2 Quantum mechanical description of identical particles

2.1 Symmetrical and antisymmetrical states

We will now make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics.

For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical Hilbert spaces. If we denote the Hilbert space of a single particle as H, then the Hilbert space of the combined system is formed by the tensor product H×H.

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.) Suppose that one particle is in the state n₁, and another is in the state n₂. What is the quantum state of the system? We might guess that it is

which is simply the canonical way of constructing a basis for a tensor product space from the individual spaces. However, this expression implies that we can identify the particle with n₁ as "particle 1" and the particle with n₂ as "particle 2", which conflicts with the ideas about indistinguishability discussed earlier.

Actually, it is an empirical fact that identical particles occupy special types of multi-particle states, called symmetric states and antisymmetric states. Symmetric states have the form

Antisymmetric states have the form

Note that if n₁ and n₂ are the same, our equation for the antisymmetric state gives the zero ket, which cannot be a state vector as it cannot be normalized. In other words, in an antisymmetric state the 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.

2.2 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 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).

The eigenvalues of P are +1 and -1, and 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.

2.3 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.

2.4 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 possible permutations p acting on N elements. The square root on the right hand side is a normalizing constant. The quantity N_j 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 Π_jN_j term, because each single-particle state can appear only once in a fermionic state.

These states have been normalized so that

2.5 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.

2.6 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 that antisymmetric wavefunction can be written as the determinant of a matrix, known as a Slater determinant:

3 Statistical properties

3.1 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_js 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 factorize 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 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 = 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.

3.2 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 by the Pauli exclusion principle from sharing quantum states, giving rise to systems such as the Fermi gas.

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 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:

Table 1: Statistics of two 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

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.

<Hide>