Indistinguishability of Identical Particles

a fundamental principle of quantum mechanics. According to the principle of the indistinguishability of identical particles, if identical particles in a given system of particles are interchanged, the resulting states of the system cannot be distinguished in any experiment and must be regarded as the same physical state.

The principle of indistinguishability constitutes one of the basic differences between classical mechanics and quantum mechanics. In classical mechanics, the motion of individual particles can, in principle, always be traced along the particles’ trajectories, and the particles can thus be distinguished from each other. In quantum mechanics, identical particles have no individuality. In quantum mechanics, the state of a particle is described by the wave function ψ, which permits determination of just the probability ψ² of detecting the particle at a given point in space. When the wave functions of two or more identical particles overlap in space (that is, when the possible regions where the particles may be detected overlap), it is meaningless to discuss which of the particles is located at a given point. It is meaningful, in this case, to speak only of the probability of detecting one of the identical particles at the point.

The essence of the principle of indistinguishability lies in the empirical fact that for systems of identical particles only two classes of wave functions are realized in nature: symmetric wave functions and antisymmetric wave functions. A symmetric wave function remains unchanged when the space and spin coordinates of any pair of identical particles are interchanged. An antisymmetric wave function changes sign when such an interchange is carried out. A theorem in quantum field theory asserts that symmetric wave functions describe particles with integral spin (such as photons and pions), whereas antisymmetric wave functions describe particles with half-integral spin (such as electrons, protons, and neutrons), which are subject to the Pauli exclusion principle. In the first case, the particles obey Bose-Einstein statistics; in the second case, they obey Fermi-Dirac statistics.

An important quantum effect results from the principle of indistinguishability and the corollary requirements on the symmetry of wave functions for systems of identical particles. This effect, which has no counterpart in classical theory, is the existence of the exchange interaction. One of the first successes of quantum mechanics was W. Heisenberg’s explanation for the existence of two states of the helium atom—the ortho and para states—on the basis of the principle of indistinguishability. The exchange interaction underlies present-day theory on atomic, molecular, and nuclear structures, on the solid state, on the chemical bond, and on other aspects of the structure of matter.