PX284 - L1 - exchange symmetry

considering a two particle system with wavefunctions:

ψ ({\vec{r}}_{1}, {\vec{r}}_{2})

defining the exchange operator as:

{\hat{P}}_{12} ψ ({\vec{r}}_{1}, {\vec{r}}_{2)} = ψ ({\vec{r}}_{2}, {\vec{r}}_{1})

but if the particles are identical, swapping them should make no measurable difference, ie: $| ψ ({\vec{r}}_{1}, {\vec{r}}_{2}) |^{2} = | ψ ({\vec{r}}_{2}, {\vec{r}}_{1}) |^{2}$

{\hat{P}}_{12} ψ ({\vec{r}}_{1}, {\vec{r}}_{2}) = ψ ({\vec{r}}_{2}, {\vec{r}}_{1}) = \pm ψ ({\vec{r}}_{1}, {\vec{r}}_{2})

there are two types of exchange symmetry

type 1

$ψ$ symmetric under particle exchange: $ψ ({\vec{r}}_{1}, {\vec{r}}_{2}) = ψ ({\vec{r}}_{2}, {\vec{r}}_{1})$
particles with this symmetry have integer values of quantum numbers and are called bosons

type 2

$ψ$ antisymmetric under particle exchange: $ψ ({\vec{r}}_{1}, {\vec{r}}_{2}) = - ψ ({\vec{r}}_{2}, {\vec{r}}_{1})$
particle with this symmetry have half-integer quantum numbers, and are called fermions