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arXiv:2603.09560

An elementary proof of symmetrization postulate in quantum mechanics for a system of particles

Diganta Parai, Nikhilesh Maity

quant-phcond-mat.quant-gasSubmitted Mar 10, 2026• Announced Mar 11, 2026• 0 citations
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Abstract

According to symmetrization postulate for a system of identical particles, wave function has to be completely symmetric or completely anti-symmetric. In this paper we want to mathematically justify this postulate ignoring the spin part of wave function in three dimension. For a system of N identical particles, if the solution to the governing Schrodinger equation meets these criteria: a) the probability density remains invariant when any two particle positions are exchanged over time, b) the wave function is continuous and has a continuous gradient, and the system exhibits the following characteristics: c) the configuration space, which is 3N dimensional, is connected, and d) the potential term in the Hamiltonian is invariant under the exchange of any two particle positions, then the wave function must be either totally symmetric or totally antisymmetric over time.

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Abstract. Reasonable requirements of (a) physical invariance under particle permutation and (b) physical completeness of state descriptions [1], enable us to deduce a Symmetric Permutation Rule(SPR): that by taking care with our state descriptions, it is always possible to construct state vectors (or wave functions) that are purely symmetric under pure permutation for all particles, regardless of type distinguishability or spin. The conventional exchange antisymmetry for two identical half-integer spin particles is shown to be due to a subtle interdependence in the individual state descriptions arising from an inherent geometrical asymmetry. For three or more such particles, however, antisymmetrization of the state vector for all pairs simultaneously is shown to be impossible and the SPR makes observably different predictions, although the usual pairwise exclusion rules are maintained. The usual caveat of fermion antisymmetrization – that composite integer spin particles (with fermionic consitituents) behave only approximately like bosons – is no longer necessary. I TERMINOLOGY

•Jan 1, 2000•4 citations
According to symmetrization postulate for a system of identical particles, wave function has to be completely symmetric or completely anti-symmetric. In this paper we want to mathematically justify this postulate ignoring the spin part of wave function in three dimension. For a system of N identical particles, if the solution to the governing Schrodinger equation meets these criteria: a) the probability density remains invariant when any two particle positions are exchanged over time, b) the wave function is continuous and has a continuous gradient, and the system exhibits the following characteristics: c) the configuration space, which is 3N dimensional, is connected, and d) the potential term in the Hamiltonian is invariant under the exchange of any two particle positions, then the wave function must be either totally symmetric or totally antisymmetric over time.
The symmetrization postulate (SP) states that wave functions are either completely symmetric or completely antisymmetric under permutations of identical particles. It is shown by one-dimensional counter-examples that SP is not demanded by the usual physical interpretation of the mathematical formalism of wave mechanics unless one makes use of further physical properties of real systems; the error in a standard proof of SP which ignores these properties is pointed out. It is then proved that SP is true for those systems of spinless particles which have the following properties: (a) probability densities are permutation-invariant, (b) allowable wave functions are continuous with continuous gradient, (c) the 3n3n3n-dimensional configuration space is connected, (d) the Hamiltonian is symmetric, and (e) the nodes of allowed wave functions have certain properties. The counterexamples show that SP is not a necessary property of those systems which do not have property (c). The proof is extended to particles with internal degrees of freedom (including spin) by noting that any observable commutes with every permutation and making use of a particular observable acting only on internal variables. Extraneous mathematical assumptions, such as that of the existence of a "complete" set of commuting observables, criticized by Messiah and Greenberg, are not employed. Some comments are made on the conventional nature of the concept of identity for similar particles; the equivalence between the usual formulation in which different species of similar particles are treated as distinct, and that in which they are regarded as identical particles in different internal states, is emphasized.
On the basis of reasonable physical assumptions, it is proved that for systems of identical spinless particles, the permissible nonrelativistic Schr\"odinger wave functions are either all symmetric or all antisymmetric. The proof is a generalization of that given by Girardeau. Although connectivity properties of the configuration space play an important role in the proof, it is not true, as was previously believed, that a connected space is always necessary. In particular, for one-dimensional systems, connectivity of the configuration space is not necessary for the proof to hold.
Symmetries and wave functions An important difference between the classical and quantum perspectives is their different criteria of distinguishability. Identical particles are classically distinguishable when separated in phase space. On the other hand, identical particles are always quantum mechanically indistinguishable for the purpose of counting distinct microstates. But these concepts and these distinctions do not tell the whole story of how we count the microstates and determine the multiplicity of a quantized system. There are actually two different ways of counting the accessible microstates of a quantized system of identical, and so indistinguishable, particles. While these two ways were discovered in the years 1924–1926 independently of Erwin Schrödinger’s (1887–1961) invention of wave mechanics in 1926, their most convincing explanation is in terms of particle wave functions. The following two paragraphs may be helpful to those familiar with the basic features of wave mechanics. A system of identical particles has, as one might expect, a probability density that is symmetric under particle exchange, that is, the probability density is invariant under the exchange of two identical particles. But here wave mechanics surprises the classical physicist. A system wave function may either keep the same sign or change signs under particle exchange. In particular, a system wave function may be either symmetric or antisymmetric under particle exchange.
Reasonable requirements of (a) physical invariance under particle permutation and (b) physical completeness of state descriptions [1], enable us to deduce a Symmetric Permutation Rule(SPR): that by taking care with our state descriptions, it is always possible to construct state vectors (or wave functions) that are purely symmetric under pure permutation for all particles, regardless of type distinguishability or spin. The conventional exchange antisymmetry for two identical half-integer spin particles is shown to be due to a subtle interdependence in the individual state descriptions arising from an inherent geometrical asymmetry. For three or more such particles, however, antisymmetrization of the state vector for all pairs simultaneously is shown to be impossible and the SPR makes observably different predictions, although the usual pairwise exclusion rules are maintained. The usual caveat of fermion antisymmetrization—that composite integer spin particles (with fermionic constituents) behave only approximately like bosons—is no longer necessary.
Abstract. Reasonable requirements of (a) physical invariance under particle permutation and (b) physical completeness of state descriptions [1], enable us to deduce a Symmetric Permutation Rule(SPR): that by taking care with our state descriptions, it is always possible to construct state vectors (or wave functions) that are purely symmetric under pure permutation for all particles, regardless of type distinguishability or spin. The conventional exchange antisymmetry for two identical half-integer spin particles is shown to be due to a subtle interdependence in the individual state descriptions arising from an inherent geometrical asymmetry. For three or more such particles, however, antisymmetrization of the state vector for all pairs simultaneously is shown to be impossible and the SPR makes observably different predictions, although the usual pairwise exclusion rules are maintained. The usual caveat of fermion antisymmetrization – that composite integer spin particles (with fermionic consitituents) behave only approximately like bosons – is no longer necessary. I TERMINOLOGY