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Bosons and Fermions: Indistinguishable Particles with Intrinsic Spin

Duncan G. Steel

Published Apr 30, 2021• 0 citations
Publisher

Abstract

Abstract If we imagine a Hamiltonian, H^(r1,r2), describing two identical particles at positions r1 and r2 and then we interchange the particles, the Hamiltonian will be unaffected, i.e. H^(r1,r2)=H^(r2,r1). If we introduce an exchange operator P^r1,r2 such that P^r1,r2H^(r1,r2)=H^(r2,r1)P^r1,r2=H^(r1,r2)P^r1,r2, we see that they commute, or [P^r1,r2,H^(r1,r2)]=0. We know then that P^r1,r2andH^(r1,r2) have common eigenfunctions. We can then easily show that the eigenfunctions of the exchange operator must be either even or odd. Experiments show that odd exchange symmetry corresponds to half-integer spin particles called fermions, while even exchange symmetry corresponds to integer spin particles called bosons. The notes then discuss the implications of the new postulate and then presents the Heitler–London theory and the Heisenberg exchange Hamiltonian which has been so successful in predicting molecular structure.

DOI
10.1093/oso/9780192895073.003.0012

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•Oct 11, 2018•0 citations
Abstract If we imagine a Hamiltonian, H^(r1,r2), describing two identical particles at positions r1 and r2 and then we interchange the particles, the Hamiltonian will be unaffected, i.e. H^(r1,r2)=H^(r2,r1). If we introduce an exchange operator P^r1,r2 such that P^r1,r2H^(r1,r2)=H^(r2,r1)P^r1,r2=H^(r1,r2)P^r1,r2, we see that they commute, or [P^r1,r2,H^(r1,r2)]=0. We know then that P^r1,r2andH^(r1,r2) have common eigenfunctions. We can then easily show that the eigenfunctions of the exchange operator must be either even or odd. Experiments show that odd exchange symmetry corresponds to half-integer spin particles called fermions, while even exchange symmetry corresponds to integer spin particles called bosons. The notes then discuss the implications of the new postulate and then presents the Heitler–London theory and the Heisenberg exchange Hamiltonian which has been so successful in predicting molecular structure.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
Two particles are said to be identical if all their intrinsic properties, e.g., mass, electrical charge, spin, colour are exactly the same. Similarly, all protons are identical to one another, and so are all quarks. They possess the same defining properties under similar interactions with each other. The exchange in position of identical particles has a so-called permutation symmetry. Clearly, this permutation symmetry is extremely important for quantum mechanical calculations of the multi-indistinguishable particles. So knowing how a collection of fermions (or bosons) behave is kinematically determined by their permutation symmetry. If we have different cycles in a permutation and there is no element common to any two different cycles, then all these cycles are termed disjoint. In fact any permutation of a finite set is the product of disjoint cycles. A cycle of length two is called a transposition.
For a quantum mechanical system of N particles, the Hamiltonian is a sum of the total kinetic energy and the potential energy. In classical physics, it is possible to keep track of individual particles. A system of N identical particles is said to satisfy Bose-Einstein statistics if its wave function is symmetric under the exchange of particles, and these particles are known as bosons. In the nonrelativistic quantum mechanics, this principle must be accepted as an empirical postulate. In the relativistic quantum theory, it can be proven that half-integer-spin particles cannot be bosons and integer-spin particles cannot be fermions. The Pauli exclusion principle plays a critical role in programming physical properties of electrons or more generally fermions.
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions in all cases. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate (Dirac P 1926 Proc. R. Soc. A 112 661 , Heisenberg W 1926 Z. Phys. 38 411 ), which restricts a system of identical particles to the set of symmetric states ('bosons') or the set of antisymmetric states ('fermions'). However, the physical basis and range of validity of the symmetrization postulate has not been established. A well-known topological derivation of the postulate implies that its validity depends on the dimensionality of the space in which the particles move (Laidlaw M and DeWitt C 1971 Phys. Rev. D 3 1375–8 , Leinaas J M and Myrheim J 1977 Il Nuovo Cimento B 37 1–23 ). Here we show that the symmetrization postulate can be derived by strictly adhering to the informational requirement that particles which cannot be experimentally distinguished from one another are not labelled. Our key novel postulate is the operational indistinguishability postulate, which posits that the amplitude of a process involving several indistinguishable particles is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behavior in two dimensions is excluded.
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor (−1)2s(-1)^{2s}(−1)2s, which is a two-particle effect.