Two-Particle System: Boson and Fermion Eigen Energies and Wave Functions

[kasse]

Homework Statement

Assume that we have a system where the two lowest one-particle states are \psi _{1} (r) with eigen energy E1 and \psi _{2} (r) with eigen energy E2. What is the lowest eigen energy E and the wave function \psi (r_{1},r_{2}) for a two-particle system if

a) they are bosons
b) they are fermions



2. The attempt at a solution

a) The symmetric (boson) wave function becomes

\psi (r_{1},r_{2}) = \frac{1}{\sqrt{2}}[\psi _{1} (r1)\psi _{1} (r2) + \psi _{1} (r2)\psi _{1} (r1)] = \frac{2}{\sqrt{2}}[\psi _{1} (r1)\psi _{1} (r2)]

The energy is of course 2E1 because bosons can be in the same quantum state.

According to my book, the answer is \psi (r_{1},r_{2}) =\psi _{1} (r1)\psi _{1} (r2) only, so before proceeding with b) I want to know what I have done wrong.

 

[weejee]

To ensure proper normalization, the prefactor for N-body bosonic wavefunction is not 1/sqrt(N!) but 1/sqrt(n_1!*n_2!*...*N!), where n_1, n_2, ... are number of particles in state 1, state 2, ...