Distinguishability in Statistical Mechanics

"Consider now a system that contains two or more similar particles, say, for definiteness, an atom with two electrons. denote by (mn) that state of the atom in which one electron is in an orbit labelled m, and the other in the orbit n. The question arises whether the two states (mn) and (nm), which are physically indistinguishable as they differ only by the interchange of the two electrons, are to be counted as two different states or as only one state, i.e., do they give rise to two rows and columns in the matrices or to only one? If the first alternative is right, then the theory would enable one to calculate the intensities due to the two transitions (mn) —> (m'n') and (mn) —> (n'm') separately, as the amplitude corresponding to either would be given by a definite element in the matrix representing the total polarisation. The two transitions are, however, physically indistinguishable, and only the sum of the intensities of the two together could be determined experimentally. Hence, in order to keep the essential characteristic of the theory that it shall enable one to calculate only observable quantities, one must adopt the second alternative that (mn) and (nm) count as only one state." Dirac, "On the Theory of Quantum Mechanics", 1926.

"In the classical mechanics, the procedure of taking the interchange of similar particles between different individual states as leading to a new state of the system as a whole was justified, not because the particles were thought of as carrying labels which would distinguish between them, but because it would in principle be possible to think of an observer who could follow the motions of the individual particles — without disturbing them — and actually determine whether two similar particles had interchanged roles or not. In the quantum mechanics, however, the possibility of following the behaviour of the individual particles is limited by the Heisenberg uncertainty principle, and this can make it impossible to determine whether such an interchange has taken place. Moreover, such a limitation is regarded in the quantum mechanics, not as an accident due to an unsatisfactory choice of the tools of observation, but as a limitation in principle which can make the question of such interchange meaningless. As a result, in the quantum statistics we do not count as different two states which cannot at least by conceivable observations be distinguished from the other." - Tolman, Principles of Statistical Mechanics, p. 323.

"our n elements could be n oscillators, all having the same energy spectrum because of the same intrinsic frequency f, but distinguishable from each other by spatial location or orientation. Or the n elements could be n particles, all having the same energy spectrum because of the same mass m and spin s, but not permanently distinguishable from each other on account of their free motion inside a common container." -Tolman, Principles of Statistical Mechanics, p. 367

"Operationally, two particles are distinguishable if they can always be selectively separated by a filter." -Hestenes, D. (1970). Entropy and indistinguishability. American Journal of Physics 38 (7), 840-845.