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Proceedings of the National Academy of Sciences

Bose–Einstein condensation

John M. Doyle

Published Apr 1, 1997• 17 citations
Publisher

Abstract

In quantum mechanics, elementary particles of the same type are considered identical. For example, an electron in an atom could be replaced by another electron and the atom would behave in exactly the same way. Nature has taken particular advantage of this, allowing only two types of elementary particles: bosons and fermions. They differ fundamentally in how a collection of identical particles interact. Identical fermions cannot occupy the same quantum state (only one per state!). Identical bosons can occupy the same quantum state. The statistics of identical particles (how they act in groups) has profound affects. For example, the rich and complex structure of the periodic table arises from the principle that only one electron can occupy each quantum state. The spin and the statistics of particles are closely connected: (fermions) bosons have (half-)integer spin. (Spin is an internal angular momentum measured in units of Planck’s constant.)

DOI
10.1073/pnas.94.7.2774

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In quantum mechanics, elementary particles of the same type are considered identical. For example, an electron in an atom could be replaced by another electron and the atom would behave in exactly the same way. Nature has taken particular advantage of this, allowing only two types of elementary particles: bosons and fermions. They differ fundamentally in how a collection of identical particles interact. Identical fermions cannot occupy the same quantum state (only one per state!). Identical bosons can occupy the same quantum state. The statistics of identical particles (how they act in groups) has profound affects. For example, the rich and complex structure of the periodic table arises from the principle that only one electron can occupy each quantum state. The spin and the statistics of particles are closely connected: (fermions) bosons have (half-)integer spin. (Spin is an internal angular momentum measured in units of Planck’s constant.)
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.
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.
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.
Abstract Two concepts, of fundamental importance to quantum mechanics, will be discussed in this chapter. The first is that of symmetry. Even though the concept of symmetry is familiar in a wide range of natural sciences, the way symmetry constrains the consequences of quantum mechanics is rather subtle and, at the same time, far-reaching. The second is the symmetry property of the states under the exchange of two identical particles. Identical particles with a half-integer spin (fermions) obey the Fermi–Dirac statistics: the wave functions are totally antisymmetric under the exchange of the particles. Identical particles with an integer spin (bosons) are instead subject to the Bose–Einstein statistics. Their wave functions are totally symmetric.
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).