Pauli exclusion principle

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, whereas helium-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 the chemical behavior of atoms.

Half-integer spin means that the intrinsic angular momentum value of fermions is ${\displaystyle \hbar =h/2\pi }$ (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 (bosons) have symmetric wave functions and 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, which they obey, and bosons take theirs from the Bose–Einstein distribution.

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 he assumed to be typically arranged symmetrically at the eight corners of a cube.[3] 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.[4] 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".[5]^: 203

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.[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 the symmetrical class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the antisymmetrical 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 be antisymmetric with respect to exchange. If ${\displaystyle |x\rangle }$ and ${\displaystyle |y\rangle }$ range over the basis vectors of the Hilbert space describing a one-particle system, then the tensor product produces the basis vectors ${\displaystyle |x,y\rangle =|x\rangle \otimes |y\rangle }$ of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as a superposition (i.e. sum) of these basis vectors:

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

where each A(x,y) is a (complex) scalar coefficient. 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

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

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

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

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

${\displaystyle \langle \psi |x,x\rangle +\langle \psi |x,y\rangle +\langle \psi |y,x\rangle +\langle \psi |y,y\rangle .}$

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:

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

or

${\displaystyle A(x,y)=-A(y,x).}$

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

General

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