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 diagonalization(numerical method) b) mean field aproximation. 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:intgral over y of a function: U(x-y)ro(y) where ro(y) is density of remaining 3 particles... harmonic pot is given in the usual form, one half omegasqr mass xsqr. contact pot is deltafunc(particle1-particle2) times some parameter u.
plot dependency of ground state energy on u/hcross*omega for both methods. plot overlap of wave funct from the first method with the one from the second method depending again on u/hcross*omega.