Identical particles, or
indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include
elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s such as
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s, as well as composite microscopic particles such as
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s and
moleculeA molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense...
s.
There are two main categories of identical particles:
bosonIn particle physics, bosons are particles which obey Bose–Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein. In contrast to fermions, which obey Fermi-Dirac statistics, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the...
s, which can share
quantum stateIn quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be...
s, and
fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s, which are forbidden from sharing quantum states (this property of fermions is known as the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
.) Examples of bosons are
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s,
gluonGluons are elementary expressions of quark interaction, and are indirectly involved with the binding of protons and neutrons together in atomic nuclei...
s,
phononIn physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's...
s, and
helium-4Helium-4 is a non-radioactive and light isotope of helium. It is by far the most abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on earth. Its nucleus is the same as an alpha particle, consisting of two protons and two neutrons. The total spin of...
atoms. Examples of fermions are
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s,
neutrinoNeutrinos are elementary particles that often travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect. Neutrinos have a minuscule, but nonzero mass...
s,
quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never found in...
s,
protonThe proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H
+...
s and
neutronThe neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutron are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a...
s, and
helium-3Helium-3 is a light, non-radioactive isotope of helium with two protons and one neutron. It is rare on Earth, and is sought for use in nuclear fusion research...
atoms.
The fact that particles can be identical has important consequences in
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
. Calculations in statistical mechanics rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behavior from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs'
mixing paradoxOriginally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances, the Gibbs paradox applies to thermodynamics. It involves the discontinuous nature of the entropy of mixing...
.
Distinguishing between particles
There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
,
electric chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
, and
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
. If differences exist, we can distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why we can speak of such a thing as "
the charge of the electronThe elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. This is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the "elementary positive...
".
Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as we can measure the position of each particle with infinite precision (even when the particles collide), there would be no ambiguity about which particle is which.
The problem with this approach is that it contradicts the principles of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by
wavefunctionA wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics describe how the wave function evolves over time...
s that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be
indistinguishable.
Symmetrical and antisymmetrical states
We will now make the above discussion concrete, using the formalism developed in the article on the
mathematical formulation of quantum mechanicsThe mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such...
.
For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s. If we denote the Hilbert space of a single particle as
H, then the Hilbert space of the combined system is formed by the
tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this...
.
Let
n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the
particle in a boxIn quantum mechanics, the particle in a box model describes a particle, which is free to move in a space surrounded by impenetrable barriers...
problem we can take
n to be the quantized
wave vectorA wave vector is a vector representation of a wave. The wave vector has magnitude indicating wavenumber , and the direction of the vector indicates the direction of wave propagation....
of the wavefunction.) Suppose that one particle is in the state
n1, and another is in the state
n2. What is the quantum state of the system? Intuitively, it should be
which is simply the canonical way of constructing a basis for a tensor product space from the individual spaces. However, this expression implies the ability to identify the particle with
n1 as "particle 1" and the particle with
n2 as "particle 2". If the particles are indistinguishable, this is impossible by definition; either particle can be in either state:
States where this is a sum are known as
symmetric; states involving the difference are called
antisymmetric. More completely, symmetric states have the form
while antisymmetric states have the form
Note that if
n1 and
n2 are the same, the antisymmetric expression gives zero, which cannot be a state vector as it cannot be normalized. In other words, in an antisymmetric state the particles cannot occupy the same single-particle states. This is known as the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
, and it is the fundamental reason behind the
chemicalChemistry is the science concerned with the composition, behavior, structure, and properties of matter, as well as the changes it undergoes during chemical reactions...
properties of atoms and the stability of
matterThe term matter traditionally refers to the substance that all objects are made of. One common way to identify this "substance" is through its physical properties; a common definition of matter is anything that has mass and occupies a volume...
.
Exchange symmetry
The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. It appears to be a fact of nature that identical particles do not occupy states of a mixed symmetry, such as
There is actually an exception to this rule, which we will discuss later. On the other hand, we can show that the symmetric and antisymmetric states are in a sense special, by examining a particular symmetry of the multiple-particle states known as
exchange symmetry.
Let us define a linear operator
P, called the exchange operator. When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors:
P is both Hermitian and
unitaryIn functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfying...
. Because it is unitary, we can regard it as a symmetry operator. We can describe this symmetry as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces).
