The operators corresponding to the spin along the x, y, and z direction, denoted S < sub > x </ sub >, S < sub > y </ sub >, and S < sub > z </ sub > respectively, can be represented using the Pauli matrices:

These matrices were used by, then named after, the Austrian-born physicist Wolfgang Pauli ( 1900 – 1958 ), in his 1925 study of spin in quantum mechanics.

Each Pauli matrix is Hermitian, and together with the identity I ( sometimes considered the zeroth Pauli matrix ), the Pauli matrices span the full vector space of 2x2 Hermitian matrices.

In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.

The Pauli matrices ( after multiplication by i to make them anti-hermitian ), also generate transformations in the sense of Lie algebras: the matrices form a basis for, which exponentiates to the spin group, and for the identical Lie algebra, which exponentiates to the Lie group of rotations of 3-dimensional space.

* The determinants and traces of the Pauli matrices are:

* Together with the identity matrix I ( which is sometimes written as σ < sub > 0 </ sub >), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

Each of the ( hermitian ) Pauli matrices has two eigenvalues, + 1 and − 1.

Note that in this vector dotted with Pauli vector operation the Pauli matrices are treated in a scalar like fashion, commuting with the vector basis elements.

The Pauli matrices obey the following commutation and anticommutation relations:

It is an abstract version of the algebra generated by the gamma or Pauli matrices.

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced spin matrices.

More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero.

Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from the other electrons in the atom.

* The electrons are bound into Cooper pairs, and these pairs are correlated due to the Pauli exclusion principle for the electrons, from which they are constructed.

Baryons are strongly interacting fermions — that is, they experience the strong nuclear force and are described by Fermi − Dirac statistics, which apply to all particles obeying the Pauli exclusion principle.

There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which do not share quantum states due to the Pauli exclusion principle.

Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum multiplicity ( only 2 electrons, having opposite spins, per orbital ; place as many unpaired electrons on one energy level as possible before starting to pair them ).

While paired electrons are required by the Pauli exclusion principle to have their intrinsic (' spin ') magnetic moments pointing in opposite directions, causing their magnetic fields to cancel out, an unpaired electron is free to align its magnetic moment in any direction.

Neutron stars are very hot and are supported against further collapse by quantum degeneracy pressure due to the Pauli exclusion principle.

This means they are fermions not bosons, and therefore, like electrons, they are subject to the Pauli exclusion principle.

The equilibrium between electromagnetic forces and Pauli repulsion of electrons ( they are fermions obeying Fermi statistics ) is slightly violated resulting in a small returning force.

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.

The Pauli exclusion principle governs the behavior of all fermions ( particles with " half-integer spin "), while bosons ( particles with " integer spin ") are not subject to it.

In addition, protons and neutrons ( subatomic particles composed from three quarks ) and some atoms are fermions, and are therefore subject to the Pauli exclusion principle as well.

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 per state, if the electron states are defined using four quantum numbers.

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.

Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

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.

However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory.

Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:

By the Pauli exclusion principle, no quantum state can be occupied by more than one fermion with an identical set of quantum numbers.

A devout Roman Catholic throughout his life, Curtis would repair to a convent church in the early hours of Sunday morning after finishing the Saturday night set at the Star Club, St. Pauli, near Hamburg.

Kleine Freiheit, the German title of the movie, (" Little Freedom ", translated literally ) is a wordplay on Große Freiheit ( literally " Great Freedom "), the rather famous name of a street in the red light St. Pauli district where the plot is set.

Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:

Due to the Pauli exclusion principle, two electrons cannot share the same set of quantum numbers within the same system ; therefore, there is room for only two electrons in each spatial orbital.

Perhaps the first remark that quarks should possess an additional quantum number was made as a short footnote in the preprint of Boris Struminsky in connection with Ω < sup >-</ sup > hyperon composed of three strange quarks with parallel spins ( this situation was peculiar, because since quarks are fermions, such combination is forbidden by the Pauli exclusion principle ): Boris Struminsky was a PhD student of Nikolay Bogolyubov.

Here A and φ represent the electromagnetic field, and the three sigmas are the Pauli matrices.

The Stokes parameters contain all of the information of the coherency matrix, and are related to it linearly by means of the identity matrix plus the three Pauli matrices:

The township contains these three cemeteries: Clinton, Long Island and Saint Pauli.

It has been translated into English by Benjamin Thorpe as History of England under the Anglo-Saxon Kings ( 1845 ), and History of England under the Norman Kings ( Oxford, 1857 ), and was continued in three additional volumes from 1154 to 1509 by R Pauli.

It has an alternative basis given by the identity matrix and the three Pauli matrices.

For SU ( 2 ) gauges, for example, one has ; is an arbitrary real function of, and are the three Pauli matrices ; as usual, a sum over repeated indices is implied.

Born in Mitchell, Ontario to William Frederick Morenz and Rose Pauli, Howie Morenz had three sisters and two brothers.

The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix.

The board consisted of three people: Mads Andrias Winther, Joan Pauli Joensen and Jakup Mouritsen.

They negotiated a contract with the Star-Club in the St. Pauli district Hamburg for 128 days, with three one-hour performances a night, starting in July 1962.

In 2004 he returned to the FC St. Pauli in the Regionalliga Nord, playing 88 times and scoring 24 goals during his three years in Hamburg ; the team twice won the Hamburger Pokal.

Robert's theology is expressed in his three surviving works, the Quaestiones de divina pagina, Quaestiones de epistolis Pauli, and the unfinished Sententiae.