# What Are the Eigenvalues and Eigenvectors of This Spin Hamiltonian?

• dreamspy
So the eigenvalue will be (1/2)(0)(0+1) = 0. In summary, the conversation discusses a question related to spin in quantum mechanics. The Hamilton operator for two stationary particles is given and the question asks to find the eigenvalues and eigenvectors. The solution provided involves using a triplet state and calculating the eigenvalues for different spin operators. The conversation also briefly touches upon a singlet state and how to calculate the eigenvalue for it.
dreamspy

## Homework Statement

I'm struggling with this question here from my QM class. I have read all my material on Spin (both Griffiths, and a chapter in an Icelandic book). I'we done some problems, but I really have no Idea where to start with this one. It goes something like this:

Two particles have spin 1/2 and are stationary, but their spins interact with this Hamilton operator:

$$\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)}$$

where $$\bf{S}^{(j)}$$ is the spin operator for particle j, and j=1,2.

As a basis in the state space (hope that's the right word) you can f.x. take $$u_s^{(1)}u_r^{(2)}$$ where$$r,s,=\pm\frac{1}{2}$$, and $$\hat S_3^{1}u_s^{1}=shu_s^{j}$$ and $$\hat S_3^{2}u_r^{2}=rhu_r^{j}$$

Questions:
(i) Find the eigenvalues and eigenvektors of the Hamilton operator.

(ii) How would the result be if we used this Hamilton operator instead:

$$\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)} + \lambda \hat{\underline S}^{(1)}\cdot \lambda \hat{\underline S}^{(2)}$$

## The Attempt at a Solution

Now here is a quick solution I got from my teacher:

(i)
Triplet:

$$u_{\frac{1}{2}}u_{\frac{1}{2}}$$
$$(u_{\frac{1}{2}}u_{-\frac{1}{2}}+u_{-\frac{1}{2}}u_{\frac{1}{2}})\frac{1}{\sqrt 2}$$
$$u_{-\frac{1}{2}}u_{-\frac{1}{2}}$$

$$\underlince{\hat S}^2 = s(s+1)$$

The Eigenvalues:

$$(\gamma S_3^{(1)}-\gamma S_3^{(2)})u_{\frac{1}{2}}u_{\frac{1}{2}} = \gamma \hbar(s+r)u_{\frac{1}{2}}u_{\frac{1}{2}}$$
(the others should follow the same procedure)

(ii)
Two spin operators:

$$\underline{\hat S}^{(1)}, \underline{\hat S}^{(2)}$$

$$\underline{\hat S}^{(1)}\cdot \underline{\hat S}^{(2)} = \frac{1}{2}(\underline{\hat S}^{2}-(\underline{\hat S}^{(1)})^2-(\underline{\hat S}^{(1)})^2)$$

$$\underline{\hat S} = \underline{\hat S}^{(1)} + \underline{\hat S}^{(2)}$$

$$\underline{\hat S}}$$ has eigenvalue $$s(s+1) \hbar ^2, s=0,1$$

Now I almost have no clue on what's going on here.

Now I suppose the part in the Triplet section, is all possible linear combinations of the u vektors. And the eigenvalue can be read from the right side of the formula below. But could anyone care to comment on this? I'm standing on very shaky ground here :) This is the solution my teacher gave us, nobody has a clue what's going on, and were taking the exam tomorrow :)

Frímann

My basic quesion is probably, how do I choose the part in the Triplet section, are those the eigenfunctions? If not then how would I find them?

Well if I am not mistaken for two fermions the eigenstate should be anti symmetrical, i.e singlet and not triplet which is symmetric.

But I myself in a shaky ground... :-)

$$\underline{\hat S}^{(1)}\cdot \underline{\hat S}^{(2)} = \frac{1}{2}(\underline{\hat S}^{2}-(\underline{\hat S}^{(1)})^2-(\underline{\hat S}^{(1)})^2) = (1/2)[s(s+1) - (3/2)]$$

remember

$$S_{1}^{2} = S_{2}^{2} = (1/2)[(1/2) + 1]$$

But what if we have $$u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$

How do we calculate $$\underline{\hat S}^2u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$ ?

[QUOTE said:
dreamspy;2493358]But what if we have $$u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$

How do we calculate $$\underline{\hat S}^2u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$ ?
[/QUOTE]

That is a singlet state which has s = 0.

## 1. What is spin in physics?

Spin is an intrinsic property of particles in quantum physics that describes their angular momentum. It is a quantum mechanical property that cannot be fully explained by classical physics.

## 2. How is spin represented in equations?

Spin is represented by the symbol 's' and is often included in equations as a quantum number. It can also be represented as a vector with magnitude and direction.

## 3. What are the possible values of spin?

The possible values of spin are half-integer or integer numbers, such as 1/2, 1, 3/2, etc. These values determine the behavior of particles and their interactions with other particles.

## 4. How is spin measured in experiments?

Spin can be measured using various techniques, such as Stern-Gerlach experiments or scattering experiments. These experiments involve measuring the deflection or polarization of particles, which can then be used to determine their spin.

## 5. Can spin be changed or manipulated?

Spin can be changed or manipulated through various processes, such as interactions with other particles or external fields. This is an important aspect of spin-related research in fields such as quantum computing and spintronics.

Replies
1
Views
786
Replies
0
Views
744
Replies
9
Views
1K
Replies
3
Views
1K
Replies
3
Views
1K
• Quantum Physics
Replies
2
Views
575
Replies
5
Views
2K
Replies
3
Views
1K