QUANTUM MECHANICS Homework set #19: Identical particles ...

(d) Assuming that perturbation theory is valid, discuss what happens to the energy levelsas λ is increased and find an expression (valid to first order in λ) for the dependenceon λ of the critical field for the level crossing you found in (c).Problem # 19.4 : Bosons and fermionsFour non-interacting equal-mass particles are placed in a one-dimensional square-well potentialof width L and of infinite depth. Two of the particles are identical spin 0 bosons,while the other two are identical spin 1/2 fermions.(a) Find the ground state energy and the corresponding wave function (wave functions ifthere is degeneracy).(b) Find the energy of the first excited state and the corresponding wave functions.(c) Use first order perturbation techniques to calculate the change in the ground stateenergy of the system if the two fermions interact through the potential V 0S1· ⃗ ⃗S 2 , whereV 0 is a constant and S ⃗ 1 and S ⃗ 2 are the spin operators of the respective fermions.(d) Use first order perturbation techniques to calculate the change in the energy of thefirst excited state if the two fermions interact through the potential V 0S1 ⃗ · ⃗S 2 . Discussthe degree to which these energy shifts remove the degeneracies found in part (b).(e) What degeneracies are lifted in the presence of a magnetic field by the Zeeman effect?