The exclusion principle is a consequence of the indiscernibility of identical particles. Suppose we measure a two-value observable W on two identical particles. Call a and b the possible outcomes, and and the corresponding eigenvectors. Now suppose that the two particles are perfectly correlated: if the measurement of W yields a for one particle then it yields b for the other. Assuming no interaction between the particles, their global state vector must have the form:

Each term of this linear combination represents a possible result of the joint measurement of W: either we find a for particle 1 and b for particle 2 (first term); or we find b for particle 1 and a for particle 2 (second term). But, if we are dealing with two quantum particles, their identity implies their indiscernibility, which means that particle permutation must not produce any observable effect. The probabilities derived from their global state must therefore be invariant under label exchange (see the discussion of the indiscernibility principle). This reduce state (1) to either one of the following symmetric or anti-symmetric superpositions:

Particles that, when brought together, behave as predicted by symmetric states (like (2)) are called bosons; particles that behave according to anti-symmetric states (like (3)) are called fermions. Notice that, in this definition, the symmetry constraint applies to the global state, including all degrees of freedom of the particles: position, spin, etc. This means that, for example, two fermions are allowed to have a symmetric wave function, insofar as their spin state is anti-symmetric (since the tensor product of a symmetric and an anti-symmetric state is anti-symmetric).

The exclusion principle for fermions follows straightforwardly from equation (3). For the two fermions be found in the same configuration, one should have:

in equation (3). But then the global state would get reduced to the null vector, which, according to the quantum formalism, is not admissible (one cannot extract probabilities from such a state, since it is not normalised !).

As an illustration, let's consider the spatial anti-symmetric or symmetric state. In this case we can rewrite equations (2) and (3) using wave functions:

Given a system of two particles described by these wave functions, it is interesting to ask what is the probability to find the two particles close to each other. We have to evaluate eq. (4) and (5) for the case in which the space coordinates of the two electrons happen to have almost the same value: In this case

.

Inserting in equations (4) and (5) we get:

and

We can compare these results with the case of distinguishable particles, for which:

We see that in the case of a symmetric wave function, the probability of finding the particles close to each other is enhanced, whereas it is essentially zero in the case of an anti-symmetric wave function. In the former case the particles act as if they like to keep together, whereas in the latter they act as if they repel each other. This phenomenon is connected with the exchange forces discussed in the section on experimental evidence.