Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, as well as composite microscopic particles such as atoms.There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which are forbidden from sharing quantum states (this property of fermions is known as the Pauli exclusion principle.) Examples of bosons are photons, gluons, phonons, and helium-4 atoms. Examples of fermions are electrons, neutrinos, quarks, protons and neutrons, 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.
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2 Quantum mechanical description of identical particles 3 Statistical properties of identical 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 mass, 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".
A second method for distinguishing between particles is to track the trajectory of each particle, and this ought to work even if the particles are equivalent. As long as we can measure the position of each particle with infinite precision (especially 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, between each measurement performed on the particles, the position of each particle is not definite, but is given by a probabilistic wavefunction. As time passes, the wavefunctions tend to spread out and overlap one another. 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 thus said to be indistinguishable.
The argument given above indicates that it is incorrect to regard particles as individual entities if they have equivalent physical properties, and if quantum effects are important. This can be made concrete by studying the symmetry of quantum states under the interchange of particle labels, or "exchange symmetry". The following discussion relies on formalism developed in the article mathematical formulation of quantum mechanics.
Consider a system with two identical particles. Suppose the state vector of one particle is |ψ>, and the state vector of the other particle is |ψ′>. Let us represent the state of the combined system, which is some unspecified combination of the single-particle states, by
In the above discussion, we did not prove that total symmetric or antisymmetric states are the only way to satisfy exchange symmetry. However, it is an empirical fact that particles in Nature have quantum states that are either totally symmetric or totally antisymmetric, with a single minor exception that will be discussed later. Furthermore, the choice of symmetry or antisymmetry is determined entirely by a particle's species. For example, photons always form totally symmetric states, and electrons always form totally antisymmetric ones.
Particles which exhibit totally antisymmetric states are called fermions. Total antisymmetry gives rise to the Pauli exclusion principle, which forbids identical fermions from sharing the same quantum state; this is the reason for the stability of matter. The Pauli exclusion principle leads to Fermi-Dirac statistics, which describes systems of many identical fermions.
Particles which exhibit totally symmetric states are called bosons. Unlike fermions, identical bosons can share quantum states. Because of this, systems of many identical bosons are described by Bose-Einstein statistics. This gives rise to such varied phenomena as the laser, Bose-Einstein condensation, and superfluidity.
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyons, and obey fractional statistics. This phenomenon has been observed in the two-dimensional electron gases that form the inversion layer of MOSFETs, when subjected to intense magnetic fields. There is also yet another statistic called braid statistics, with corresponding particles called plektons, which are a mathematical generalization of anyons.
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.
Earlier, we stated that the two-particle state |ψψ′> is some combination of the single-particle states |ψ> and |ψ′>. However, we have not stated what the combination is. The natural guess is (since it is the canonical way to define the basis of the Hilbert space of two particles from one-particle states; in what follows we assume all states refer to some basis)
The procedure of symmetrization readily generalizes to the case of N particles. Suppose we have N single-particle states |ψ1>, |ψ2>, ..., |ψN>. If the particles are bosons, the multi-particle state is
The inner product of two symmetrized states is given by, as can be checked by explicit computation:
where ni is the number of times the state ψi appear in the N-particle state. This precaution is unnecessary for fermions since in this case all ni! are always unity. With this latter definition, one has for the closure relation (also valid for fermions):
where the sum extends other all basis states of the N-particles Hilbert space.
Earlier, we noted that distinguishable particles, fermions, and bosons give rise to different statistics. This can be demonstrated using a toy model of two particles. (We do not consider anyons.)
Suppose we have a composite system consisting of two 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 2-1/2(|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 2-1/2(|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.Distinguishing between particles
Quantum mechanical description of identical particles
Exchange symmetry
If the particles are identical, then (i) their state vectors occupy mathematically identical Hilbert spaces, and (ii) |ψψ′> and |ψ′ ψ> must have equal probability to collapse to any other multi-particle state |φ>:
This property is referred to as exchange symmetry. One way of satisfying this symmetry is for permutation to introduce only a phase:
However, two permutations are the identity, so we require e2iα = 1. Then either
which is called a totally symmetric state, or
which is called a totally antisymmetric state.Fermions and bosons
Symmetrization and antisymmetrization
One can readily verify that this choice is generally neither totally symmetric nor totally antisymmetric. To satisfy these conditions, we must construct the multi-particle state more carefully. For bosons,
and for fermions,
This method of constructing multi-particle states from single-particle states is referred to as symmetrization (for bosons) and antisymmetrization (for fermions.) Since particles of a given species always stick to the same symmetrization requirement, one can work with a Hilbert space properly symmetrized, i.e., which states |ψ1ψ2...ψN>ζ all obey either bosonic (ζ=+) or fermionic (ζ=-) statistics. Such a space is called a Fock space.
and for fermions,
Here, the sum is taken over all permutations p acting on N elements, and 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.)
where is either the determinant (ζ=-) or the permanent (ζ=+). In the case where the state is dotted with
of
the position basis, so that the inner product yields the N-particle wavefunction, the inner product for fermions is called the Slater determinant.Statistical properties of identical particles
Particles
Both 0
Both 1
One 0 and one 1
Distinguishable
0.25
0.25
0.5
Bosons
0.33
0.33
0.33
Fermions
0
0
1