Identical particles
Encyclopedia
Identical particles, or indistinguishable particles, are particle
s that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particle
s such as electron
s, and, with some clauses, composite particles such as atom
s and molecule
s.
There are two main categories of identical particles: boson
s, which can share quantum states, and fermion
s, which do not share quantum states due to the Pauli exclusion principle
. Examples of bosons are photon
s, gluon
s, phonon
s, and helium-4
atoms. Examples of fermions are electron
s, neutrino
s, quark
s, proton
s and neutron
s, and helium-3
atoms.
The fact that particles can be identical has important consequences in statistical mechanics
. 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 paradox
.
, electric charge
, and spin
. 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 electron
".
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 mechanics
. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunction
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.
We will now make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics
.
Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box
problem we can take n to be the quantized wave vector
of the wavefunction.) For simplicity, consider a system composed of two identical particles. 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
of the combined system 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. It turns out, for reasons ultimately based in quantum field theory
, that we must have:
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 two identical particles cannot occupy the same single-particle states. This is known as the Pauli exclusion principle
, and it is the fundamental reason behind the chemical
properties of atoms and the stability of matter
.
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 unitary
. 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 Hamiltonian
can be written in a symmetrical form, such as
It is possible to show that such Hamiltonians satisfy the commutation relation
According to the Heisenberg equation
, 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 space
.
s or helium-4
atoms, and antisymmetric states when describing electron
s or proton
s.
Particles which exhibit symmetric states are called boson
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 fermion
s. As we have seen, antisymmetry gives rise to the Pauli exclusion principle
, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by Fermi–Dirac statistics.
Parastatistics
are also possible.
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyon
s, and they obey fractional statistics. Experimental evidence for the existence of anyons exists in the fractional quantum Hall effect, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of MOSFET
s. There is another type of statistic, known as braid statistics
, which are associated with particles known as plektons.
The spin-statistics theorem
relates the exchange symmetry of identical particles to their spin
. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.
Here, the sum is taken over all different states under permutation
s p acting on N elements. The square root left to the sum is a normalizing constant
. 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 signature
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. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the Pauli exclusion principle
for many particles.
These states have been normalized so that
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.
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 function
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 wavefunction
,
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, ..., nN, 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 antisymmetric wavefunction can be written as the determinant
of a matrix
, known as a Slater determinant
:
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 function
of the system is
where k is Boltzmann's constant and T is the temperature
. We can factor
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 [n1, ..., nN]. 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 paradox
. Gibbs showed that if we use the equation Z = ξN, the entropy of a classical ideal gas
is
where V is the volume
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 thermodynamics
.
Gibbs also showed that using Z = ξN/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.
and Fermi–Dirac statistics respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the laser
, Bose–Einstein condensation
, and superfluid
ity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the Fermi gas
. 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 shells
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 entanglement
.) 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:
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 statistics and Bose–Einstein statistics
, these principles are extended to large number of particles, with qualitatively similar results.
What does this all mean?
Let's first look at the case . The universal covering space of which is none other than itself, only has two points which are physically indistinguishable from , namely itself and . So, the only permissible interchange is to swap both particles. Performing this interchange twice gives us 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 has infinitely many points that are physically indistinguishable from . This is described by the infinite cyclic group
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 anyon
ic statistics. In fact, even with two DISTINGUISHABLE particles, even though is now physically distinguishable from , 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 plektons.
The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.
Particle
A particle is, generally, a small localized object to which can be ascribed physical properties. It may also refer to:In chemistry:* Colloidal particle, part of a one-phase system of two or more components where the particles aren't individually visible.In physics:* Subatomic particle, which may be...
s that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particle
Elementary particle
In 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 electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, and, with some clauses, composite particles such as atom
Atom
The atom is a basic unit of matter that consists 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 molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s.
