In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electronatom it is impossible for any two electrons to have the same two values ofall four of theirquantum numbers, which are:n, theprincipal quantum number;ℓ, theazimuthal quantum number;mℓ, themagnetic quantum number; andms, thespin quantum number. For example, if two electrons reside in the sameorbital, then their values ofn,ℓ, andmℓ are equal. In that case, the two values ofms (spin) pair must be different. Since the only two possible values for the spin projectionms are +1/2 and −1/2, it follows that one electron must havems = +1/2 and onems = −1/2.
Particles with an integer spin (bosons) are not subject to the Pauli exclusion principle. Any number of identical bosons can occupy the same quantum state, such as photons produced by alaser, or atoms found in aBose–Einstein condensate.
A rigorous statement which justifies the exclusion principle is: under the exchange of two identical particles, the total (many-particle)wave function isantisymmetric for fermions and symmetric for bosons. This means that if thespaceandspin coordinates of two identical particles are interchanged, then the total wave function changessign (from positive to negative or vice versa) for fermions, but does not change sign for bosons. So, if hypothetically two fermions were in the same state—for example, in the sameatom in the same orbital with the same spin—then interchanging them would change nothing and the total wave function would be unchanged. However, the only way a total wave function can both change sign (which is required for fermions), and also remain unchanged, is that such a function must bezero everywhere, which means such a state cannot exist. This reasoning does not apply to bosons because the sign does not change.
The Pauli exclusion principle describes the behavior of allfermions (particles with half-integerspin), whilebosons (particles with integer spin) are subject to other principles. Fermions includeelementary particles such asquarks,electrons andneutrinos. Additionally,baryons such asprotons andneutrons (subatomic particles composed from three quarks) and someatoms (such ashelium-3) are fermions, and are therefore described by 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, whereashelium-4 has spin 0 and is a boson.[2]: 123–125 The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to thechemical behavior of atoms.
In the early 20th century it became evident that atoms and molecules with even numbers of electrons are morechemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" byGilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in any given shell, and especially to hold eight electrons, which he assumed to be typically arranged symmetricallyat the eight corners of a cube.[3] In 1919 chemistIrving Langmuir suggested that theperiodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set ofelectron shells around the nucleus.[4] In 1922,Niels Bohr updatedhis model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".[5]: 203
Pauli looked for an explanation for these numbers, which were at first onlyempirical. At the same time he was trying to explain experimental results of theZeeman effect in atomicspectroscopy and inferromagnetism. He found an essential clue in a 1924 paper byEdmund C. Stoner, which pointed out that, for a given value of theprincipal quantum number (n), the number of energy levels of a single electron in thealkali metal spectra in an external magnetic field, where alldegenerate energy levels are separated, is equal to the number of electrons in the closed shell of thenoble gases for the same value ofn. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule ofone electron per state if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified bySamuel Goudsmit andGeorge Uhlenbeck aselectron spin.[6][7]
In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:[8]
Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are thesymmetrical class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and theantisymmetrical class, in which for such a permutation the wave function changes its sign...[The antisymmetrical class is] the correct and general wave mechanical formulation of the exclusion principle.
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to beantisymmetric with respect to exchange. If and range over the basis vectors of theHilbert space describing a one-particle system, then the tensor product produces the basis vectors of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as asuperposition (i.e. sum) of these basis vectors:
where eachA(x,y) is a (complex) scalar coefficient. Antisymmetry under exchange means thatA(x,y) = −A(y,x). This impliesA(x,y) = 0 whenx =y, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric.
Conversely, if the diagonal quantitiesA(x,x) are zeroin every basis, then the wavefunction component
is necessarily antisymmetric. To prove it, consider the matrix element
This is zero, because the two particles have zero probability to both be in the superposition state. But this is equal to
The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
or
For a system withn > 2 particles, the multi-particle basis states becomen-fold tensor products of one-particle basis states, and the coefficients of the wavefunction are identified byn one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged: for any. The exclusion principle is the consequence that, if for any then This shows that none of then particles may be in the same state.
According to thespin–statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.In relativisticquantum field theory, the Pauli principle follows from applying arotation operator inimaginary time to particles of half-integer spin.