Clearly,
P² = 1 (the identity operator), so the eigenvalues of
P are +1 and −1. The corresponding eigenvectors are the symmetric and antisymmetric states:
In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of +1 or −1, rather than being "rotated" somewhere else in the Hilbert space. This indicates that the particle labels have no physical meaning, in agreement with our earlier discussion on indistinguishability.
We have mentioned that
P is Hermitian. As a result, it can be regarded as an observable of the system, which means that we can, in principle, perform a measurement to find out if a state is symmetric or antisymmetric. Furthermore, the equivalence of the particles indicates that the
HamiltonianIn quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
can be written in a symmetrical form, such as
It is possible to show that such Hamiltonians satisfy the
commutation relationIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
According to the
Heisenberg equationIn physics, the Heisenberg picture is that formulation of quantum mechanics where the operators are time-dependent and the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which operators are constant and the states evolve in time...
, this means that the value of
P is a constant of motion. If the quantum state is initially symmetric (antisymmetric), it will remain symmetric (antisymmetric) as the system evolves. Mathematically, this says that the state vector is confined to one of the two eigenspaces of
P, and is not allowed to range over the entire Hilbert space. Thus, we might as well treat that eigenspace as the actual Hilbert space of the system. This is the idea behind the definition of
Fock spaceThe Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
.
Fermions and bosons
The choice of symmetry or antisymmetry is determined by the species of particle. For example, we must always use symmetric states when describing
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s or
helium-4Helium is the chemical element with atomic number 2, and is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert monatomic gas that heads the noble gas group in the periodic table...
atoms, and antisymmetric states when describing
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s or
protonThe proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H
+...
s.
Particles which exhibit symmetric states are called
bosonIn particle physics, bosons are particles which obey Bose–Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein. In contrast to fermions, which obey Fermi-Dirac statistics, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the...
s. As we will see, the nature of symmetric states has important consequences for the statistical properties of systems composed of many identical bosons. These statistical properties are described as Bose-Einstein statistics.
Particles which exhibit antisymmetric states are called
fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s. As we have seen, antisymmetry gives rise to the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
.
ParastatisticsIn quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models...
are also possible.
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as
anyonIn mathematics and physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
s, and they obey fractional statistics. Experimental evidence for the existence of anyons exists in the
fractional quantum Hall effectThe quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductivity σ takes on the quantized valueswhere e is the elementary charge and h is Planck's constant...
, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of
MOSFETThe metal–oxide–semiconductor field-effect transistor is a device used to amplify or switch electronic signals. The basic principle of the device was first proposed by Julius Edgar Lilienfeld in 1925...
s. There is another type of statistic, known as
braid statisticsIn mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group. A similar notion exists using a loop braid group....
, which are associated with particles known as
plektonIn physics, a plekton is a theoretical kind of elementary particle, which obeys a different style of statistics with respect to the interchange of identical particles. That is, it would be neither a boson nor a fermion, but subject to a braid statistics. Such particles have been discussed as a...
s.
The
spin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics obeyed by it. The spin of a particle is its intrinsic angular momentum...
relates the exchange symmetry of identical particles to their
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.
N particles
The above discussion generalizes readily to the case of
N particles. Suppose we have
N particles with quantum numbers
n1,
n2, ..., n
N. If the particles are bosons, they occupy a
totally symmetric state, which is symmetric under the exchange of
any two particle labels:
Here, the sum is taken over all different states under
permutationIn several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging elements of a set.- Definitions :The general concept of permutation can be...
s
p acting on
N elements (the resulting duplicate states will appear only once). The square root on the right hand side is a
normalizing constantThe concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...
. The quantity
Nj stands for the number of times each of the single-particle states appears in the
N-particle state.
In the same vein, fermions occupy
totally antisymmetric states:
Here, sgn(
p) is the
signatureIn mathematics, the symmetric group on a set is the group consisting of all automorphisms of the set with function composition as the group operation....
of each permutation (i.e. +1 if
p is composed of an even number of transpositions, and −1 if odd.) Note that we have omitted the
ΠjNj term, because each single-particle state can appear only once in a fermionic state.
