Quantum Statistics

FIELDS OF STUDY: Particle Physics; Quantum Physics

ABSTRACT: Quantum statistics describes the probable distribution of identical particles within a system with reference to their quantized energy states. Particles that have identical energy characteristics cannot be differentiated in an absolute manner. This makes the probability that they occupy a certain energy level an important aspect of their distribution functions.

PRINCIPAL TERMS

• Boltzmann factor: the ratio of the probability of finding a particle or system at a certain energy level to the probability of finding it at another energy level; proportional to the probability of the system being in a particular quantum state.
• boson: an elementary particle that carries a specific type of force, or a composite particle containing an even number of fermions, having an integer spin.
• bra-ket notation: a system of mathematical notation developed by physicist Paul Dirac to manipulate very large vector equations more easily.
• distinguishability: the ability of particles with the same energy to be differentiated from one another.
• distribution function: a mathematical function that describes the probability that a certain variable, such as the energy state of a particle, will have a given value or range of values.
• fermion: an elementary particle, either a quark or a lepton, that is a fundamental unit of matter, or a composite particle containing an odd number of other fermions, and that has a half-integer spin.
• phase space: a space containing all possible states of the particles in a given system, wherein each state is represented by a single point.
• quantum state: the particular condition of a physical system as defined by its various quantum attributes.

Particles and Probability

In a system containing a large number of particles, the average overall behavior of the system may be known. However, it is not possible to know for certain the exact state of each particle. In order to understand the behavior of the system on a macroscopic scale, the behavior of the various particles can be described in terms of probability. Each possible microscopic state, or microstate, of the system is represented as a single point in a multidimensional abstract space known as phase space. A distribution function can then be used to calculate which possible states correspond to the observed macroscopic state, or macrostate, and thus are most likely to be true. These probable states make up what is known as a "statistical ensemble."

The disconnect between microstates and macrostates can be illustrated using the example of an inflated balloon. At a constant temperature, the total energy of all the air molecules in the balloon can be known, as can the average energy of each molecule. However, because the air a person exhales, and thus the air within the balloon, consists of different molecules with different masses, the individual molecules will in fact have different energies that correspond to their respective momenta and positions. Collisions between molecules can impart greater momentum, and therefore greater energy, to one molecule while decreasing the momentum and energy of the other. Thus, while the total energy of the system remains constant, it is the sum of the different energies possessed by the air molecules within the balloon. The actual distribution of energies among the different molecules would be difficult, if not impossible, to determine. Despite this, the most probable distributions can be calculated given what is known about the system.

This use of probability theory to determine probable microstates of macroscopic systems (or to extrapolate macroscopic behavior from microstates) is called "statistical mechanics." The rules that govern particle behavior in a system are known as "statistics." In a classical mechanical system, where quantum effects are negligible, the system can be described using Maxwell-Boltzmann statistics, named after Scottish physicist James Clerk Maxwell (1831–79) and Austrian physicist Ludwig Boltzmann (1844–1906).

Below a certain level of complexity, however, quantum effects must be taken into account. This is particularly true with elementary and other subatomic particles. According to the uncertainty principle, proposed by German physicist Werner Heisenberg (1901–76), one can know either the position or the momentum of a quantum particle, but not both at once. Thus, the behavior of particles in the quantum realm is not readily describable by classical mechanics. Instead, such particles are best described by a wave function, which can be used to determine the probability that a particle will be in a given position (or, if its position is known, have a given momentum) at a particular time. Quantum particles cannot be described using Maxwell-Boltzmann statistics. Instead, they must be described using one of two systems of quantum statistics: Bose-Einstein statistics or Fermi-Dirac statistics.

Distribution Functions

Particles in a system may be distributed among various possible quantum states, and the corresponding energy values of those states, in many different ways. Each potential distribution forms a microstate. While these microstates contain a massive amount of information, the most important is how many particles are in a given quantum state (and therefore at a given energy level). In this respect, "quantum state" and "energy level" mean the same thing.

