p\right\rangle =\left\vert n\right\rangle$$, $$T_{3}\left\vert p\right\rangle =\frac{1}{2}\left\vert p\right\rangle$$ and $$T_{3}\left\vert n\right\rangle =-\frac{1}{2}\left\vert n\right\rangle$$, satisfy the SU(2) algebra.

2. Relevant equations
I belive that's all we need.

3. The attempt at a solution

First of all, I'm havig a little difficulty using the general form of the SU(2) algebra $$\left[ A_{ij},A_{kl}\right] =\delta _{jk}A_{il}-\delta _{il}A_{kj}$$ (found this in wikipedia) But at least I know I have to calculate the commutators between these 3 operators!
I managed to calculate the first one:
$$\left[ T_{+},T_{-}\right] \left\vert p\right\rangle =\left(T_{+}T_{-}-T_{-}T_{+}\right) \left\vert p\right\rangle =\left\vert p\right\rangle =\frac{1}{2}T_{3}$$
The same result can be achieved using $$\left\vert n\right\rangle$$
Any suggestions as how to calculate the other commutators? (Involving the $$T_{3}$$)