EXCLUSION > Origins The exclusion principle is linked to the indiscernibility of identical particles. Consider a system of two identical particles, whose individual quantum states are characterized by an observable Ω. Suppose that the particles are prepared in such a way that the measurement of Ω always yields the result α for one particle and the result β for the other. (We do not specify α and β: they label two results, which can either be equal or different.) This is a physically significant situation, which obtains whenever the sum of the values of Ω found for the two particles has to obey some constraint. If we call and the eigenvectors of Ω that correspond to the values α and β, the global state vector of the two particles (which is the state one needs in order to predict the correlations between them) must have the form:



Each term of this linear combination represents a possible result of the joint measurement of Ω. Either we find α for particle 1 and β for particle 2 (first term); or we find β for particle 1 and α for particle 2 (second term). However, if we are dealing with 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). Imposing this constraint upon state (1) we get only two possible (entangled) states:





Particles that behave as predicted by symmetric states (like (2)) are called bosons. Particles whose behaviour is described by anti-symmetric states (like (3)) are called fermions. It is important to stress that, when considering real physical situations, the symmetry condition must take into account all the degrees of freedom that enter the definition of the quantum state of the particles: position, spin, etc. This means that, for example, two fermions are allowed to have a symmetric wave function, provided that their spin state is anti-symmetric (since in this case the global quantum state, obtained by combining the spatial and the spin components, is indeed anti-symmetric).

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



in equation (3). But this would imply that the global state of the two particles is the null vector, which is not permissible within the quantum formalism (one cannot derive probabilities from such a state vector, since it is not a unit vector!).

Let us now focus on spatial degrees of freedom. Here is an example of symmetric and anti-symmetric wave functions:





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

.

Using this identity in equations (4) and (5) we get:





We can compare these results to 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 liked to keep together, in the latter case as if they repelled each other. This phenomenon is connected to the exchange forces discussed in the section on experimental evidence.