Show that Isospin operators satisfy the SU(2) algebra

[diegzumillo]

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

Homework Statement

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.

Homework Equations

I believe that's all we need.

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 $$T_{3}$$)

 

[Jhero]

Hi there! It's great that you are tackling this problem. The SU(2) algebra can be a bit tricky to work with, but I'm sure you'll get the hang of it. To calculate the other commutators, you can use the same approach you used for the first one. Remember that the commutator is defined as \left[ A,B\right] =AB-BA, so you can use this formula to calculate the remaining commutators involving T_3. Good luck!