Two identical spin fermions move in one dimension under the influence of the infinite-wall potential V

Question:

Two identical spin ½ fermions move in one dimension under the influence of the infinite-wall potential V = ∞ for x < 0, x > L, and V = 0 for 0 ≤ x ≤ L.

a. Write the ground-state wave function and the ground-state energy when the two particles are constrained to a triplet spin state (ortho state).

b. Repeat (a) when they are in a singlet spin state (para state).

c. Let us now suppose that the two particles interact mutually via a very short-range attractive potential that can be approximated by V= – λδ(x1 – x2) (λ > 0). Assuming that perturbation theory is valid even with such a singular potential, discuss semi quantitatively what happens to the energy levels obtained in (a) and (b).