Indistinguishable particles in quantum mechanics: an introduction

Abstract

In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This is, for electrons, the Pauli Exclusion Principle, or in general, the Symmetrization Postulate. Then, we introduce fermions and bosons and the distributions respectively describing their statistical behaviour in indistinguishable situations. Following that, we discuss the spin-statistics connection, as well as alternative statistics and experimental evidence for all these results, including the use of bunching and antibunching of particles emerging from a beam splitter as a signature for some bosonic or fermionic states.

Acknowledgments

I would like to thank L. Hardy, P. Knight, J. Jones, V. Vedral and V. Vieira for their valuable comments about this article, as well as M. Massimi and G. Mil-hano for some useful remarks. I would also like to thank Fundaçăo para a Ciência e a Tecnologia (Portugal) and the 3rd Community Support Framework of the European Social Fund for financial support under grant SFRH/BPD/9472/2002, and FCT and EU FEDER through project POCI/MAT/55796/2004 QuantLog.

Notes

†A wrong idea, as we shall discuss later.

†The reader unfamiliar with this formalism is referred to appendix A for a brief introduction.

†The usual special relativity restriction that the measurement of a physical system cannot influence another if the two are space-like separated.

†The S-matrix formalism is an alternative approach to relativistic quantum physics based on the unitary S-matrix that encodes all the information on all possible scattering processes. Formally, the S-matrix is the realization of the isomorphism between the in and out Fock spaces. For more details, see for instance 21.

†As opposed to the spin-statistics connection, a property that we believe can be derived in the context of relativistic Quantum Mechanics, as we saw in section 3.3.

†Note that at even lower temperatures, below 3 mK, ³He can also exhibit a superfluid phase 33,34, as the fermionic helium atoms pair up to form bosonic quasiparticles.