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In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Quasiparticles also behave in this way. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by Werner Heisenberg and Paul Dirac in 1926. [1]
There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.
The fact that particles can be identical has important consequences in statistical mechanics, where calculations 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 behaviour from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing paradox.
There are two methods for distinguishing between particles. The first method relies on differences in the intrinsic physical properties of the particles, such as mass, electric charge, and spin. If differences exist, it is possible to distinguish between the particles by measuring the relevant properties. However, as far as can be determined, microscopic particles of the same species have completely equivalent physical properties.[ citation needed ] For instance, every electron has the same electric charge.
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 the position of each particle can be measured with infinite precision (even when the particles collide), then there would be no ambiguity about which particle is which.
The problem with the second 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 wavefunctions 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.
What follows is an example to 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, take n to be the quantized wave vector of the wavefunction.) For simplicity, consider a system composed of two particles that are not interacting with each other. Suppose that one particle is in the state n1, and the other is in the state n2. The quantum state of the system is denoted by the expression
where the order of the tensor product matters ( if , then the particle 1 occupies the state n2 while the particle 2 occupies the state n1). This is the canonical way of constructing a basis for a tensor product space of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since and as a result of exchanging the particles are generally different states.
Two states are physically equivalent only if they differ at most by a complex phase factor. For two indistinguishable particles, a state before the particle exchange must be physically equivalent to the state after the exchange, so these two states differ at most by a complex phase factor. This fact suggests that a state for two indistinguishable (and non-interacting) particles is given by following two possibilities: [2] [3] [4]
States where it is a sum are known as symmetric, while 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 since it cannot be normalized. In other words, more than one identical particle cannot occupy an antisymmetric state (one antisymmetric state can be occupied only by one particle). 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.
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 will be discussed later. On the other hand, it can be shown 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.
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, it can be regarded as a symmetry operator. This symmetry may be described as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces).
Clearly, (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 the earlier discussion on indistinguishability.
It will be recalled that P is Hermitian. As a result, it can be regarded as an observable of the system, which means that, in principle, a measurement can be performed 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, that eigenspace might as well be treated as the actual Hilbert space of the system. This is the idea behind the definition of Fock space.
The choice of symmetry or antisymmetry is determined by the species of particle. For example, symmetric states must always be used when describing photons or helium-4 atoms, and antisymmetric states when describing electrons or protons.
Particles which exhibit symmetric states are called bosons. 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 fermions. 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 mathematically possible, but no examples exist in nature. [5]
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyons, 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 MOSFETs. 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.
The above discussion generalizes readily to the case of N particles. Suppose there are 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 permutations p acting on N elements. The square root left to the sum is a normalizing constant. The quantity mn stands for the number of times each of the single-particle states n appears in the N-particle state. Note that Σnmn = N.
In the same vein, fermions occupy totally antisymmetric states:
Here, sgn(p) is the sign of each permutation (i.e. if is composed of an even number of transpositions, and if odd). Note that there is no 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
Suppose there is a system of N bosons (fermions) in the symmetric (antisymmetric) state
and a measurement is performed on some other set of discrete observables, m. In general, this yields 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
It can be shown that
which verifies that the total probability is 1. The sum has to be restricted to ordered values of m1, ..., mN to ensure that each multi-particle state is not counted more than once.
So far, the discussion has included only discrete observables. It can be extended 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 instead of unity:
Symmetric and antisymmetric multi-particle states can be constructed from continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:
A many-body wavefunction can be written,
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 a position measurement is performed, 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, the normalizing constant has been chosen to reflect this.
Finally, antisymmetric wavefunction can be written as the determinant of a matrix, known as a Slater determinant:
The Hilbert space for particles is given by the tensor product . The permutation group of acts on this space by permuting the entries. By definition the expectation values for an observable of indistinguishable particles should be invariant under these permutation. This means that for all and
or equivalently for each
Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent
The equivalence classes are in bijective relation with irreducible subspaces of under .
Two obvious irreducible subspaces are the one dimensional symmetric/bosonic subspace and anti-symmetric/fermionic subspace. There are however more types of irreducible subspaces. States associated with these other irreducible subspaces are called parastatistic states. [6] Young tableaux provide a way to classify all of these irreducible subspaces.
The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, 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 energy 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 the Boltzmann constant and T is the temperature. This expression can be factored 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. Thus, the number of states has been over-counted.
If the possibility of overlapping states is neglected, which is valid if the temperature is high, then the number of times each state is counted 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 in 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 N and V are doubled, 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
There are important differences between the statistical behavior of bosons and fermions, which are described by Bose–Einstein statistics 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 superfluidity. 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.
The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled and , which have the same energy.
The composite system can 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. The particle states are then measured.
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 the experiment is performed, 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 the experiment is performed, one particle is always 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, these principles are extended to large number of particles, with qualitatively similar results.
To understand why particle statistics work the way that they do, note first 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 directly, then their locations must belong to the space [M × M] \ {coincident points}, the subspace with coincident points removed. The element (x, y) describes the configuration with particle I at x and particle II at y, while (y, x) describes the interchanged configuration. With identical particles, the state described by (x, y) ought to be indistinguishable from the state described by (y, x). Now consider the homotopy class of continuous paths from (x, y) to (y, x), within the space [M × M] \ {coincident points} . If M is where d ≥ 3, then this homotopy class only has one element. If M is , 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 , then this homotopy class is empty.
Suppose first that 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. This interchange is an involution, so its only effect is to multiply the phase by a square root of 1. If the root is +1, then the points have Bose statistics, and if the root is –1, the points have Fermi statistics.
In the case the universal covering space of [M × M] ∖ {coincident points} has infinitely many points that are physically indistinguishable from (x, y). 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 recover the original state; so such an interchange can generically result in a multiplication by exp(iθ) for any real θ (by unitarity, the absolute value of the multiplication must be 1). This is called anyonic statistics. In fact, even with two distinguishable particles, even though (x, y) is now physically distinguishable from (y, x), the universal covering space still contains infinitely many points which are physically indistinguishable from the original point, now generated by a counterclockwise rotation by one full turn. This generator, then, results in a multiplication by exp(iφ). This phase factor here is called the mutual statistics.
Finally, in the case the space [M × M] ∖ {coincident points} is not connected, so even if particle I and particle II are identical, they can still be distinguished via labels such as "the particle on the left" and "the particle on the right". There is no interchange symmetry here.
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.
In quantum physics, a wave function is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ. Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung".
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles. These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics.
The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin and obey Bose-Einstein statistics or half-integer spin and obey Fermi-Dirac statistics.
In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical properties intermediate between fermions and bosons. In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons, detected by two experiments in 2020, play a major role in the fractional quantum Hall effect.
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields are thought of as field operators, in a manner similar to how the physical quantities are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons. Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. They were introduced by Paul Dirac.
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible.
In quantum mechanics, the exchange operator, also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state . Since the particles are identical, the notion of exchange symmetry requires that the exchange operator be unitary.
In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot always be predicted based on classical ideas of force. Both bosons and fermions can experience the exchange interaction.
In quantum mechanics, an antisymmetrizer is a linear operator that makes a wave function of N identical fermions antisymmetric under the exchange of the coordinates of any pair of fermions. After application of the wave function satisfies the Pauli exclusion principle. Since is a projection operator, application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect, acting as the identity operator.
In quantum mechanics, a raising or lowering operator is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-body Schrödinger equation for identical particles, the field theory is more suitable for situations where the particle number changes.
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.