1. The problem statement, all variables and given/known data
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.
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.