Eigenvalues of 4x4 Hermitian Matrix (Observable)

[Gunthi]

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 $det(\hat{H}-\lambda I)=0$?

 

[tiny-tim]

Hi Gunthi!

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.

 

[Gunthi]

Hi Gunthi!


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

Hi tiny-tim!

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

 

[tiny-tim]

Hi Gunthi!

… 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 ​

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?

 

[paranormal]

I understand that the calculation of eigenvalues for a 4x4 Hermitian matrix can be time-consuming and tedious. However, it is important to go through this process in order to accurately determine the allowed energies for a spin-3/2 particle in the given Hamiltonian. The eigenvalues of a Hermitian matrix represent the energy levels of a system, and it is crucial to have precise values in order to make accurate predictions and interpretations in physics.

That being said, there are some methods that can make the calculation of eigenvalues more efficient. One approach is to use the diagonalization method, where the matrix is transformed into a diagonal form using unitary transformations. This can simplify the calculation of eigenvalues and make it more manageable.

Another approach is to use numerical methods such as the Jacobi algorithm or the QR algorithm, which can quickly and accurately compute the eigenvalues of a matrix. These methods are especially useful for larger matrices where the traditional method of solving det(\hat{H}-\lambda I)=0 becomes more complicated.

In conclusion, while there are alternative methods for finding eigenvalues of a 4x4 Hermitian matrix, it is important to go through the traditional calculation in order to accurately determine the allowed energies in the given system. However, utilizing these alternative methods can make the process more efficient and less tedious.