Wolfram Alpha:

```Pauli Exclusion Principle
-------------------------
No two electrons in an atom can have identical quantum numbers. This is an example of a
general principle which applies not only to electrons but also to other particles of
half-integer spin (fermions). It does not apply to particles of integer spin (bosons).

The nature of the Pauli exclusion principle can be illustrated by supposing that electrons
1 and 2 are in states a and b respectively. The wavefunction for the two electron system
would be:

ψ = ψ1(a)ψ2(b)

but this wavefunction is unacceptable because in quantum mechanics the particles are
identical and indistinguishable. To account for this we must use a linear combination
of the two possibilities since the determination of which particle is in which state
is not possible to determine.  The wavefunction for the state in which both states
"a" and "b" are occupied by the particles can be written as:

Bosons: ψ = ψ1(a)ψ2(b) + ψ1(b)ψ2(a)

Fermions:  ψ = ψ1(a)ψ2(b) - ψ1(b)ψ2(a)

Particles of half-integer spin must have antisymmetric wavefunctions, and particles of
integer spin must have symmetric wavefunctions. The minus sign in the above relationship
forces the wavefunction to vanish identically if both states are "a" or "b", implying
that it is impossible for both fermions to occupy the same state. ```