Exclusion Principle

• Type of physical science: Atomic physics
• Field of study: Nonrelativistic quantum mechanics

The Pauli exclusion principle states that no two electrons can exist in the same quantum mechanical state. Not only does this principle effectively reduce chemistry to physics but also, taken as applying to fermions in general, it plays a fundamental role in the development of quantum theory and elementary particle physics.

Overview

In quantum mechanics, the overall state of a system, such as an electron, is given in terms of a set of "quantum numbers," corresponding to the allowed values of certain properties of the system, such as its energy, orbital angular momentum, the component of this angular momentum in a specified direction, and spin. According to Wolfgang Pauli's exclusion principle, no two electrons can possess the same set of quantum numbers; in other words, no two electrons can exist in the same overall state.

An immediate and fundamental consequence of this principle is the explanation of the properties of the chemical elements, as systematized in the periodic table. An atom of hydrogen, for example, consists of a single proton and an electron, occupying the lowest possible energy level. With two protons (and a neutron) in the nucleus, two electrons can be attracted to form a stable atom, as in the case of helium. According to the exclusion principle, however, they cannot both occupy the lowest energy level unless their overall quantum states are different. This can be achieved if the electrons' spins are different. Since the spins of all electrons have the same magnitude (they are said to possess spin 1/2), the directions must be different. There are only two directions possible, "up" and "down." Thus, in a helium atom, there are two electrons in the lowest energy level, one with spin up and one with spin down. This energy level is now fully occupied.

In an atom with three protons in its nucleus, the third electron cannot occupy the lowest energy level since it would then be in the same overall quantum state as one of the other electrons, and this is forbidden by the exclusion principle. Hence, it must occupy the next energy level. Since this level is farther from the nucleus, the electron occupying it is less tightly bound and it is much easier to remove it than it is to remove one of the electrons from a helium atom.

This explains why helium is so much more chemically inert than lithium. Proceeding in the above manner, the entire periodic table can be built up.

Pauli's principle can, in fact, be generalized to apply not only to electrons but also to all principles of a certain kind, known as fermions. All the particles known to exist can be classified as either fermions or bosons. Fermions include electrons, positrons, protons, and neutrons, and possess half-integral values of spin (1/2, 2/2, 5/2, and so on). They obey the exclusion principle and, considered collectively, their behavior is described in terms of Fermi-Dirac statistics (named for Enrico Fermi and Paul Adrien Maurice Dirac, two physicists who developed the theoretical basis of this form of particle statistics).

Bosons, on the other hand, include photons (the quanta of light) and mesons and possess integral values of spin (0, 1, and so on). They do not obey the exclusion principle and, therefore, more than one boson may occupy the same state. This results in some peculiar collective behavior, where bosons are said to condense into one state; this behavior is expressed in terms of Bose-Einstein statistics (named for Satyendra N. Bose and Albert Einstein).

The difference between these two kinds of particles can be roughly imagined in the following terms. Consider a large concert hall. Fermions are analogous to those members of the audience who occupy only one seat and who fill up the auditorium from the most desirable seats, nearest the stage, outward. Bosons, on the other hand, are like those members of the audience who rush to the front of the stage and try to cram themselves into the smallest space possible.

Bosons may exhibit a condensation effect, in which there is a tendency for the particles to occupy the same quantum state, whereas fermions remain apart, each occupying a distinct quantum state.

It is this latter behavior that is captured by the exclusion principle.

Understood in these general terms, the exclusion principle can be put on a sound theoretical footing in the context of quantum mechanics. Consider a system of two particles, for example. The quantum mechanical wave function for this two-particle system is given by a combination of the wave functions of each particle. This combination can be either symmetric or antisymmetric. A wave function is said to be symmetric if its sign (plus or minus) does not change when the particles are interchanged or permuted. If such an interchange leads to a change of sign, the wave function is said to be antisymmetric. If two particles are in the same overall quantum state, so that they have the same energy, spin, and the like, they would possess the same set of quantum numbers and their wave functions would be identical. If the overall wave function of the two-particle system is antisymmetric, then, with identical single-particle wave functions as its components, it simply vanishes. Since the squares of the amplitudes of the wave functions in quantum mechanics give the probabilities of finding the particles in the corresponding states, a vanishing wave function corresponds to a zero probability of finding the particles in that overall state. Thus, if the wave function for the system is antisymmetric, the particles cannot occupy the same quantum mechanical state. This is the case for fermions.

A system of bosons, however, is described by a symmetric wave function. In this case, the wave function for the entire system does not vanish if both particles are in exactly the same state. Bosons, unlike fermions, can occupy the same state. Hence, Pauli's exclusion principle is a result of the fundamental symmetry properties of wave functions in quantum mechanics.