There are two main categories of identical particles: boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s, which can share quantum states, and fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, which do not share quantum states due to the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
. Examples of bosons are photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s, gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
s, phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s, and helium-4
Helium-4
Helium-4 is a non-radioactive 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. Alpha decay of heavy...
atoms. Examples of fermions are electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s, quark
Quark
A 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 directly...
s, proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s and neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s, and helium-3
Helium-3
Helium-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 mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. Calculations in statistical mechanics rely on probabilistic
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
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 paradox
Gibbs paradox
In statistical mechanics, a semi-classical derivation of the entropy that doesn't take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive...
.
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 massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
, and spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
. 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 electron
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
".
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 mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
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
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
.
Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
problem we can take n to be the quantized wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
of the wavefunction.) For simplicity, consider a system composed of two identical particles. 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
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
space
of the combined system 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. It turns out, for reasons ultimately based in quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, that we must have:
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 two identical particles cannot occupy the same single-particle states. This is known as the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
, and it is the fundamental reason behind the chemical
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
properties of atoms and the stability of matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies 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 asThere 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 unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
. 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 Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, 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 relation
Commutator
In 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 equation
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the 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 space
Fock space
The 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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction 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-4
Helium-4
Helium-4 is a non-radioactive 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. Alpha decay of heavy...
atoms, and antisymmetric states when describing electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s or proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s.
Particles which exhibit symmetric states are called boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
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
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
.
Particles which exhibit antisymmetric states are called fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s. As we have seen, antisymmetry gives rise to the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by Fermi–Dirac statistics.
Parastatistics
Parastatistics
In 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 anyon
Anyon
In 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 effect, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of MOSFET
MOSFET
The metal–oxide–semiconductor field-effect transistor is a transistor used for amplifying or switching electronic signals. The basic principle of this kind of transistor was first patented by Julius Edgar Lilienfeld in 1925...
s. There is another type of statistic, known as braid statistics
Braid statistics
In 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.-See also:* Braid symmetry* Parastatistics...
, which are associated with particles known as plektons.
The spin-statistics theorem
Spin-statistics theorem
In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...
relates the exchange symmetry of identical particles to their spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
. 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, ..., nN. 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 permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s p acting on N elements. The square root left to the sum is a normalizing constant
Normalizing constant
The 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 signature
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
for many particles.
These states have been normalized so that
Measurements of identical particles
Suppose we have a system of N bosons (fermions) in the symmetric (antisymmetric) stateand 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 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 function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
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 wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
,
 |  |
 | |
 |  |
 |
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, ..., nN, 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 antisymmetric wavefunction can be written as the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, known as a Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently 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 energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
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 function
Partition function (statistical mechanics)
Partition functions describe 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 temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. We can factor
Factorization
In 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 [n1, ..., nN]. 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 paradox
Gibbs paradox
In statistical mechanics, a semi-classical derivation of the entropy that doesn't take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive...
. Gibbs showed that if we use the equation Z = ξN, the entropy of a classical ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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 normal conditions such as...
is
where V is the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
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 thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
.
Gibbs also showed that using Z = ξN/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 statisticsBose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
and Fermi–Dirac statistics respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
, Bose–Einstein condensation
Bose–Einstein condensate
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
, and superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
ity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
. 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 shells
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....
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
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
.) 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:
| 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 statistics and Bose–Einstein statistics
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
, 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 the subspace with coincident points removed. describes the configuration with particle I at x and particle II at y. describes the interchanged configuration. With identical particles, the state described by ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by . Let's look at the homotopy class of continuous paths from to . 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 . The universal covering space of which is none other than itself, only has two points which are physically indistinguishable from , namely itself and . So, the only permissible interchange is to swap both particles. Performing this interchange twice gives us 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 has infinitely many points that are physically indistinguishable from . This is described by the infinite cyclic group
Cyclic group
In 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 anyon
Anyon
In 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 is now physically distinguishable from , 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 plektons.
The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.
External Links
- Exchange of Identical and Possibly Indistinguishable Particles by John S. Denker
- Identity and Individuality in Quantum Theory (Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
)