Pauli exclusion principle

Wolfgang Pauliduring a lecture in Copenhagen (1929).Wolfgang Pauli formulated the Pauli exclusion principle.

Inquantum mechanics,thePauli exclusion principlestates that two or moreidentical particleswithhalf-integer spins(i.e.fermions) cannot simultaneously occupy the samequantum statewithin a system that obeys the laws ofquantum mechanics.This principle was formulated by Austrian physicistWolfgang Pauliin 1925 forelectrons,and later extended to all fermions with hisspin–statistics theoremof 1940.

In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values ofallfour of theirquantum numbers,which are:n,theprincipal quantum number;ℓ,theazimuthal quantum number;m_ℓ,themagnetic quantum number;andm_s,thespin quantum number.For example, if two electrons reside in the sameorbital,then their values ofn,ℓ,andm_ℓare equal. In that case, the two values ofm_s(spin) pair must be different. Since the only two possible values for the spin projectionm_sare +1/2 and −1/2, it follows that one electron must havem_s= +1/2 and onem_s= -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 more rigorous statement is: under the exchange of two identical particles, the total (many-particle)wave functionisantisymmetricfor fermions and symmetric for bosons. This means that if the spaceandspin coordinates of two identical particles are interchanged, then the total wave function changes sign for fermions, but does not change sign for bosons.

So, if hypothetically two fermions were in the same state—for example, in the same atom 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 (required for fermions), and also remain unchanged is that such a function must be zero everywhere, which means such a state cannot exist. This reasoning does not apply to bosons because the sign does not change.

Overview

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 particlessuch asquarks,electronsandneutrinos.Additionally,baryonssuch asprotonsandneutrons(subatomic particlescomposed 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-3has spin 1/2 and is therefore a fermion, whereashelium-4has spin 0 and is a boson.The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to thechemical behavior of atoms.

Half-integer spin means that the intrinsicangular momentumvalue of fermions is$\hbar =h/2\pi$(reducedPlanck's constant) times ahalf-integer(1/2, 3/2, 5/2, etc.). In the theory ofquantum mechanics,fermions are described byantisymmetric states.In contrast, particles with integer spin (bosons) have symmetric wave functions and may share the same quantum states. Bosons include thephoton,theCooper pairswhich are responsible forsuperconductivity,and theW and Z bosons.Fermions take their name from theFermi–Dirac statistical distribution,which they obey, and bosons take theirs from theBose–Einstein distribution.

History

In the early 20th century it became evident that atoms and molecules with even numbers of electrons are morechemically stablethan 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.In 1919 chemistIrving Langmuirsuggested that theperiodic tablecould 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 shellsaround the nucleus.In 1922,Niels Bohrupdatedhis model of the atomby assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".

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 effectin atomicspectroscopyand 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 metalspectra in an external magnetic field, where alldegenerate energy levelsare separated, is equal to the number of electrons in the closed shell of thenoble gasesfor 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 ofoneelectron 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 GoudsmitandGeorge Uhlenbeckaselectron spin.

Connection to quantum state symmetry

In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:

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$|x\rangle$and$|y\rangle$range over the basis vectors of theHilbert spacedescribing a one-particle system, then the tensor product produces the basis vectors$|x,y\rangle =|x\rangle \otimes |y\rangle$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:

$|\psi \rangle =\sum _{x,y}A(x,y)|x,y\rangle,$

where eachA(x,y)is a (complex) scalar coefficient. Antisymmetry under exchange means thatA(x,y) = −A(y,x).This impliesA(x,y) = 0whenx=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

$A(x,y)=\langle \psi |x,y\rangle =\langle \psi |{\Big (}|x\rangle \otimes |y\rangle {\Big )}$

is necessarily antisymmetric. To prove it, consider the matrix element

$\langle \psi |{\Big (}(|x\rangle +|y\rangle )\otimes (|x\rangle +|y\rangle ){\Big )}.$

This is zero, because the two particles have zero probability to both be in the superposition state$|x\rangle +|y\rangle$.But this is equal to

$\langle \psi |x,x\rangle +\langle \psi |x,y\rangle +\langle \psi |y,x\rangle +\langle \psi |y,y\rangle.$

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:

$\langle \psi |x,y\rangle +\langle \psi |y,x\rangle =0,$

or

$A(x,y)=-A(y,x).$

For a system withn> 2particles, the multi-particle basis states becomen-fold tensor products of one-particle basis states, and the coefficients of the wavefunction$A(x_{1},x_{2},\ldots,x_{n})$are identified bynone-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged:$A(\ldots,x_{i},\ldots,x_{j},\ldots )=-A(\ldots,x_{j},\ldots,x_{i},\ldots )$for any$i\neq j$.The exclusion principle is the consequence that, if$x_{i}=x_{j}$for any$i\neq j,$then$A(\ldots,x_{i},\ldots,x_{j},\ldots )=0.$This shows that none of thenparticles may be in the same state.

Advanced quantum theory

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 operatorinimaginary timeto 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,as well as forinteracting spinsandHubbard modelin one dimension, and for other models solvable byBethe ansatz.Theground statein models solvable by Bethe ansatz is aFermi sphere.

Applications

Atoms

The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborateelectron shell structureof atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. Anelectrically neutralatom 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 alithiumatom (Li), with three bound electrons, the third electron cannot reside in a1sstate and must occupy a higher-energy state instead. The lowest available state is2s,so that theground stateof Li is1s²2s.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.

To test the Pauli exclusion principle for the helium atom, Gordon Drakecarried out very precise calculations for hypothetical states of the He atom that violate it, which are calledparonic states.Later, K. Deilamian et al.used an atomic beam spectrometer to search for the paronic state 1s2s¹S₀calculated 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.)

Solid state properties

Inconductorsandsemiconductors,there are very large numbers ofmolecular orbitalswhich effectively form a continuousband structureofenergy levels.In strong conductors (metals) electrons are sodegeneratethat they cannot even contribute much to thethermal capacityof a metal.Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

Stability of matter

Further information:Stability of matter

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 principleof Heisenberg.However, stability of large systems with many electrons and manynucleonsis a different question, and requires the Pauli exclusion principle.

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.

The first rigorous proof was provided in 1967 byFreeman Dysonand 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. A much simpler proof was found later byElliott H. LiebandWalter Thirringin 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.

Astrophysics

Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in someastronomicalobjects. In 1995Elliott Lieband 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.It is a consequence ofgeneral relativitythat, 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 dwarfandneutron stars.In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held inhydrostatic equilibriumbydegeneracy 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 pressurecaused by heat produced inthermonuclear fusionin 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 higherdensitythan 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 fieldof a neutron star mass exceeding theTolman–Oppenheimer–Volkoff limit,leading to the formation of ablack hole.

See also

• Spin-statistics theorem
• Exchange interaction
• Exchange symmetry
• Fermi–Dirac statistics
• Fermi hole
• Hund's rule
• Pauli effect

References

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.
• Liboff, Richard L.(2002).Introductory Quantum Mechanics.Addison-Wesley.ISBN0-8053-8714-5.
• Massimi, Michela(2005).Pauli's Exclusion Principle.Cambridge University Press.ISBN0-521-83911-4.
• Tipler, Paul; Llewellyn, Ralph (2002).Modern Physics(4th ed.). W. H. Freeman.ISBN0-7167-4345-0.
• Scerri, Eric (2007).The periodic table: Its story and its significance.New York: Oxford University Press.ISBN978-0-19-530573-9.

External links

• Nobel Lecture: Exclusion Principle and Quantum MechanicsPauli's account of the development of the Exclusion Principle.