# Eigenvalues of 4x4 Hermitian Matrix (Observable)

• Gunthi
In summary, the conversation discusses finding the allowed energies for a spin-3/2 particle with a given Hamiltonian and the question of whether there is a quicker way to find the eigenvalues of a 4x4 hermitian matrix. It is suggested to rearrange the rows and columns of the matrix to get two separate 2x2 matrices, which can then be calculated separately to find the eigenvalues. This is based on the concept that the matrix can be separated into two parts that operate independently.
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 $det(\hat{H}-\lambda I)=0$?

Hi Gunthi!
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.

tiny-tim said:
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

Hi Gunthi!
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

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?

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.

## 1. What is a 4x4 Hermitian matrix?

A 4x4 Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the matrix is symmetric about its main diagonal and the elements in the matrix are complex conjugates of each other.

## 2. What are eigenvalues of a 4x4 Hermitian matrix?

Eigenvalues of a 4x4 Hermitian matrix are the set of numbers that, when multiplied by the identity matrix and subtracted from the original matrix, result in a determinant of 0. They represent the characteristic values of the matrix and are important in understanding the behavior of the matrix.

## 3. How are eigenvalues of a 4x4 Hermitian matrix calculated?

The eigenvalues of a 4x4 Hermitian matrix can be calculated by solving the characteristic polynomial of the matrix. This involves finding the roots of the polynomial equation using techniques such as factoring or the quadratic formula.

## 4. What is the significance of eigenvalues in a 4x4 Hermitian matrix?

The eigenvalues of a 4x4 Hermitian matrix play an important role in understanding the behavior of the matrix. They are used in various mathematical and scientific applications, including quantum mechanics, signal processing, and data analysis.

## 5. How are eigenvalues of a 4x4 Hermitian matrix related to observables in quantum mechanics?

In quantum mechanics, observables are represented by Hermitian operators, which have real eigenvalues. The eigenvalues of these operators correspond to the possible outcomes of a measurement of the observable in a physical system. Therefore, the eigenvalues of a 4x4 Hermitian matrix can be interpreted as the possible values of an observable in quantum mechanics.

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