Pauli exclusion principle
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The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state simultaneously. It is one of the most important principles in physics, primarily because the three types of particles from which ordinary matter is made—electrons, protons, and neutrons—are all subject to it. The Pauli exclusion principle underlies many of the characteristic properties of matter, from the large-scale stability of matter to the existence of the periodic table of the elements.
Pauli exclusion principle follows mathematically from definition of wave function for a system of identical particles - it can be either symmetric or antisymmetric (depending on particles' spin).
Particles with antisymmetric wave function are called fermions - they have to obey the Pauli exclusion principle. Apart from the familiar electron, proton and neutron, these include the neutrinos, the quarks (from which protons and neutrons are made), as well as some atoms like helium-3. All fermions possess "half-integer spin", meaning that they possess an intrinsic angular momentum whose value is <math>\hbar = h/2\pi</math> (Planck's constant divided by 2π) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics, fermions are described by "antisymmetric states", which are explained in greater detail in the article on identical particles.
Particles with integer spin have symmetric wave function and are called bosons, in contrast to fermions they share same quantum states. Examples of bosons include the photon and the W and Z bosons.
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Connection to quantum state symmetry
The Pauli exclusion principle was originally formulated as an empirical principle. It was invented by Pauli in 1924 to explain experimental results in the Zeeman effect in atomic spectroscopy, ferromagnetism, and how the periodic table is regulated by the electron structure of atoms, well before the 1925 formulation of the modern theory of quantum mechanics by Werner Heisenberg and Erwin Schrödinger. However, this does not mean that the principle is in any way approximate or unreliable; in fact, it is one of the most well-tested and commonly-accepted results in physics.
The Pauli exclusion principle can be derived starting from the assumption that a system of particles occupy antisymmetric quantum states. 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.
As discussed in the article on identical particles, an antisymmetric two-particle state in which one particle exists in state <math>\left|\psi_1\right\rangle</math> (nota) and the other in state <math>\left|\psi_2\right\rangle</math> is
- <math> |\psi_1, \psi_2\rangle = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle|\psi_2\rangle - |\psi_2\rangle|\psi_1\rangle \right) </math>
However, if <math>\left|\psi_1\right\rangle</math> and <math>\left|\psi_2\right\rangle</math> are just the same state, the above formula gives the zero set:
- <math> |\psi_1, \psi_2\rangle = 0 </math>
This does not represent a valid quantum state, because the state vectors representing quantum states must have norm 1. In other words, we can never find the particles in this system occupying the same quantum state.
Consequences
The Pauli exclusion principle plays a profound role in a huge number of physical phenomena. One of the most important, it results in "rigidness" or "stiffness" of fermions as contrast to bosons - impossibility to squeeze identical fermions into each other - thus in "rigidness" of ordinary matter (Young and bulk moduli of solids are direct consequences of Pauli principle). Another consequence is the elaborate electron shell structure of atoms and of the way atoms share electron(s) - thus variety of chemical elements and of their combinations (chemistry). (An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus. Since electrons are fermions, the Pauli exclusion principle forbids them from occupying the same quantum state, so electrons have to "pile on top of each other" in atom).
For example, consider a neutral helium atom, which has two bound electrons. Both of these electrons can occupy the lowest-energy (1s) states by acquiring opposite spin. This does not violate the Pauli principle because spin is part of the quantum state of the electron, so the two electrons are occupying different quantum states. However, the spin can take only two different values (or eigenvalues). In a lithium atom, which contains three bound electrons, the third electron cannot fit into a 1s state, and has to occupy one of the higher-energy 2s states instead. Similarly, successive elements produce successively higher-energy shells. The chemical properties of an element largely depend on the number of electrons in the outermost shell, which gives rise to the periodic table of the elements.
Astronomy provides another spectacular demonstrations of this effect, in the form of white dwarf stars and neutron stars. In both types of objects, the usual atomic structures are disrupted by large gravitational forces, leaving the constituents supported only by a "degeneracy pressure" produced by the Pauli exclusion principle. This exotic form of matter is known as degenerate matter. In white dwarfs, the atoms are held apart by the degeneracy pressure of the electrons. In neutron stars, which exhibit even larger gravitational forces, the electrons have merged with the protons to form neutrons, which produce a larger degeneracy pressure. Neutrons are the most "rigid" objects known - their Young modulus (or more accurate, bulk modulus) is 20 orders of magnitude larger than that of diamond.
Another physical phenomenon for which the Pauli principle is responsible is ferromagnetism, in which the exclusion effect results in exchange energy that induces neighboring electron spins to align (whereas classically they would anti-align).
Exceptions
According to general relativity, gravity in the vicinity of point mass rises without limit so it can overcome degeneracy pressure of neutrons (or any other known fermions) and thus collapse matter into a black hole.
References
See also
ca:Principi d'exclusió de Pauli cs:Pauliho vylučovací princip da:Paulis udelukkelsesprincip de:Pauli-Prinzip es:Principio de exclusión de Pauli fr:Principe d'exclusion de Pauli ko:파울리 배타 원리 it:Principio di esclusione di Pauli he:עקרון האיסור של פאולי hu:Pauli-elv nl:Uitsluitingsprincipe van Pauli ja:パウリの排他原理 pl:Reguła Pauliego pt:Princípio de exclusão de Pauli ru:Принцип Паули sl:Paulijevo izključitveno načelo sr:Паулијев принцип fi:Paulin kieltosääntö sv:Paulis uteslutningsprincip vi:Nguyên lý loại trừ tr:Pauli dışlama prensibi zh:泡利不相容原理