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Skrien
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Homework Statement
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 [itex]V=A\delta(\mathbf{r}_1-\mathbf{r}_2)[/itex] and are in their ground state (1s). Use first order perturbation theory to find the systems energy in two cases:
[itex]
\begin{cases}
a ) \text{ The particles are fermions with anti-parallel spins}\\
b) \text{ The particles are fermions with parallel spins}
\end{cases}
[/itex]
Homework 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 [itex]''\chi(s_1,s_2)''[/itex]. What is this function in my case?
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
[itex]
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).
[/itex]
where H' is the interacting potential which is considered the perturbation. The first order correction to the energy is
[itex]E_{n_1n_2}^{(1)}=<\psi_{n_1n_2}|H'|\psi_{n_1n_2}>[/itex]
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.
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