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CRC Press eBooks

Identical Particles

Gordon Baym

Published Mar 5, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429499265-18

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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.
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.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.
Abstract “Identical particles and the helium atom” introduces bosons and fermions. Fermion states are expressed in terms of Slater determinants and the Pauli Principle. Helium is presented in such a way as to show what properties are and are not due to electron identity. Quantum states are described according as the space wave function is symmetric or antisymmetric under interchange of labels attached to the electrons. These in turn form singlet and triplet spin states when the electrons’ fermion identity is taken into account. Helium is an example of LS coupling, but a rather stunted example.
Why is it that particles with half-integer spin are Fermi particles whereas particles with integer spin are Bose particles? An explanation has been worked out by Pauli from complicated arguments from quantum field theory and relativity. He has shown that the two must necessarily go together … but we have not been able to reproduce his arguments on an elementary level. This probably means we do not have a complete understanding of the fundamental principle involved… R. P. Feynman, R.B. Leighton and M. Sands, Feynman Lectures on Physics , Volume 3, Chapter 4, Section 1, Addison-Wesley, Reading, MA (1963) Introduction Particles with half-integer angular momentum obey the Pauli exclusion principle (PEP) – a restriction that a non-degenerate single-particle quantum state can have occupation number of only 0 or 1. This restriction was announced by W. Pauli in 1924 for which, in 1945, he received the Nobel Prize in Physics. Soon after Pauli, the exclusion principle was generalized by P. Dirac and E. Fermi who – independently – integrated it into quantum mechanics. As a consequence half-integer spin particles are called Fermi–Dirac particles or fermions . PEP applies to electrons, protons, neutrons, neutrinos, quarks – and their antiparticles – as well as composite fermions such as He 3 atoms. Thermodynamic properties of metals and semiconductors are largely determined by electron (fermion) behavior.
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.
Most of the visible matter around us is made of fermions, e.g. neutrons, protons and electrons. Their properties are thus relevant to understand the behavior of matter in different situations and at very different scales. Fermions, in contrast to bosons, are described by antisymmetric wavefunctions, as explained in Chapter 1.3. This has an important consequence already for non-interacting fermionic systems. Namely, the way a set of non-interacting identical fermions builds its ground state is quite peculiar as compared to bosons and gives rise to the so-called Fermi sea.