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, ...