It might appear from the above explanation of the properties of the chemical elements as if there were some kind of force that existed between two electrons with parallel spins, keeping them apart. Indeed, physicists and chemists often refer to these forces as "exchange forces," since they are described, in mathematical terms, as arising from the permutation, or exchange, of indistinguishable particles. Nevertheless, these are not genuine forces, like gravitational or electromagnetic forces. Exchange forces are merely the expression of the way in which quantum mechanical probabilities combine. Consider a pair of dice, for example.

Mathematically, it can be shown that there is a much higher probability of rolling a seven than a two or a twelve (there are six different ways of obtaining a seven but only one way of obtaining a two or a twelve). If the dice are rolled repeatedly, it may seem that they are "attracted" to forming the number seven and "repelled" from two or twelve. There is, however, no force doing the attracting or repelling; there is only the playing out of the probabilities. Likewise, there is no real exchange force keeping the electrons apart but only the vanishing of their combined wave function, giving a zero probability of finding them in the same overall quantum state.

Applications

As well as explaining the periodic table and thus effectively reducing chemistry to physics, Pauli's exclusion principle helps to explain the properties of atomic nuclei. Experimental evidence shows that nuclei tend to contain only certain numbers of protons or neutrons. These are known as the "magic numbers": 2, 8, 14, 20, 28, 50, 82, and 126. According to the shell model of the nucleus, neutrons and protons are distributed over energy shells according to the exclusion principle. Since two protons or two neutrons cannot possess the same set of quantum numbers, each energy state is occupied by only a limited number of particles. Two protons, for example, can be in the same energy state only if their spins are opposite. The pattern of nuclear energy levels is different from that of the atom because the forces acting within the nucleus are very different from those acting between the nucleus and the atom's electrons.

Nevertheless, in both cases, Pauli's exclusion principle can be used to explain many of the properties that atoms and nuclei have.

Another important application of the exclusion principle concerns the explanation of chemical bonding. Consider two atoms, for example. If their outer electrons have opposite spins, then they can come together and form a bond, with both electrons in the same energy state.

According to the exclusion principle, no more electrons can occupy this state. Thus, the strongest such binding occurs when there are two electrons in the bond. Using the exclusion principle in this manner, Linus Pauling and others developed the theory of the molecular bond. This marked the beginning of quantum chemistry, which explains how atoms are bound together in certain molecular configurations. Knowing how such bonds are formed allows physicists and chemists to calculate the distances and angles between atoms in molecules. By comparing these calculations with the results of X-ray scattering experiments, detailed molecular models can be built up.

Francis Crick and James D. Watson produced the double-helix model of DNA (deoxyribonucleic acid). Without such a model, genetic engineering would be impossible. To be able to construct and reconstruct genetic material, a detailed knowledge of how atoms are arranged to form molecules is required. This knowledge is, in turn, based on quantum mechanics and, in particular, Pauli's exclusion principle.

The conduction of electrons in a metal is also usefully explained with the help of the exclusion principle. A metal consists of a crystal lattice of atoms whose outermost electrons are weakly bound only to the nucleus. According to the "free electron" model, these electrons become detached from the atoms and roam around freely. If an electric field is applied across the metal, creating a potential difference, the conduction electrons will produce a current. Pauli's principle explains why the conduction electrons do not become more tightly bound to the nuclei in the lattice: The energy states corresponding to such tight binding are already fully occupied.

This phenomenon also explains why metals are so strong: Because these states are fully occupied, the electrons, and hence the atoms of the lattice, cannot be squeezed tighter together.

Furthermore, the interactions between the conduction electrons themselves are comparatively unimportant for similar reasons. If a collision between these electrons occurred, they would be scattered from a pair of initial single-particle states to a pair of final single-particle states. It can be shown, however, on the basis of quantum mechanical considerations, that most such final states are already occupied. Thus, according to the exclusion principle, this form of scattering is forbidden and the collisions simply do not take place. In this way, one can justify regarding the conduction electrons as being "free." Such a collection of electrons that are free and noninteracting is known as a "free electron Fermi gas." Many of the most important properties of metals can be understood in these terms, and thus the exclusion principle plays a fundamental role in solid-state physics.

In particular, it helped to resolve a long-standing problem concerning the heat capacity of metals, understood as the heat required to raise the temperature of a quantity of the material by one degree. According to classical physics, if a metal absorbs heat energy, not only should the vibrations of the ions in the crystal lattice increase but also should the velocities of the free electrons. Nevertheless, this is not what is observed to happen: Almost all the energy absorbed by the metal goes to increase the lattice vibrations.

Quantum theory explains this discrepancy in the following way. For a Fermi gas of electrons at absolute zero, all the energy states below a certain value, known as the "Fermi energy," are completely filled, in accordance with the exclusion principle. The states can be thought of as rungs on a ladder, and the exclusion principle ensures that there can be only two electrons on each rung. As the temperature of the metal is increased, there is a tendency for the electrons to gain energy by collisions with the thermally agitated ions of the lattice. Yet, the exclusion principle dictates that an electron can move up only to a higher energy state if that state is not already completely filled. Only those states that lie above the Fermi energy are unoccupied and hence available, but at normal temperatures, most of the conduction electrons cannot gain enough energy to reach them. They are, in effect, out of bounds. Only those electrons in states close to the Fermi energy can be given enough energy to make the transition to an unfilled state. Consequently, only a small fraction of the conduction electrons gain energy and increase their velocities when the metal is heated, and so most of the heat energy is used to increase the thermal vibrations of the lattice ions, as observed experimentally.

