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Changing the basis of Pauli spin matrices

  1. Jul 21, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the matrix representation of [itex]S_z[/itex] in the [itex]S_x[/itex] basis for spin [itex]1/2[/itex].

    2. Relevant equations

    I have the Pauli matrices, and I also have the respective kets derived in each basis. There aren't really any relevant equations, other than the eigenvalue equations for the operators.

    3. The attempt at a solution

    In the [itex]S_x[/itex] basis, the [itex]S_x[/itex] operator is just

    [tex]
    \frac{\hbar}{2}
    \begin{pmatrix}
    1 & 0\\
    0 & -1
    \end{pmatrix}
    [/tex]

    So isn't the [itex]S_z[/itex] operator in the [itex]S_x[/itex] basis just equal to the [itex]S_x[/itex] operator in the [itex]S_z[/itex] basis? That is,

    [tex]
    S_z=
    \frac{\hbar}{2}
    \begin{pmatrix}
    0 & 1\\
    1 & 0
    \end{pmatrix}
    [/tex]

    in the [itex]S_x[/itex] basis? If so, I don't really understand how to show this in any meaningful way... I've also tried solving the eigenvalue equations for [itex]S_z[/itex] and converting the kets to the [itex]S_x[/itex] basis but that just lead me in circles. I can see that the question is a very basic one but something is just not letting me get a satisfactory answer. Any help would be much appreciated, thanks!
     
  2. jcsd
  3. Jul 21, 2014 #2
    After some thought, I've considered that maybe I can just use the vector representation of [itex]S_z[/itex] kets in the [itex]S_x[/itex] basis. That is,

    [tex]
    |+>_x
    \doteq
    \begin{pmatrix}
    1\\
    0
    \end{pmatrix}
    [/tex]

    in the [itex]S_x[/itex] basis, and the same for the spin down ket, so I can just plug these vector representations into the eigenvalue equation and solve for the [itex]S_z[/itex] matrix. This will indeed give me the matrix which I asked about above (namely, the same as the [itex]S_x[/itex] the [itex]S_z[/itex] basis). I'm going to assume this is correct unless anyone tells me otherwise!
     
  4. Jul 27, 2014 #3
    Phruizler - to find the required matrix, you just have to find the four matrix elements involving the eigenvectors of [itex]S_x[/itex].

    That is, you need to find [itex]\langle x,\pm \lvert S_z \lvert x,\pm \rangle[/itex].
     
  5. Jul 29, 2014 #4
    Thanks! This yields the answer I got doing it the above way but is much more satisfying. I didn't even think of solving the matrix elements individually like that for some reason. Quantum mechanical problem solving simply doesn't adhere to the same intuition as other problems in physics. I can tell it will be a while before I will have useful insight into even some of the easier problems!

    Thanks again!
     
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