An example for two non-interacting identical particles will illustrate the point.
The probability density of the the two particle wave function
must be identical to that of the the wave function
where the particles have been interchanged.

(55) |

(56) |

(57) |

It turns out that particles whose wave functions which are symmetric under particle interchange have integral or zero intrinsic spin, and are termed bosons. Particles whose wave functions which are anti-symmetric under particle interchange have half-integral intrinsic spin, and are termed fermions. Experiment and quantum theory place electrons in the fermion category. Any number of bosons may occupy the same state, while no two fermions may occupy the same state. This result, which establishes the behaviour of many-electron atoms, is proved below.