Such a gas of electrons occupying discrete energy states is said to be "degenerate."

Degenerate gases are also found in the interiors of white dwarf stars. These are small, faint stars that are in the process of cooling down and dying. The material of such a star is extremely compressed and the atoms have been stripped of their electrons, which form a free electron gas.

These electrons are distributed among the available energy states according to the exclusion principle, and since their energy is much higher than the average thermal energy inside the star, the degeneracy pressure of the electron gas prevents the star from collapsing into a neutron star or even a black hole.

Finally, the phenomenon of superconductivity can also be explained in terms of the symmetry properties of the wave functions of electrons. Through the interactions between electrons and the ions of the crystal lattice in metals, two electrons may actually weakly attract each other and form a pair. Such a bound pair acts as a boson, since if the electrons are both exchanged, the sign of the wave function changes twice and thus returns to the value it had before the interchange. The wave function for the pair is, therefore, symmetric under exchange of both electrons in the pair.

Since the electron pairs are bosons, there is a tendency for them to condense into the same state and, consequently, there is a very low probability for any such pair to be in an unoccupied state. This explains why there is no resistance to the flow of current in a superconductor. Electrical resistance is a result of electrons being knocked out of the conducting state by interactions with the thermally agitated lattice ions. There is very little chance of this happening with an electron pair because of the high probability for all such pairs to collect together into the same state. The energy of pairing that binds the electrons into boson pairs is extremely weak, and any thermal agitation will tend to break up the pairs. This explains why metals are superconductors only below certain very low temperatures.

Context

Wolfgang Pauli proposed his exclusion principle in 1925 to resolve certain difficulties faced by the model of the atom then current, which was based on the work of Niels Bohr. According to this model, the electrons in an atom occupy discrete energy levels and the absorption and emission of light by the atom corresponds to electrons "jumping" between these levels. Although Bohr's model successfully explained the spectrum of hydrogen, it could not, for example, explain why certain spectral lines were split into a number of closely spaced lines, indicating the existence of closely spaced energy levels in the atom, nor could it explain why all the electrons did not occupy the lowest energy level.

Pauli realized that the splitting of the lines in atomic spectra could be explained if a fourth quantum number were assigned to the electrons, in addition to the three given by Bohr's model. This fourth quantum number was subsequently given the name "spin," although the property it designates is quite unlike the classical spin of a child's top (an electron has to "spin" around twice to get back to its starting point). With four quantum numbers, Pauli perceived that the number of electrons in each shell around the atom corresponded exactly to the number of different sets of quantum numbers belonging to that shell. On this basis, he formulated the principle that no two electrons can possess the same set of quantum numbers and was able to explain why the shells of the atoms of the various elements fill up the way they do.

The exclusion principle stood as an empirical result having no deeper theoretical motivation until it came to be understood, merely a few years later, in terms of the symmetry properties of the particle wave functions described by the new quantum mechanics.

Mathematically, these properties span a range from the completely symmetric, through various mixed symmetry combinations, to the completely antisymmetric. Physically, however, only those particles corresponding to the symmetric (bosons) and antisymmetric (fermions) combinations appear to exist. Nevertheless, during the 1960s, the behavior of particles obeying the mixed symmetry statistics, or "parastatistics" as it came to be known, was theoretically investigated and it was suggested that quarks could be "paraparticles" in this sense. This suggestion played a key role in the development of "quantum chromodynamics," the theory of quark interactions, according to which quarks are regarded as fermions possessing an extra quantum number known as "color."

Pauli's exclusion principle has been at the center of developments in quantum physics since it was proposed. Understood in terms of the symmetry properties of quantum mechanical wave functions, it represents a fundamental feature of the subatomic world.

Principal terms

BOSONS: particles whose quantum mechanical wave functions are symmetric under particle exchange; they possess integral values of spin and do not obey Pauli's exclusion principle

EXCHANGE FORCES: forces postulated to explain the tendency of fermions and bosons to remain apart or cluster together, respectively; they are merely an expression of the symmetry properties of the wave functions of the two kinds of particles

FERMIONS: particles with half-integral values of spin and that obey the exclusion principle; examples include protons, neutrons, and electrons

QUANTUM NUMBERS: a set of numbers representing certain fundamental properties of a system and that define its overall state

SPIN: a fundamental quantum property; although it has an obvious classical analog, it can be truly understood only in terms of relativistic quantum mechanics

WAVE FUNCTION: a mathematical function representing the state of the system; the square of the amplitude of this function gives the probability of finding the system in a particular state

Bibliography

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