- #1

Gunthi

- 65

- 1

## Homework Statement

Find the allowed energies for a spin-3/2 particle with the given Hamiltonian:

[tex]\hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z}[/tex]

## The Attempt at a Solution

The final matrix I get is:

\begin{pmatrix}

\frac{3}{2} & 0 & \hbar\sqrt{3} & 0\\

0& \frac{\hbar}{2}-\frac{1}{2} & 0 &\hbar\sqrt{3} \\

\hbar\sqrt{3}& 0 & \frac{\hbar}{2}+\frac{1}{2} & 0\\

0& \hbar\sqrt{3} & 0 & \frac{3}{2}

\end{pmatrix}

My question is: Is there a more quick way to find the eigenvalues of a 4x4 hermitian matrix than going trough the tedious calculation of [itex]det(\hat{H}-\lambda I)=0[/itex]?