In classical mechanics, all particles have distinguishability. Even particles of the same type with identical intrinsic properties, such as mass and spin, can be distinguished based on their trajectory or other aspects of their behavior. As a result, exchanging any two particles in a classical system causes changes to the system. In such a system, the distribution of identical (but distinguishable) particles among different energy states is described by the Maxwell-Boltzmann distribution function

where f(E) is the probability that a given particle has the energy E, A is a normalization constant, e is Euler’s number, k is the Boltzmann constant (roughly 1.38 joules per kelvin), and T is the absolute temperature of the system. This function can also be written as follows:

f(E) = Ae⁻E^/kT

Here, the element "e⁻E^/kT" is the Boltzmann factor of the system. It is proportional to the probability of finding the system in a given microstate of energy E.

In quantum mechanics, particles do not have distinguishability. Particles are classified as either bosons, which have whole-integer spins, or fermions, which have half-integer spins. The primary difference between them, aside from their spins, is that no two identical fermions can occupy the same quantum state at the same time. This principle, known as the "Pauli exclusion principle," does not apply to bosons. In a quantum mechanical system, bosons cannot be distinguished from other bosons, and fermions cannot be distinguished from other fermions. Thus, the Maxwell-Boltzmann distribution does not apply.

Bosons are so named because they follow Bose-Einstein statistics, named for Bengali physicist Satyendra Nath Bose (1894–1974) and German-born physicist Albert Einstein (1879–1955). The energy distribution of bosons is described by the Bose-Einstein distribution function:

While very similar to the first form of the Maxwell-Boltzmann distribution function, this function subtracts 1 from the denominator to account for particle indistinguishability. As bosons are not subject to the Pauli exclusion principle, this function allows an unlimited number of bosons to occupy the same energy level.

Like bosons, fermions got their name from the statistics they follow. Fermi-Dirac statistics was named for Italian physicist Enrico Fermi (1901–54) and English physicist Paul Dirac (1902–84). In some respects, fermions may be thought of as the opposite of bosons. This is reflected in the Fermi-Dirac energy distribution function:

This function is almost identical to the Bose-Einstein function, with the only difference being that the 1 is added, not subtracted. Because fermions are subject to the Pauli exclusion principle, this function only permits one fermion to occupy each energy level.

Probability and Distinguishability

In classical systems, identical particles are distinguishable by their position and momentum. This normally requires a frame of reference or a coordinate system in which the positions and movements of particles can be described as vectors. A vector is a quantity that describes both the magnitude of a property and its direction.

It is possible to perform mathematical operations on such classical systems independently of the origin point of any coordinate system. The equations for such operations state the relationship between a set of operators and the vector characteristics on which they operate. This enables vector operations within a system to be described in a relative sense rather than an absolute sense. In other words, the effect of an operation on a vector property can be determined regardless of where that property is located within the overall system. Though such mathematical relations are highly complex, they can be greatly simplified using bra-ket notation, introduced by Dirac.

Bra-ket notation is also commonly applied to quantum systems, although a number of restrictions arise. In matrix operations, for example, only certain values are allowed. Such values, called "eigenvalues," produce an answer that is simply a multiple of the original vector. This relationship is what determines the opposite designations of bosons and fermions in mathematical expressions and limits elementary fermions to spin values of +1/2 and −1/2. (Composite fermions can also have spin values of +3/2 or −3/2.) It is also what prevents two identical fermions from occupying the same energy level.

It is important to understand how the number of particles in different energy levels relates to observable physical properties, such as absorption and emission spectra. Such spectra reflect the transitions of electrons (a type of lepton, or elementary fermion) from one allowed energy level to another. Knowing the probability that electrons of a certain energy occupy certain allowed energy levels enables one to predict whether a specific energy transition will be observed in a system at a given temperature or energy level. Similar considerations are important in the study of subatomic particle interactions during high-energy nuclear collision experiments.

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