Bosons and Fermions -interactions

1. The problem statement, all variables and given/known data



A particle of mass m is confined to the region |x| < a in one dimension by an infinite

square-well potential. Solve for the energies and corresponding normalized energy

eigenfunctions of the ground and first excited states.



(b) Two particles are confined in the same potential. The particles are bosons and do not

interact. What is the two-particle wavefunction, ψ(x1 , x2 ), of lowest energy? Is it an

eigenfunction of total energy? Explain.



(c) Answer part (b) with the two bosons replaced by two fermions (neglect spin).



(d) For each case [(b) and (c)] write down the probability density to find the two particles

at the same location in the potential well.





2. Relevant equations





3. The attempt at a solution



So I solved for the single particle in an infinite well and I get a sin function



For b) I think it should be 1/sqrt(2) *(2psi(x1)psi(x2))? but what confuses me here is, do I need the normalisation constant? and since psi 1 and psi 2 are already normalised, I feel my normalisation constant is not right...





and c) I take the anti symetric state 1/sqrt(2) (psi1(x1)psi2(x2)-psi2(x1)psi1(x2))



now d) is where I'm REALLY confused. so due to paulis exclusion principal the antisymetric case can't exist right (ie fermions can be at the same location in the potential well)



but for bosons... what do I integrate between? and do I do a double integral? dx1, dx2



Thanks!