Spin eigenfunctions for two particles

[SonOfOle]

Homework Statement

Consider two identical particles of mass $$m$$ and spin 1/2. They interact via a potential given by

$$V = \frac{g}{r} \sigma_{1} \sigma{2}$$

where $$g>0$$ and $$\sigma_{j}$$ are Pauli spin matrices which operate on the spin of particle j.

(a) Construct the spin eigenfunctions for the two particle states. What is the expectation value of V for each of these states?

(b) Give eigenvalues of all the bounded states.

Homework Equations

$$\sigma_{1} = \left( \stackrel{0}{1} \stackrel{1}{0}\right)$$
$$\sigma_{2} = \left( \stackrel{0}{i} \stackrel{-i}{0}\right)$$
$$\sigma_{3} = \left( \stackrel{1}{0} \stackrel{0}{-1}\right)$$

The Attempt at a Solution

Other than finding the Pauli Spin matrices, I don't know how to go about solving this problem. I have Griffiths QM text, so feel free to refer to that when giving pointers as how to proceed. Thanks.

 

[Jhero]

Thank you for your post. This is an interesting problem in quantum mechanics that involves the concept of spin and spin operators. In order to solve this problem, we first need to understand the spin eigenfunctions for the two particle states. These eigenfunctions can be written as:

|+> = |+\frac{1}{2}>|+\frac{1}{2}>
|-> = |-\frac{1}{2}>|-\frac{1}{2}>
|\uparrow> = |+\frac{1}{2}>|-\frac{1}{2}>
|\downarrow> = |-\frac{1}{2}>|+\frac{1}{2}>

where |+\frac{1}{2}> and |-\frac{1}{2}> are the spin eigenstates of a single particle with spin 1/2. These eigenfunctions are also known as singlet and triplet states, respectively. Now, using the given potential, we can calculate the expectation value of V for each of these states. The expectation value is given by:

<\psi|V|\psi> = \frac{g}{r} <\psi|\sigma_{1}\sigma_{2}|\psi>

For the singlet state, we have <\psi|V|\psi> = 0, since the spin operators act on different particles and the singlet state is antisymmetric. For the triplet states, we have <\psi|V|\psi> = \frac{g}{r} <\psi|\sigma_{1}\sigma_{2}|\psi>, where <\psi|\sigma_{1}\sigma_{2}|\psi> = 1. Therefore, the expectation value of V is given by <\psi|V|\psi> = \frac{g}{r}.

Moving on to part (b) of the problem, we need to find the eigenvalues of all the bounded states. Since the potential V is given by the product of the spin operators, the eigenvalues of V will be determined by the eigenvalues of the spin operators. The eigenvalues of the spin operators are given by the eigenvalues of the Pauli matrices, which are $\pm 1$. Therefore, the eigenvalues of the potential V are given by $\pm \frac{g}{r}$. This means that there are two bounded states with eigenvalue $\frac{g}{r}$ and two bounded states with eigenvalue $