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Homework Help: Show that Isospin operators satisfy the SU(2) algebra

  1. Dec 2, 2008 #1
    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 [tex]T_{+}\left\vert p\right\rangle =0[/tex], [tex]T_{-}\left\vert n\right\rangle =0[/tex], [tex]T_{+}\left\vert n\right\rangle =\left\vert p\right\rangle[/tex], [tex]T_{-}\left\vert p\right\rangle =\left\vert n\right\rangle[/tex], [tex]T_{3}\left\vert p\right\rangle =\frac{1}{2}\left\vert p\right\rangle[/tex] and [tex]T_{3}\left\vert n\right\rangle =-\frac{1}{2}\left\vert n\right\rangle[/tex], 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 [tex]\left[ A_{ij},A_{kl}\right] =\delta _{jk}A_{il}-\delta _{il}A_{kj}[/tex] (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:
    [tex]\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}[/tex]
    The same result can be achieved using [tex]\left\vert n\right\rangle[/tex]
    Any suggestions as how to calculate the other commutators? (Involving the [tex]T_{3}[/tex])
  2. jcsd
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