Question

Consider two electrons moving in a central potential well in which there are only three single-particle states \(\psi_1\), \(\psi_2\) and \(\psi_3\).

(a) Write down all of the wave functions \(\psi\)(r₁, r₂) for the two-electron system.

(b) If now the electrons interact with a Hamiltonian \(\delta H = V'(r_1, r_2)= V'(r_2, r_1)\), show that the following expression for the matrix element is correct:

Answer 1

(a) The wave functions for a fermion system are antisymmetric for interchange of particles, so the possible wave functions for the system are

(b) We can write

Since the particles are identical, r₁ and r₂ may be interchanged in each term. Do this and as V'(r₁, r₂) = V'(r₂, r₁), we again obtain the same expression, showing its correctness.