The Pauli exclusion principle is the quantum
mechanical principle which states that two
or more identical fermions (particles with
half-integer spin) cannot occupy the same
quantum state within a quantum system simultaneously.
In the case of electrons in atoms, 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:
n, the principal quantum number, ℓ, the
angular momentum quantum number, mℓ, the
magnetic quantum number, and ms, the spin
quantum number. For example, if two electrons
reside in the same orbital, and if their n,
ℓ, and mℓ values are the same, then their
ms must be different, and thus the electrons
must have opposite half-integer spin projections
of 1/2 and −1/2. This principle was formulated
by Austrian physicist Wolfgang Pauli in 1925
for electrons, and later extended to all fermions
with his spin–statistics theorem of 1940.
Particles with an integer spin, or 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.
A more rigorous statement is that with respect
to exchange of two identical particles the
total wave function is antisymmetric for fermions,
and symmetric for bosons. This means that
if the space and spin co-ordinates of two
identical particles are interchanged, then
the wave function changes its sign for fermions
and does not change for bosons.
== Overview ==
The Pauli exclusion principle describes the
behavior of all fermions (particles with "half-integer
spin"), while bosons (particles with "integer
spin") are subject to other principles. Fermions
include elementary particles such as quarks,
electrons and neutrinos. Additionally, baryons
such as protons and neutrons (subatomic particles
composed from three quarks) and some atoms
(such as helium-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, 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
ℏ
=
h
/
2
π
{\displaystyle \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.)
== 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
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 any given shell, and especially
to hold eight electrons which are normally
arranged symmetrically at the eight corners
of a cube (see: cubical atom). 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 (for example 2, 8 and
18) 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 (n), 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
with respect to exchange. An antisymmetric
two-particle state is represented as a sum
of states in which one particle is in state
|
x
⟩
{\displaystyle \scriptstyle |x\rangle }
and the other in state
|
y
⟩
{\displaystyle \scriptstyle |y\rangle }
, and is given by:
|
ψ
⟩
=
∑
x
,
y
A
(
x
,
y
)
|
x
,
y
⟩
,
{\displaystyle |\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 local changes of basis
keep antisymmetric matrices antisymmetric.
Conversely, if the diagonal quantities A(x,x)
are zero in every basis, then the wavefunction
component
A
(
x
,
y
)
=
⟨
ψ
|
x
,
y
⟩
=
⟨
ψ
|
(
|
x
⟩
⊗
|
y
⟩
)
{\displaystyle A(x,y)=\langle \psi |x,y\rangle
=\langle \psi |(|x\rangle \otimes |y\rangle
)}
is necessarily antisymmetric. To prove it,
consider the matrix element
⟨
ψ
|
(
(
|
x
⟩
+
|
y
⟩
)
⊗
(
|
x
⟩
+
|
y
⟩
)
)
.
{\displaystyle \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
⟩
+
|
y
⟩
{\displaystyle |x\rangle +|y\rangle }
. But this is equal to
⟨
ψ
|
x
,
x
⟩
+
⟨
ψ
|
x
,
y
⟩
+
⟨
ψ
|
y
,
x
⟩
+
⟨
ψ
|
y
,
y
⟩
.
{\displaystyle \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:
⟨
ψ
|
x
,
y
⟩
+
⟨
ψ
|
y
,
x
⟩
=
0
,
{\displaystyle \langle \psi |x,y\rangle +\langle
\psi |y,x\rangle =0,}
or
A
(
x
,
y
)
=
−
A
(
y
,
x
)
.
{\displaystyle A(x,y)=-A(y,x).}
=== 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.
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 ===
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 (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 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 occupied electron 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 ===
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 (metals)
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 (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.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 ===
Freeman Dyson and Andrew Lenard did not consider
the extreme magnetic or gravitational forces
that 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 bodies, atomic structure is disrupted
by extreme pressure, but the stars are held
in hydrostatic equilibrium by degeneracy pressure,
also known as Fermi pressure. This exotic
form of matter is known as degenerate matter.
The immense gravitational force of a star's
mass is normally held in equilibrium by thermal
pressure caused by heat produced in thermonuclear
fusion in the star's core. In white dwarfs,
which do not undergo nuclear fusion, an opposing
force to gravity is provided 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, neutron 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.
Neutron stars are the most "rigid" objects
known; their Young modulus (or more accurately,
bulk 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
Fermi–Dirac statistics
Fermi hole
Hund's rule
Pauli effect
