# Eigenstates, Eigenvalues & Multicplity of Hamiltonian w/ Spin 1/2

• LCSphysicist
In summary, the conversation discussed the Hamiltonian for two particles with spin 1/2 interacting via a constant A and asked about the eigenstates, eigenvalues, and multiplicity of the Hamiltonian. The conversation concluded that the Hamiltonian is a 4x4 matrix with 4 eigenvalues, which passed a quick test for correctness.
LCSphysicist
Homework Statement
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Relevant Equations
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> Consider two particle with spin 1/2 interacting via the hamiltonian $H = \frac{A}{\hbar^2}S_{1}.S_{2}$, Where A is a constant. What aare the eigenstates, eigenvalues and its multicplity?

$H = \frac{A}{\hbar^2}S_{1}.S_{2} = A\frac{(SS-S_{1}S_{1}-S_{2}S_{2})}{2\hbar^2 } = A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }$

Now, $S_{1}²$, for example, has the same eigenvectors as S1z, that is, $11,10,1-1,00$
And all these states are eigenvectors of S², so we have:

$$H|11\rangle= A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }|11\rangle = A\frac{(2 \hbar^2- 3 \hbar^2/4 -3 \hbar^2/4 )}{2\hbar^2 }|11\rangle = \frac{A}{4}|11\rangle$$

$$H|10\rangle= A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }|10\rangle = A\frac{(2 \hbar^2- 3 \hbar^2/4 -3 \hbar^2/4 )}{2\hbar^2 }|10\rangle = \frac{A}{4}|10\rangle$$

$$H|1-1\rangle= A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }|1-1\rangle = A\frac{(2 \hbar^2- 3 \hbar^2/4 -3 \hbar^2/4 )}{2\hbar^2 }|1-1\rangle = \frac{A}{4}|1-1\rangle$$

$$H|00\rangle= A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }|00\rangle = A\frac{(- 3 \hbar^2/4 -3 \hbar^2/4 )}{2\hbar^2 }|00\rangle = \frac{-3A}{4}|00\rangle$$

I want to know i this is right. Is it? To be honest, i think it is, but what worries me is that i am not sure i these are all the eigenvalues/eigenvectors. I believe H would be something like a "4x4" matrix, so i think it is. But want to hear your answer too.

Delta2
Your work looks correct to me.

For inline Latex, use double hashtag rather than single dollar sign.

Delta2
Just making it easier to read> Consider two particle with spin 1/2 interacting via the hamiltonian $$H = \frac{A}{\hbar^2}S_{1}.S_{2}$$, Where A is a constant. What aare the eigenstates, eigenvalues and its multicplity?

$$H = \frac{A}{\hbar^2}S_{1}.S_{2} = A\frac{(SS-S_{1}S_{1}-S_{2}S_{2})}{2\hbar^2 } = A\frac{(S^2-S_{1}^2-S_{2}^2)}{2\hbar^2 }$$

Now, $$S_{1}²$$, for example, has the same eigenvectors as S1z, that is, $$11,10,1-1,00$$
And all these states are eigenvectors of S², so we have:

Herculi said:
I want to know i this is right. Is it? To be honest, i think it is, but what worries me is that i am not sure i these are all the eigenvalues/eigenvectors. I believe H would be something like a "4x4" matrix, so i think it is. But want to hear your answer too.
The Hamiltonian is indeed a 4×4 matrix. You have found 4 eigenvalues, so what's your concern? A quick test on your eigenvalues to check whether if the Hamiltonian matrix is traceless (the sum of diagonal elements is zero), the sum of the eigenvalues must be zero. That is true in this case which does not guarantee the correctness of your eigenvalues but at least they pass this test so there is no error in the arithmetic.

## 1. What are eigenstates and eigenvalues in the context of spin 1/2 particles?

Eigenstates are the possible states that a spin 1/2 particle can exist in, while eigenvalues are the corresponding energy values associated with each eigenstate. In other words, eigenstates describe the possible orientations of the spin of a particle, while eigenvalues represent the energy levels associated with each orientation.

## 2. How are eigenstates and eigenvalues related to the Hamiltonian of a spin 1/2 particle?

The Hamiltonian of a spin 1/2 particle is a mathematical operator that describes the total energy of the particle. The eigenstates and eigenvalues of the Hamiltonian represent the possible energy states and corresponding energy values that the particle can have.

## 3. What is the multiplicity of the Hamiltonian for a spin 1/2 particle?

The multiplicity of the Hamiltonian for a spin 1/2 particle is 2. This means that there are two possible eigenstates and two corresponding eigenvalues for the Hamiltonian.

## 4. How do eigenstates and eigenvalues affect the behavior of a spin 1/2 particle?

Eigenstates and eigenvalues play a crucial role in determining the behavior of a spin 1/2 particle. The eigenstates represent the possible orientations of the particle's spin, and the eigenvalues represent the associated energy levels. The behavior of the particle will depend on which eigenstate it is in and the corresponding energy value.

## 5. Can the eigenstates and eigenvalues of a spin 1/2 particle change over time?

No, the eigenstates and eigenvalues of a spin 1/2 particle are constant and do not change over time. However, the particle can transition between different eigenstates, resulting in a change in its behavior and energy level.

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