We say that two particles are identical when they have the same physical characteristics (mass, electric charge, spin,…). Placed in an external field of force, two identical particles will behave exactly the same way. Therefore, for a system of N identical particles, the equations of motion are invariant under permutation of the labels attached to the particles.
In classical mechanics, two identical particles can always be distinguished, at least in principle. If we know the position of the two particles at some instant t0, then at any time t > t0 we can assign an exact position to each of them by simply calculating their trajectory. In quantum mechanics, however, this possibility is prevented by the uncertainty relations, which implies that the more precisely the initial position of a particle is known, the larger is the spread of its velocity. This means that, even if the position of a particle is known accurately at a given instant, such information is replaced by a probability distribution shortly afterwards. As soon as the distributions of the two particles overlap, the position can no longer work as a label. We say therefore that in quantum mechanics two identical particles are, in general, indiscernible.
Because of its symmetry properties under the permutation of particle labels, an ensemble of identical quantum particles can only be found in either a symmetric or an antisymmetric quantum state (although this constraint does not apply to distinguishable quantum systems, e.g. atoms in a crystal lattice, whose quasi-fixed position acts as a label). In the first case the particles are called bosons (in honour of Satyendra Nath Bose), in the second case they are called fermions (in honour of Enrico Fermi).
BOSE :
FERMI :Integer spin particles always behave like bosons, whereas half-integer spin particles always behave like fermions. Photons and mesons are examples of bosonic elementary particles; electrons, neutrinos and quarks are all fermions. Compound systems, like atoms or ions, have a fermionic or bosonic character depending on their total spin (calculated by adding the spins of their parts).
The most important consequences of the indiscernibility principle are found in the physics of aggregates. Indeed, the statistical pattern of an ensemble of identical particles is radically different for distinguishable and indistinguishable particles.
The statistical approach to ensembles of distinguishable particles was developed by Wolfgang Boltzmann in his kinetic theory of ideal gas. For ensembles of indistinguishable particles, there exist two kinds of statistics exist: the Bose-Einstein statistics for bosons and the Fermi-Dirac statistics for fermions. The two display very different patterns at low temperatures.
A peculiar feature of bosons is that they like to keep together: N bosons already in a quantum state will enhance the probability of one more joining them. The opposite is true for fermions, which must obey Pauli's exclusion principle. In the table above we consider the case in which a two-value observable with equiprobable outcomes is measured on a pair of identical particles (the right and left halves of the boxes represent the two states accessible to the particles, each corresponding to an eigenstate of the observable). The toss of two coins provides a good model for such a situation. If we compare the statistical distributions expected for a pair of bosonic coins, a pair of fermionic coins and a pair of distinguishable (classical) coins, we see that using fermionic coins one never gets two tails nor two heads, whereas with bosonic coins this happens more likely than it does with distinguishable coins.
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