In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta-function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, the exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantumnonlinear Schrödinger equation. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions,[9] as well as forinteracting spins andHubbard model in one dimension, and for other models solvable byBethe ansatz. Theground state in models solvable by Bethe ansatz is aFermi sphere.
The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborateelectron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. Anelectrically neutral atom contains bound electrons equal in number to the protons in thenucleus. Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below.
An example is the neutralhelium atom (He), which has two bound electrons, both of which can occupy the lowest-energy (1s) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values (eigenvalues). In alithium atom (Li), with three bound electrons, the third electron cannot reside in a 1s state and must occupy a higher-energy state instead. The lowest available state is 2s, so that theground state of Li is 1s22s. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of occupied electron shells but the same number of electrons in the outermost shell have similar properties, which gives rise to theperiodic table of the elements.[10]: 214–218
To test the Pauli exclusion principle for the helium atom, Gordon Drake of theUniversity of Windsor[11] carried out very precise calculations for hypothetical states of the He atom that violate it, which are calledparonic states. Later, K. Deilamian et al.[12] used an atomic beam spectrometer to search for the paronic state 1s2s1S0 calculated by Drake. The search was unsuccessful and showed that the statistical weight of this paronic state has an upper limit of5×10−6. (The exclusion principle implies a weight of zero.)
The stability of each electron state in an atom is described by the quantum theory of the atom, which shows that close approach of an electron to the nucleus necessarily increases the electron's kinetic energy, an application of theuncertainty principle of Heisenberg.[14] However, stability of large systems with many electrons and manynucleons is a different question, and requires the Pauli exclusion principle.[15]
It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. This suggestion was first made in 1931 byPaul Ehrenfest, who pointed out that the electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms, therefore, occupy a volume and cannot be squeezed too closely together.[16]
The first rigorous proof was provided in 1967 byFreeman Dyson and Andrew Lenard (de), who considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.[17][18]A much simpler proof was found later byElliott H. Lieb andWalter Thirring in 1975. They provided a lower bound on the quantum energy in terms of theThomas-Fermi model, which is stable due to atheorem of Teller. The proof used a lower bound on the kinetic energy which is now called theLieb–Thirring inequality.
The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsiveexchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic orCoulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.
Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in someastronomical objects. In 1995Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as inneutron stars, although at a much higher density than in ordinary matter.[19] It is a consequence ofgeneral relativity that, in sufficiently intense gravitational fields, matter collapses to form ablack hole.
Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form ofwhite dwarf andneutron stars. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held inhydrostatic equilibrium bydegeneracy pressure, also known as Fermi pressure. This exotic form of matter is known asdegenerate matter. The immense gravitational force of a star's mass is normally held in equilibrium bythermal pressure caused by heat produced inthermonuclear fusion in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided byelectron degeneracy pressure. Inneutron stars, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure,neutron degeneracy pressure, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higherdensity than a white dwarf. Neutron stars are the most "rigid" objects known; theirYoung modulus (or more accurately,bulk modulus) is 20 orders of magnitude larger than that ofdiamond. However, even this enormous rigidity can be overcome by thegravitational field of a neutron star mass exceeding theTolman–Oppenheimer–Volkoff limit, leading to the formation of ablack hole.[20]: 286–287
^Lieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics".arXiv:math-ph/0209034.
^This realization is attributed byLieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics".arXiv:math-ph/0209034. and byG. L. Sewell (2002).Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press.ISBN0-691-05832-6. to F. J. Dyson and A. Lenard:Stability of Matter, Parts I and II (J. Math. Phys.,8, 423–434 (1967);J. Math. Phys.,9, 698–711 (1968) ).
^As described by F. J. Dyson (J.Math.Phys.8, 1538–1545 (1967)), Ehrenfest made this suggestion in his address on the occasion of the award of theLorentz Medal to Pauli.
^F. J. Dyson and A. Lenard:Stability of Matter, Parts I and II (J. Math. Phys.,8, 423–434 (1967);J. Math. Phys.,9, 698–711 (1968) )
^Martin Bojowald (5 November 2012).The Universe: A View from Classical and Quantum Gravity. John Wiley & Sons.ISBN978-3-527-66769-7.
General
Dill, Dan (2006). "Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps".Notes on General Chemistry (2nd ed.). W. H. Freeman.ISBN1-4292-0068-5.
