The Pauli exclusion principle is the quantum mechanical principle that says that two identical
fermions cannot occupy the same quantum state simultaneously. In the case of electrons,
it can be stated as follows: it is impossible for two electrons of a poly-electron atom
to have the same values of the four quantum numbers. For two electrons residing in the
same orbital, n, ℓ, and mℓ are the same, so ms must be different and the electrons
have opposite spins. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925.
A more rigorous statement is that the total wave function for two identical fermions is
anti-symmetric with respect to exchange of the particles. This means that the wave function
changes its sign if the space and spin co-ordinates of any two particles are interchanged.
Integer spin particles, bosons, are not subject to the Pauli exclusion principle: any number
of identical bosons can occupy the same quantum state, as with, for instance, photons produced
by a laser and Bose–Einstein condensate.
Overview The Pauli exclusion principle governs the
behavior of all fermions, while bosons are not subject to it. Fermions include elementary
particles such as quarks, electrons and neutrinos. In addition, protons and neutrons 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 times
a half-integer. In the theory of quantum mechanics fermions are described by antisymmetric states.
In contrast, particles with integer spin 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..
History In the early 20th century it became evident
that atoms and molecules with even numbers of electrons are more chemically stable than
those with odd numbers of electrons. In the famous 1916 article "The Atom and the Molecule"
by Gilbert 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 the shell and especially
to hold eight electrons which are normally arranged symmetrically at the eight corners
of a cube. In 1919 chemist Irving Langmuir suggested that the periodic 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 of electron shells around the nucleus. In
1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons
corresponded to stable "closed shells". Pauli looked for an explanation for these
numbers, which were at first only empirical. At the same time he was trying to explain
experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism.
He found an essential clue in a 1924 paper by Edmund C. Stoner which pointed out that
for a given value of the principal quantum number, the number of energy levels of a single
electron in the alkali metal spectra in an external magnetic field, where all degenerate
energy levels are separated, is equal to the number of electrons in the closed shell of
the noble gases for the same value of n. This led Pauli to realize that the complicated
numbers of electrons in closed shells can be reduced to the simple rule of one 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 by Samuel Goudsmit and George Uhlenbeck
as electron spin. Connection to quantum state symmetry
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 and the other in state :
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:
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 on the right hand side are diagonal elements and are zero, and
the whole sum is equal to zero. So the wavefunction matrix elements obey:
. or
Pauli principle in advanced quantum theory According to the spin-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 relativistic quantum field theory, the
Pauli principle follows from applying a rotation operator in imaginary time to particles of
half-integer spin. Since, nonrelativistically, particles can have any statistics and any
spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.
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, exchange of particles requires
that they pass through each other; for infinitely strong repulsion this cannot happen. This
model is described by a quantum nonlinear 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 for interacting spins and Hubbard model in one dimension, and for
other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz
is a Fermi sphere. Consequences
Atoms and the Pauli principle The Pauli exclusion principle helps explain
a wide variety of physical phenomena. One particularly important consequence of the
principle is the elaborate electron shell structure of atoms and the way atoms share
electrons, explaining the variety of chemical elements and their chemical combinations.
An electrically neutral atom contains bound electrons equal in number to the protons in
the nucleus. 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 neutral helium atom, which
has two bound electrons, both of which can occupy the lowest-energy 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. In a lithium atom, with three bound electrons, the third
electron cannot reside in a 1s state, and must occupy one of the higher-energy 2s states
instead. 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 shells but the same number of electrons in the outermost
shell have similar properties, which gives rise to the periodic table of the elements.
Solid state properties and the Pauli principle In conductors and semiconductors, there are
very large numbers of molecular orbitals which effectively form a continuous band structure
of energy levels. In strong conductors electrons are so degenerate that they cannot even contribute
much to the thermal capacity of a metal. Many mechanical, electrical, magnetic, optical
and chemical properties of solids are the direct consequence of Pauli exclusion.
Stability of matter The stability of the electrons in an atom
itself is unrelated to the exclusion principle, but is described by the quantum theory of
the atom. The underlying idea is that close approach of an electron to the nucleus of
the atom necessarily increases its kinetic energy, an application of the uncertainty
principle of Heisenberg. However, stability of large systems with many electrons and many
nucleons is a different matter, 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 by Paul 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.
A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered
the balance of attractive and repulsive forces and showed that ordinary matter would collapse
and occupy a much smaller volume without the Pauli principle. The consequence of the Pauli
principle here is that electrons of the same spin are kept apart by a repulsive exchange
interaction, which is a short-range effect, acting simultaneously with the long-range
electrostatic or coulombic 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 and the Pauli principle Dyson and Lenard did not consider the extreme
magnetic or gravitational forces which occur in some astronomical objects. In 1995 Elliott
Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic
fields such as in neutron stars, although at a much higher density than in ordinary
matter. It is a consequence of general relativity that, in sufficiently intense gravitational
fields, matter collapses to form a black hole. Astronomy provides a spectacular demonstration
of the effect of the Pauli principle, in the form of white dwarf and neutron stars. In
both types of body, atomic structure is disrupted by large gravitational forces, leaving the
constituents supported by "degeneracy pressure" alone. This exotic form of matter is known
as degenerate matter. In white dwarfs atoms are held apart by electron degeneracy pressure.
In neutron 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, albeit over a shorter range. This can stabilize neutron stars from further collapse,
but at a smaller size and higher density than a white dwarf. Neutrons are the most "rigid"
objects known; their Young modulus is 20 orders of magnitude larger than that of diamond.
However, even this enormous rigidity can be overcome by the gravitational field of a massive
star or by the pressure of a supernova, leading to the formation of a black hole.
See also Exchange force
Exchange interaction Exchange symmetry
Hund's rule Fermi hole
Pauli effect References
External links Nobel Lecture: Exclusion Principle and Quantum
Mechanics Pauli's own account of the development of the Exclusion Principle.