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Sakurai Chapter 1 Question 8

  1. Feb 17, 2008 #1
    1. The problem statement, all variables and given/known data

    Using the orthonormality of [itex]|+\rangle[/itex] and [itex]|-\rangle[/itex], prove

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

    where
    [itex]S_x = \frac{\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + |[/itex]
    [itex]S_y = -\frac{i\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + |[/itex]
    [itex]S_z = \frac{\hbar}{2}|+\rangle \langle + | - | - \rangle \langle - |[/itex]


    3. The attempt at a solution

    Since S_x and S_y commute, their commutator should be zero which contradicts [itex][S_x,S_y]= i S_z[/itex]. What am I missing here?
     
  2. jcsd
  3. Feb 17, 2008 #2

    malawi_glenn

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    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.
     
  4. Feb 17, 2008 #3
    Umm, is this not right?

    [itex][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[/itex]?
     
  5. Feb 17, 2008 #4
    Is this a typo or what?
     
  6. Feb 17, 2008 #5

    George Jones

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    Must be; the way it's written,

    [tex]S_y = -iS_x.[/tex]
     
  7. Feb 17, 2008 #6

    malawi_glenn

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    jdstokes You must use paranthesis more carefully!

    According to my copy of Sakurai:

    [itex]S_y = \frac{i\hbar}{2}(-|+\rangle \langle - | + | - \rangle \langle + |)[/itex]
     
  8. Feb 17, 2008 #7
    OMG why do I miss these obvious things!

    Thanks for your patience malawi_glenn and George.
     
  9. Feb 17, 2008 #8
    If you guys have a spare moment, would you please have a look at my new post in the quantum physics forum?
     
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