In classical physics, it is possible to distinguish individual particles in a system, even if they have the same mechanical properties. One might either paint each particle a unique color to distinguish it from the rest, or track the trajectory of each particle. However, this does not work for identical particles. This may be understood in the framework of quantum mechanics. Roughly speaking, the "painting" method fails because the particles are exactly specified by their quantum mechanical states, and no additional physical properties can be assigned to them. Tracking each particle is equally impossible, because the position of each particle is inherently probabilistic.
This 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. Therefore, identical particles exhibit statistical bulk behavior markedly different from classical distinguishable particles. This is further discussed below.
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2 Fermions, Bosons, Anyons and Plektons 3 Symmetrization and Antisymmetrization 4 Statistics |
We will elucidate the above statements with a little technical detail. It turns out that "identicality" is linked to a symmetry of quantum mechanical states under the interchange of particle labels. This will give rise to two types of particles which behave differently under the exchange symmetry, called fermions and bosons (there is also an unusual third type, called anyons and its generalization, plektons.) The following 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.
One exception to the above rule: in certain two-dimensional systems subjected to a strong magnetic field, 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.
There is also yet another statistic called plektons with braid statistics.
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:
Identical Particles and 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, Bosons, Anyons and Plektons
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 specy 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
ofthe position basis, so that the inner product yields the N-particle wavefunction, the inner product for fermions is called the Slater determinant.
This shows that states, although orthogonals, are not properly symmetrized in the case of bosons (ζ=-) whenever some states appear more than once (that is, if not all φi, φj are pairwise orthogonals). The proper normalized states are:
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.Statistics
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