# Pauli's exclusion principle

## Pauli's exclusion principle

We've already mentioned the Pauli's exclusion principle:each individual state can be occupied by a single fermion. We have seen that it holds for two fermions.The introduction of Slater determinant allows us to see that this principle is valid for an arbitrary number of identical fermions: since the antisymmetric ket is a slater determinant, if two states ${\displaystyle \left|a\right\rangle ,\left|b\right\rangle }$ are the same then the matrix has two identical columns and the determinant is 0. This is not true for bosons since every ${\displaystyle -}$ has been changed into a ${\displaystyle +}$.

## Occupation numbers

If we have a system of identical particles, with each particle allowed to stay in one of the ${\displaystyle \left|u_{i}\right\rangle ,\left|u_{j}\right\rangle ,\dots \left|u_{p}\right\rangle }$ individual states, we can define the occupation number ${\displaystyle n_{k}}$ as the number of particles being in the state ${\displaystyle \left|u_{k}\right\rangle }$. Occupation numbers are subjected to the restriction that ${\displaystyle \sum _{k}n_{k}=N}$. Clearly two mathematical state

{\displaystyle {\begin{aligned}\left|u_{1}(1),u_{1}(2),\dots ,u_{1}(n_{1}),u_{2}(n_{1}+1),\dots ,u_{2}(n_{1}+n_{2})\dots \right\rangle \\\left|u_{1}(2),u_{1}(1),\dots ,u_{1}(n_{1}),u_{2}(n_{1}+1),\dots ,u_{2}(n_{1}+n_{2})\dots \right\rangle \end{aligned}}}
with the same occupation numbers are connected by the action of a permutation operator and represent the same physical state, that is under the action of ${\displaystyle S}$ or ${\displaystyle A}$ they give collinear kets. For this reason we can use occupation numbers to identify physical kets (and to span the physical state ${\displaystyle \xi _{S}}$ or ${\displaystyle \xi _{A}}$):
${\displaystyle \left|n_{1},n_{2},\dots ,n_{k},\dots \right\rangle =cS\left|u_{1}(1),\dots ,u_{1}(n_{1}),u_{2}(n_{1}+1),\dots ,u_{2}(n_{1}+n_{2}),\dots \right\rangle }$
This ket describes a state with ${\displaystyle n_{1}}$ particles in the state ${\displaystyle u_{1}}$, ${\displaystyle n_{2}}$ particles in the state ${\displaystyle u_{2}}$ and so on. From Pauli's exclusion principle it follows that:

• If the particle under study are bosons occupation numbers are arbitrary
• If the particle under study are fermions occupation numbers can only be 1 or 0, since a state can be empty or occupied by a single fermion

## Statistical implications of Pauli's exclusion principle

Since statistical description of macroscopic properties is based on the number of microscopic states that corresponds to the same macroscopic state, the difference between bosons and fermions, the second being subjected to Pauli's exclusion principle, leads to different statistics:

• bosons are described by Bose-Einstein statistics,which takes into account the possibility of having an arbitrary number of particles in each individual state
• fermions are described by Fermi-Dirac statistics,which takes into account the restriction in the occupation numbers