Identical particles

Distinguishing between particles

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, 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 it is possible to 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 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.

Symmetrical and antisymmetrical states

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 identical particles. Suppose that one particle is in the state n₁, and another is in the state n₂. Intuitively, the quantum state of the system should be

This is simply the canonical way of constructing a basis for a tensor product space H ⊗ H 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, because | n 1 ⟩ | n 2 ⟩ and | n 2 ⟩ | n 1 ⟩ are two different states. Two states are physically only equivalent if they differ by a complex phase factor. Requiring this condition leads to the conclusion that the states are given by the following two possibilities:

To see this, imagine a two identical particle system. Suppose it is known that one of the particles is in state n 1 and the other is in state n 2 . Prior to the measurement, there is no way to know if particle 1 is in state n 1 and particle 2 is in state n 2 , or the other way around because the particles are indistinguishable. So, there are equal probabilities for each of the states to occur - meaning that the system is in superposition of both states prior to the measurement.

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.

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 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, P 2 = 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 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.

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 photons or helium-4 atoms, and antisymmetric states when describing electrons or protons.

Particles which exhibit symmetric states are called bosons. 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 fermions. 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 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.

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 permutations p acting on N elements. The square root left to the sum is a normalizing constant. The quantity m_n stands for the number of times each of the single-particle states n appears in the N-particle state. Note that ∑_n m_n = N.

In the same vein, fermions occupy totally antisymmetric states:

Here, sgn(p) is the sign of each permutation (i.e. + 1 if p is composed of an even number of transpositions, and − 1 if odd). Note that there is no Π n m n 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

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, ..., m_N 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 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, ..., n_N, and we perform a position measurement, the probability of finding particles in infinitesimal volumes near x₁, x₂, ..., x_N 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:

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 n_j denote the state (i.e. quantum numbers) of particle j. If the particles have the same physical properties, the n_j's 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 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, ..., n_N]. In the equation for Z, every possible permutation of the n's 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 = ξ^N, 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 = 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 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.

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 | 0 ⟩ and | 1 ⟩ , which have the same energy.

We let the composite system evolve in time, interacting with a noisy environment. Because the | 0 ⟩ and | 1 ⟩ 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: | 0 ⟩ | 0 ⟩ , | 1 ⟩ | 1 ⟩ , | 0 ⟩ | 1 ⟩ , and | 1 ⟩ | 0 ⟩ . The probability of obtaining two particles in the | 0 ⟩ state is 0.25; the probability of obtaining two particles in the | 1 ⟩ state is 0.25; and the probability of obtaining one particle in the | 0 ⟩ state and the other in the | 1 ⟩ state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: | 0 ⟩ | 0 ⟩ , | 1 ⟩ | 1 ⟩ , and 1 2 ( | 0 ⟩ | 1 ⟩ + | 1 ⟩ | 0 ⟩ ) . When we perform the experiment, the probability of obtaining two particles in the | 0 ⟩ state is now 0.33; the probability of obtaining two particles in the | 1 ⟩ state is 0.33; and the probability of obtaining one particle in the | 0 ⟩ state and the other in the | 1 ⟩ 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 1 2 ( | 0 ⟩ | 1 ⟩ − | 1 ⟩ | 0 ⟩ ) . When we perform the experiment, we inevitably find that one particle is in the | 0 ⟩ state and the other is in the | 1 ⟩ 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.

The homotopy class

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 R^d where d ≥ 3, 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.

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 M = R², 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 M = R, 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.