These states have been normalized so that
Measurements of identical particles
Suppose we have a system of
N bosons (fermions) in the symmetric (antisymmetric) state
and we perform a measurement of some other set of discrete observables,
m. In general, this would yield some result
m1 for one particle,
m2 for another particle, and so forth. If the particles are bosons (fermions), the state after the measurement must remain symmetric (antisymmetric), i.e.
The probability of obtaining a particular result for the
m measurement is
We can show that
which verifies that the total probability is 1. Note that we have to restrict the sum to
ordered values of
m1, ...,
mN to ensure that we do not count each multi-particle state more than once.
Wavefunction representation
So far, we have worked with discrete observables. We will now extend the discussion to continuous observables, such as the
positionPosition may refer to:* A location in a coordinate system, usually in two or more dimensions; the science of position and its generalizations is topology* Position...
x.
Recall that an eigenstate of a continuous observable represents an infinitesimal
range of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |ψ>, the probability of finding it in a region of volume
d³x surrounding some position
x is
As a result, the continuous eigenstates |
x> are normalized to the
delta functionDelta function may mean:* Dirac delta function, * Kronecker delta,...
instead of unity:
We can construct symmetric and antisymmetric multi-particle states out of continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:
We can then write a many-body
wavefunctionA wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics describe how the wave function evolves over time...
,
where the single-particle wavefunctions are defined, as usual, by
The most important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign. This is the manifestation of symmetry and antisymmetry in the wavefunction representation:
The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers
n1, ..., n
N, and we perform a position measurement, the probability of finding particles in infinitesimal volumes near
x1,
x2, ...,
xN is
The factor of
N! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,
Because each integral runs over all possible values of
x, each multi-particle state appears
N! times in the integral. In other words, the probability associated with each event is evenly distributed across
N! equivalent points in the integral space. Because it is usually more convenient to work with unrestricted integrals than restricted ones, we have chosen our normalizing constant to reflect this.
Finally, it is interesting to note that that antisymmetric wavefunction can be written as the
determinantIn algebra, the determinant is a special number associated to any square matrix, that is to say, a rectangular array of numbers where the number of rows and columns are equal. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear...
of a
matrixIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
, known as a
Slater determinantIn quantum mechanics, a Slater determinant is an expression which describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and subsequently the Pauli exclusion principle by changing sign upon exchange of fermions. It is named for its discoverer, John C...
:
Statistical effects of indistinguishability
The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, let us consider a system of
N distinguishable, non-interacting particles. Once again, let
nj denote the state (i.e. quantum numbers) of particle
j. If the particles have the same physical properties, the
njs run over the same range of values. Let ε
(n
) denote the energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
of a particle in state n
. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition functionIn statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
of the system is
where k
is Boltzmann's constant and T
is the temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
. We can factorizeIn mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
this expression to obtain
where
If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states [n
1, ..., n
N]. In the equation for Z
, every possible permutation of the ns occurs once in the sum, even though each of these permutations is describing the same multi-particle state. We have thus over-counted the actual number of states.
If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately
N!. The correct partition function is
Note that this "high temperature" approximation does not distinguish between fermions and bosons.
The discrepancy in the partition functions of distinguishable and indistinguishable particles was known as far back as the 19th century, before the advent of quantum mechanics. It leads to a difficulty known as the
Gibbs paradoxOriginally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances, the Gibbs paradox applies to thermodynamics. It involves the discontinuous nature of the entropy of mixing...
. Gibbs showed that if we use the equation
Z = ξN, the entropy of a classical
ideal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving point particles that interact only through elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At...
is
where
V is the
volumeThe volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space...
of the gas and
f is some function of
T alone. The problem with this result is that
S is not extensive - if we double
N and
V,
S does not double accordingly. Such a system does not obey the postulates of
thermodynamicsIn physics, thermodynamics is the study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature, volume and pressure...
.
Gibbs also showed that using
Z = ξ
N/
N! alters the result to
which is perfectly extensive. However, the reason for this correction to the partition function remained obscure until the discovery of quantum mechanics.
Statistical properties of bosons and fermions
There are important differences between the statistical behavior of bosons and fermions, which are described by Bose-Einstein statistics and
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the
laserA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
, Bose-Einstein condensation, and
superfluidSuperfluidity is a phase of matter or description of heat capacity in which unusual effects are observed when liquids, typically of helium-4 or helium-3, overcome friction by surface interaction when at a stage at which the liquid's viscosity becomes zero...
ity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the
Fermi gasFermi gas is a physical model assuming a collection of non-interacting fermions. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles. The behavior of Electrons in metals and semiconductors and neutrons in a neutron star can be approximated by treating them as...
. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within
shellsAn electron shell may be thought of as an orbit followed by electrons around an atom nucleus. Because each shell can contain only a fixed number of electrons, each shell is associated with a particular range of electron energy, and thus...
rather than all lying in the same lowest energy state.
We can illustrate the differences between the statistical behavior of fermions, bosons, and distinguishable particles using a system of two particles. Let us call the particles A and B. Each particle can exist in two possible states, labelled and , which have the same energy.
We let the composite system evolve in time, interacting with a noisy environment. Because the and states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on
quantum entanglementQuantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full...
.) After some time, the composite system will have an equal probability of occupying each of the states available to it. We then measure the particle states.
If A and B are distinguishable particles, then the composite system has four distinct states: , , , and . The probability of obtaining two particles in the state is 0.25; the probability of obtaining two particles in the state is 0.25; and the probability of obtaining one particle in the state and the other in the state is 0.5.
If A and B are identical bosons, then the composite system has only three distinct states: , , and . When we perform the experiment, the probability of obtaining two particles in the state is now 0.33; the probability of obtaining two particles in the state is 0.33; and the probability of obtaining one particle in the state and the other in the state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump."
If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state . When we perform the experiment, we inevitably find that one particle is in the state and the other is in the state.
The results are summarized in Table 1:
Table 1: Statistics of two particles | Particles | Both 0 | Both 1 | One 0 and one 1 |
| Distinguishable | 0.25 | 0.25 | 0.5 |
| Bosons | 0.33 | 0.33 | 0.33 |
| Fermions | 0 | 0 | 1 |
As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
and Bose-Einstein statistics, these principles are extended to large number of particles, with qualitatively similar results.
The homotopy class
To understand why we have the statistics that we do for particles, we first have to note that particles are point localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact (at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower), then we are dealing with the space [M × M]/{coincident points}, the subspace with coincident points removed. (x,y) describes the configuration with particle I at x and particle II at y. (y,x) describes the interchanged configuration. With identical particles, the state described by (x,y) ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by (y,x). Let's look at the homotopy class of continuous paths from (x,y) to (y,x). If M is
Rd where , then this homotopy class only has one element. If M is
R2, then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc, a clockwise interchange by half a turn, etc). In particular, a counterclockwise interchange by half a turn is NOT homotopic to a clockwise interchange by half a turn. Lastly, if M is
R, then this homotopy class is empty. Obviously, if M is not isomorphic to
Rd, we can have more complicated homotopy classes...
What does this all mean?
Let's first look at the case
d 3. The universal covering space of [
M ×
M]/{coincident points}, which is none other than [
M ×
M]/{coincident points} itself, only has two points which are physically indistinguishable from (
x,
y), namely (
x,
y) itself and (
y,
x). So, the only permissible interchange is to swap both particles. Performing this interchange twice gives us (
x,
y) back again. If this interchange results in a multiplication by +1, then we have Bose statistics and if this interchange results in a multiplication by −1, we have Fermi statistics.
Now how about
R2? The universal covering space of [M × M]/{coincident points} has infinitely many points which are physically indistinguishable from (x,y). This is described by the infinite
cyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not lead us back to the original state. So, such an interchange can generically result in a multiplication by exp(iθ) (its absolute value is 1 because of unitarity...). This is called
anyonIn mathematics and physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
ic statistics. In fact, even with two DISTINGUISHABLE particles, even though (
x,
y) is now physically distinguishable from (
y,
x), if we go over to the universal covering space, we still end up with infinitely many points which are physically indistinguishable from the original point and the interchanges are generated by a counterclockwise rotation by one full turn which results in a multiplication by exp(
iφ). This phase factor here is called the mutual statistics.
As for
R, even if particle I and particle II are identical, we can always distinguish between them by the labels "the particle on the left" and "the particle on the right". There is no interchange symmetry here and such particles are called
plektonIn physics, a plekton is a theoretical kind of elementary particle, which obeys a different style of statistics with respect to the interchange of identical particles. That is, it would be neither a boson nor a fermion, but subject to a braid statistics. Such particles have been discussed as a...
s.
The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.