Consider a system consisting of a pair of noninteracting electrons confined to a one-dimensional box of length L (extending from x = 0 to x = L). (a) What is the total ground-state energy of this system? (b) What is the only possible value of the total spin quantum number S in the ground-state of this system, which is consistent with the Pauli exclusion principle? (c) Express the ground-state wavefunction for this system as a linear combination of the wavefunctions of a single electron confined within this same box, in a way that is consistent with the fact that electrons are Fermi particles. (d) If you assumed that both electrons are in exactly the same state, so all of their quantum numbers (including spin) were identical, show that the corresponding two-electron wavefunction would necessarily be equal zero (and thus no such state could exist).