(a) Suppose you could find a solution ψ(r1,r2,…,rZ) to the Schrödinger equation (Equation 5.37), for the Hamiltonian in Equation 5.36. Describe how you would construct from it a completely symmetric function, and a completely antisymmetric function, which also satisfy the Schrödinger equation, with the same energy. What happens to the completely antisymmetric function if ψ(r1,r2,…,rZ) is symmetric in (say) its first two arguments (r1↔r2)?
(b) By the same logic, show that a completely antisymmetric spin state for Z electrons is impossible, if Z>2 (this generalizes Problem 5.10(a)).