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 particle 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.

Particles obeying the Pauli exclusion principle are called fermions. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar = h/2\pi} (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 that are not fermions can only be bosons, which are particles described using "symmetric states" in quantum theory. Bosons are allowed to share quantum states, and possess integer spin. Examples of bosons include the photon and the W and Z bosons.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_1\right\rangle} (nota) and the other in state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_2\right\rangle} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_1, \psi_2\rangle = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle\psi_2\rangle - |\psi_2\rangle\psi_1\rangle \right) }

However, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_1\right\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_2\right\rangle} are just the same state, the above formula gives the zero ket:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_1, \psi_2\rangle = 0 }

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 role in a huge number of physical phenomena. One of the most important, and the one for which it was originally formulated, is the electron shell structure of atoms. 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.

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 the most 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.

Another physical phenomenon for which the Pauli principle is responsible is ferromagnetism, in which the exclusion effect implies an exchange energy that induces neighboring electron spins to align (whereas classically they would anti-align).

Exceptions

There is a limit to the amount of "degeneracy pressure" exerted by matter. Inside a black hole, the gravitational pull is strong enough to overcome that pressure, and so Pauli's exclusion principle does not hold.

References

  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X.
  • Liboff, Richard L. (2002). Introductory Quantum Mechanics, Addison-Wesley. ISBN 0805387145.
  • Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.), W. H. Freeman. ISBN 0716743450.

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