Pauli exclusion principle

The Pauli exclusion principle is a rule of quantum mechanics, stated by Wolfgang Ernst Pauli in 1925. It states that no two fermions can be in the same quantum state (that is, with all their quantum numbers identical) within the same quantum system. Initially formulated as a principle, it was later found to be derivable from more general assumptions: in fact, it is a consequence of the spin statistics theorem.

Introduction

Historically, the Pauli exclusion principle was formulated to explain the atomic structure and the organization of the periodic table, and it consisted of imposing a restriction on the distribution of electrons in the different quantum states. Subsequently, the analysis of systems of identical particles led to the conclusion that any state must have a symmetry under peculiar particle exchange, which implied that there were two types of particles: Fermions, which would satisfy the Pauli principle, and bosons, which they would not satisfy him.

The Pauli exclusion principle stipulates that two Fermions cannot occupy the same quantum state within the same system at the same time, while for the case of electrons it stipulates that it is impossible for 2 electrons in the same atom to have the same 4 values for the quantum numbers, where those 4 numbers include the principal quantum number, the angular momentum quantum number, the magnetic quantum number, and finally, the spin quantum number. As has been said, the Pauli exclusion principle is only applicable to Fermions, that is, particles that form antisymmetric quantum states and have half-integer spin. Fermions are, for example, electrons and quarks (the latter are what form protons and neutrons). On the other hand, particles like the photon, and the (hypothetical) graviton, do not obey this principle, since they are bosons, that is, they form symmetric quantum states and have integer spin. As a consequence, a multitude of photons can be in the same particle quantum state, as in lasers.

It is easy to derive the Pauli principle, based on the spin-statistical theorem applied to identical particles. Fermions of the same species form systems with completely antisymmetric states, which in the case of two particles means that:

日本語END END (x)END END ♫(x♫) =− − 日本語END END ♫(x)END END (x♫) {displaystyle Δpsi (x)psi '(x)rangle =-Δpsi '(x)psi (x)rangle }

The permutation of one particle by another reverses the sign of the function that describes the system. If the two particles occupy the same quantum state 日本語END END {displaystyle 日本語psi rangle }, the state of the complete system is 日本語END END END END {displaystyle 日本語psi rangle }. Then:

日本語END END (x)END END (x♫) =− − 日本語END END (x♫)END END (x) =0(ket nulo){displaystyle Δpsi (x)psi (x')rangle =-Δpsi (x')psi (x)rangle =0;{hbox{(ket nulo)}}}}}

In this case, it cannot be given because the previous ket does not represent a physical state. This result can be generalized by induction to the case of more than two particles.

History

At the beginning of the 20th century it became clear that atoms and molecules with an even number of electrons are more chemically stable than those with an odd number of electrons. In the 1916 article "The Atom and the Molecule" From Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to have an even number of electrons in any shell, and especially to have eight electrons, which he assumed to be typically symmetrically arranged in the eight corners of a cube. In 1919 the chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or grouped together in some way. Clusters 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 " closed shells" stable.

Pauli sought an explanation for these numbers, which at first were only empirical. At the same time he was trying to explain the experimental results of the Zeeman effect in atomic spectroscopy and ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner, who noted that, for a given value of the principal quantum number (n), the number of energy levels of a single electron in the spectrum of alkali metals in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of 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 states of the electrons are defined using four quantum numbers. For this he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as the spin of the electron.

Consequences

The best known case, due to its wide use in the field of chemistry and atomic physics, is the quantum system of the Schrödinger atom, where the Fermions are the electrons. That is why it is the best-known version of this motto:

Two electrons in an atom cannot have the same quantum numbers.

Another physical phenomenon for which the Pauli principle is responsible is ferromagnetism, in which the exclusion principle implies an exchange energy that induces the parallel alignment of neighboring electrons (which classically would align antiparallel).