# four boson quantum system contact interaction

I have to solve this problem. Four bosons moving in 1d harmonic potential (their spin is 0) and interacting through contact interaction defined via delta function. Now, methods that I have to use: a) exact diagonalisation(numerical method) b) mean field approximation. For a) part a set of quantum states of the system must be formed using the states of non interacting system. We only work with quantum number less then 3 i.e., only 3 first solutions for harmonic pot.

For b) part we have schro eq with hamiltonian of interaction given through some Vmeanfield defined as: $$\int U(x-y)r_0(y)\mbox{ dy}$$ where $r_0(y)$ is density of remaining 3 particles... harmonic pot is given in the usual form, $\frac{1}{2}\omega^2mx^2$. contact pot is $\delta(\mbox{Particle}_1-\mbox{Particle}_2)$ times some parameter $u$.

Plot dependency of ground state energy on $\frac{u}{\hbar\omega}$ for both methods. plot overlap of wave funct from the first method with the one from the second method depending again on $\frac{u}{\hbar\omega}$.

Thank you.

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Welcome to Physics.SE @Marko! Since this is pretty clearly a homework problem, it is considered good form to 1) break it up into specific questions about general principles, and 2) show us what progress you've made so far so we can help you. As it stands, it looks like you want the community to simply provide you with the two plots in your last paragraph. This is easily done, but it won't help future users of this site (and I suspect it won't help you much either). –  wsc Sep 1 '12 at 18:34
This is Marko, in response to one and only answer so far.. –  Marko Sep 1 '12 at 22:40
oops, wrong key...so, yes...it is a homework problem and no, i am not asking you to just give me those plots. –  Marko Sep 1 '12 at 22:40
I have no idea how to begin solving this problem. Am I suppose to form some sort of diagonal matrix? What are the state kets? –  Marko Sep 1 '12 at 22:42
Ill stop doing that.I can list about 12 state kets and their energies for non interacting system and write the matrix in a diagonal form with those energies as eigenvalues and state kets as eigenvectors...but what good will that do? How can I form states of interacting system? This is my first many body problem so...what is exact diagonalization? –  Marko Sep 1 '12 at 22:50