Density functional theory/Pauli exclusion principle

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Overview[edit]

The Pauli exclusion principle governs the behavior of all fermions (particles with "half-integer spin"), while bosons (particles with "integer spin") are not subject to it. Fermions include elementary particles such as quarks (the constituent particles of protons and neutrons), electrons and neutrinos. In addition, protons and neutrons (subatomic particles composed from three quarks) and some atoms are fermions, and are therefore subject to the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example helium-3 has spin 1/2 and is therefore a fermion, in contrast to helium-4 which has spin 0 and is a boson. As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms.

"Half-integer spin" means that the intrinsic angular momentum value of fermions is \hbar = h/2\pi (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, and the W and Z bosons. (Fermions take their name from the Fermi–Dirac statistical distribution that they obey, and bosons from their Bose–Einstein distribution).

Connection to quantum state symmetry[edit]

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state \scriptstyle |x \rangle and the other in state \scriptstyle |y\rangle:


|\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle

and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies A(x,y) = 0 when x=y, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric rank-two tensor.

Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component:


A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle )

is necessarily antisymmetric.

Identical particles[edit]

Identical particles, also called indistinguishable 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 are "tiny", there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability; see particle statistics #Quantum statistics for detailed explication.

There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which do not share quantum states due to the Pauli exclusion principle. Examples of bosons are photons, gluons, phonons, and helium-4 nuclei. 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. 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. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing paradox.

Distinguishing between particles[edit]

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".

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 one can measure the position of each particle with infinite precision (even when the particles collide), then 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, 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.

Quantum mechanical description of identical particles[edit]

Symmetrical and anti-symmetrical states[edit]

Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential.
Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential.

We will now 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 we can 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. What is the quantum state of the system? Intuitively, it should be

 | n_1 \rang | n_2 \rang

which is simply the canonical way of constructing a basis for a tensor product space H \otimes 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; either particle can be in either state. It turns out that we must have:[1]

 |n_1\rang |n_2\rang \pm |n_2\rang |n_1\rang

to see this, imagine a two identical particle system. suppose we know 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. and 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

 |n_1, n_2; S\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang + |n_2\rang |n_1\rang \bigg)

while antisymmetric states have the form

 |n_1, n_2; A\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang - |n_2\rang |n_1\rang \bigg)

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[edit]

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

 |n_1, n_2; ?\rang = \mbox{constant} \times \bigg( |n_1\rang |n_2\rang + i |n_2\rang |n_1\rang \bigg)

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 \bigg(|\psi\rang |\phi\rang \bigg) \equiv |\phi\rang |\psi\rang

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).

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:

P|n_1, n_2; S\rang = + |n_1, n_2; S\rang
P|n_1, n_2; A\rang = - |n_1, n_2; A\rang

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

H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m} + U(|x_1 - x_2|) + V(x_1) + V(x_2)

It is possible to show that such Hamiltonians satisfy the commutation relation

\left[P, H\right] = 0

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.

Fermions and bosons[edit]

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[edit]

The above discussion generalizes readily to the case of N particles. Suppose we have 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:

|n_1 n_2 \cdots n_N; S\rang = \sqrt{\frac{\prod_j n_j!}{N!}} \sum_p^{N!} |n_{(p,1)}\rang |n_{(p,2)}\rang \cdots |n_{(p,N)}\rang

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 nj 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:

|n_1 n_2 \cdots n_N; A\rang = \frac{1}{\sqrt{N!}} \sum_p^{N!}\mathrm{sgn}(p) |n_{(p,1)}\rang |n_{(p,2)}\rang \cdots |n_{(p,N)}\rang\

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 \Pi_j n_j 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

 \lang n_1 n_2 \cdots n_N; S | n_1 n_2 \cdots n_N; S\rang = 1, \qquad \lang n_1 n_2 \cdots n_N; A | n_1 n_2 \cdots n_N; A\rang = 1.

Measurements of identical particles[edit]

Suppose we have a system of N bosons (fermions) in the symmetric (antisymmetric) state

|n_1 n_2 \cdots n_N; S/A \rang

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.

|m_1 m_2 \cdots m_N; S/A \rang

The probability of obtaining a particular result for the m measurement is

P_{S/A}(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) \equiv \bigg|\lang m_1 \cdots m_N; S/A \,|\, n_1 \cdots n_N; S/A \rang \bigg|^2

We can show that

 \sum_{m_1 \le m_2 \le \dots \le m_N} P_{S/A}(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) = 1

which verifies that the total probability is 1. Note that we have to restrict the sum to ordered values of m1, ..., mN to ensure that we do not count each multi-particle state more than once.

Wavefunction representation[edit]

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

 |\lang x | \psi \rang|^2 \; d^3 x

As a result, the continuous eigenstates |x⟩ are normalized to the delta function instead of unity:

 \lang x | x' \rang = \delta^3 (x - x')

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:

|x_1 x_2 \cdots x_N; S\rang = \frac{\prod_j n_j!}{N!} \sum_p |x_{p(1)}\rang |x_{p(2)}\rang \cdots |x_{p(N)}\rang
|x_1 x_2 \cdots x_N; A\rang = \frac{1}{N!} \sum_p \mathrm{sgn}(p) |x_{p(1)}\rang |x_{p(2)}\rang \cdots |x_{p(N)}\rang

We can then write a many-body wavefunction,


\begin{align}
\Psi^{(S)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \cdots x_N) & \equiv \lang x_1 x_2 \cdots x_N; S | n_1 n_2 \cdots n_N; S \rang \\[10pt]
& = \sqrt{\frac{\prod_j n_j!}{N!}} \sum_p \psi_{p(1)}(x_1) \psi_{p(2)}(x_2) \cdots \psi_{p(N)}(x_N)
\end{align}

\begin{align}
\Psi^{(A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \cdots x_N) & \equiv \lang x_1 x_2 \cdots x_N; A | n_1 n_2 \cdots n_N; A \rang \\[10pt]
& = \frac{1}{\sqrt{N!}} \sum_p \mathrm{sgn}(p) \psi_{p(1)}(x_1) \psi_{p(2)}(x_2) \cdots \psi_{p(N)}(x_N)
\end{align}

where the single-particle wavefunctions are defined, as usual, by

\psi_n(x) \equiv \lang x | n \rang

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:


\Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) =
\Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots)

\Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) = -
\Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots)

The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers n1, ..., nN, and we perform a position measurement, the probability of finding particles in infinitesimal volumes near x1, x2, ..., xN is

 N! \; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \cdots x_N) \right|^2 \; d^{3N}\!x

The factor of N! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,

 \int\!\int\!\cdots\!\int\; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \cdots x_N)\right|^2 d^3\!x_1 d^3\!x_2 \cdots d^3\!x_N = 1

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:

\Psi^{(A)}_{n_1 \cdots n_N} (x_1, \cdots x_N)
= \frac{1}{\sqrt{N!}} \left|
\begin{matrix}
\psi_{n_1}(x_1) & \psi_{n_1}(x_2) & \cdots & \psi_{n_1}(x_N) \\
\psi_{n_2}(x_1) & \psi_{n_2}(x_2) & \cdots & \psi_{n_2}(x_N) \\
\cdots & \cdots & \cdots & \cdots \\
\psi_{n_N}(x_1) & \psi_{n_N}(x_2) & \cdots & \psi_{n_N}(x_N) \\
\end{matrix}
\right|