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Two particle system

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data

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

    [tex] \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)][/tex]

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

    According to my book, the answer is [tex] \psi (r_{1},r_{2}) =\psi _{1} (r1)\psi _{1} (r2)[/tex] only, so before proceeding with b) I want to know what I have done wrong.
     
    Last edited: Nov 13, 2008
  2. jcsd
  3. Nov 13, 2008 #2
    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, .....
     
    Last edited: Nov 13, 2008
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