In this problem we consider low-energy excited states of two electrons inside a one-dimensional box with length a, which can be approximated as products of the =1 and n=2 particle-in-a- box eigenfunctions. in other words, the electronic configuration is that having one electron occupying the n=1 level and the other electron occupying the n =2 level. note that for a single n'h , and the eigenfunction ,(x) nsin 8ma particle in a box, the energy level e, = (a) (2%) the spatial part of the two-electron wavefunction can be written as 1 = (1)0, (2)t 0,(2)0, (1). explain why a simple product such as (1)e, (2) is not a valid two-electron wavefunction. (b) (4%) consider the spin part of the wavefunction. using the pauli exclusion principle, show four possible spin-adapted wavefunctions for this configuration. give the four full that there are (spatial+spin) wavefunctions. (c) (2%) separate the four states into a triplet manifold and a singlet manifold i = llo) aelx- ) dx, dx, and (d) (5%) define the integrals j (x, ,)dx, dx, . evaluate the energies of the singlet and 41 triplet states to show that 'e= e,+j+k and 'e= e, +j-k. what is e,? (e) (2%) according to the previous results, what is the multiplicity of the first excited state of the two-electron system?