Changing the basis of Pauli spin matrices

[Phruizler]

Homework Statement

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

Homework 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.

The Attempt at a Solution

In the S_x basis, the S_x operator is just

<br /> \frac{\hbar}{2}<br /> \begin{pmatrix}<br /> 1 &amp; 0\\ <br /> 0 &amp; -1<br /> \end{pmatrix}<br />

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

<br /> S_z=<br /> \frac{\hbar}{2}<br /> \begin{pmatrix}<br /> 0 &amp; 1\\ <br /> 1 &amp; 0<br /> \end{pmatrix}<br />

in the S_x 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 S_z and converting the kets to the S_x 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!

 

[Phruizler]

After some thought, I've considered that maybe I can just use the vector representation of S_z kets in the S_x basis. That is,

<br /> |+&gt;_x<br /> \doteq<br /> \begin{pmatrix}<br /> 1\\<br /> 0<br /> \end{pmatrix}<br />

in the S_x 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 S_z matrix. This will indeed give me the matrix which I asked about above (namely, the same as the S_x the S_z basis). I'm going to assume this is correct unless anyone tells me otherwise!

 

[Oxvillian]

Phruizler - to find the required matrix, you just have to find the four matrix elements involving the eigenvectors of S_x.

That is, you need to find \langle x,\pm \lvert S_z \lvert x,\pm \rangle.

 

[Phruizler]

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!