# Energy of two fermions in a three-dimensional box

1. Dec 5, 2015

### Skrien

1. The problem statement, all variables and given/known data
Two particles of mass m are placed in a rectangular box with sides a>b>c (note 3D-box). The particles interact with each other with a potential $V=A\delta(\mathbf{r}_1-\mathbf{r}_2)$ and are in their ground state (1s). Use first order perturbation theory to find the systems energy in two cases:

$\begin{cases} a ) \text{ The particles are fermions with anti-parallel spins}\\ b) \text{ The particles are fermions with parallel spins} \end{cases}$

2. Relevant equations
I'm having trouble finding the explicit spin wave-function, what is the total wave function for my two particle system? I need an explicit expression for which I can perform the integral, I have only found implicit expression such as $''\chi(s_1,s_2)''$. What is this function in my case?

3. The attempt at a solution
So far I've concluded that the state in b) is impossible because of the Pauli exclusion principle. In a) I have written down the hamiltonian as
$H=H_0+H'=-\frac{\hbar^2}{2m_1}\nabla_1^2-\frac{\hbar^2}{2m_2}\nabla_2^2+A\delta(\mathbf{r}_1-\mathbf{r}_2).$
where H' is the interacting potential which is considered the perturbation. The first order correction to the energy is
$E_{n_1n_2}^{(1)}=<\psi_{n_1n_2}|H'|\psi_{n_1n_2}>$

The wave-function for one particle in a box without consideration to spin is quite straight forward, however, I can't seem to grasp the concept of spin-wave function.

Last edited: Dec 5, 2015
2. Dec 5, 2015

### Staff: Mentor

The integral over the spin components is trivial, as it is not a function of r1 or r2. The only thing that will be important is that it is normalized and (hint) its symmetry.

My reading of the problem is that the particles are in the ground state internally, but not necessarily in the ground state of the box (because the designation 1s has nothing to do with the box). I would find an appropriate spatial wave function where the Pauli principle is obeyed.