A Modern Approach to Quantum Mechanics · Problem 3.23

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · spin-three-halves-eigenstates

Problem

For a spin-

\frac{3}{2}

particle the matrix representation of the operator

\hat{S}_x

in the

S_z

basis is given by

\hat{S}_x \to \frac{\hbar}{2}\begin{pmatrix}0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3}\\ 0 & 0 & \sqrt{3} & 0\end{pmatrix}

Pick one of the following states and verify that it is an eigenstate of

\hat{S}_x

with the appropriate eigenvalue:

\left|\tfrac{3}{2}, \tfrac{3}{2}\right\rangle_x \to \frac{1}{2\sqrt{2}}\begin{pmatrix}1\\\sqrt{3}\\\sqrt{3}\\1\end{pmatrix} \qquad \left|\tfrac{3}{2}, \tfrac{1}{2}\right\rangle_x \to \frac{1}{2\sqrt{2}}\begin{pmatrix}\sqrt{3}\\1\\-1\\-\sqrt{3}\end{pmatrix}

\left|\tfrac{3}{2}, -\tfrac{1}{2}\right\rangle_x \to \frac{1}{2\sqrt{2}}\begin{pmatrix}\sqrt{3}\\-1\\-1\\\sqrt{3}\end{pmatrix} \qquad \left|\tfrac{3}{2}, -\tfrac{3}{2}\right\rangle_x \to \frac{1}{2\sqrt{2}}\begin{pmatrix}1\\-\sqrt{3}\\\sqrt{3}\\-1\end{pmatrix}

Do you notice any property of these representations that is at least consistent with the other states being correct?