Identical Particles

If the particle were identical such that they were indistinguishable then statistics states that the system are no longer independent of each other. The wavefunction for a system of two identical particles is,

\[ \begin{equation} \psi(r_1, r_2)_\pm = A \left[\psi_a(r_1)\psi_b(r_2) \pm \psi_b(r_1)\psi_a(r_2)\right] \end{equation} \]

• \(\psi_+\) : Wavefunction of bosons

• \(\psi_-\) : Wavefunction of fermions

• The general equation for \(N\) particles is not easy to write but easy to see the pattern. For \(N=3\) identical particles, you have 3 products for each of the 3 terms. In combinatorics this is considered a sum of permutations.

Exchange Relationship

Identical particles follows a property that if the particles were to be exchange, its wavefunction follows,

\[ \begin{equation} \psi(r_a,r_b) = \pm\psi(r_b,r_a) \end{equation} \]

• \(+\) : bosons

• \(-\) : fermions