Eigenvalues of 4x4 Hermitian Matrix (Observable)

Gunthi
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Homework Statement


Find the allowed energies for a spin-3/2 particle with the given Hamiltonian:
\hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z}

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 det(\hat{H}-\lambda I)=0?
 
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Hi Gunthi! :smile:
Gunthi said:
Is there a more quick way to find the eigenvalues of a 4x4 hermitian matrix than going trough the tedious calculation of det(\hat{H}-\lambda I)=0?

If you swap the second and third rows, it becomes two 2x2 matrices. :wink:
 
tiny-tim said:
Hi Gunthi! :smile:


If you swap the second and third rows, it becomes two 2x2 matrices. :wink:

Hi tiny-tim! :smile:

I would also have to switch the 2nd and 3rd columns right? Then I would just calculate the eigenvalues of the 2x2 matrices separately?

I've been searching for properties of block matrices that could justify this, but to no avail. Is there a theorem that demonstrates this property? Or could you explain how this works?

I guess I'm rustier than I thought at my algebra :redface:
 
Hi Gunthi! :smile:
Gunthi said:
… I've been searching for properties of block matrices that could justify this, but to no avail. Is there a theorem that demonstrates this property? Or could you explain how this works?

We're only re-arranging

instead of the basis x y z t (or whatever), we're using x z y t :wink:

To put it another way, can't you immediately see, just by looking at it, that the matrix is in two parts that operate completely separately? :smile:
 
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