Pauli Exclusion Principle

In its simplest formulation, the Pauli principle states that no two electrons in an atom can be in exactly the same state. The state of an electron is understood to be the physically measurable state that an electron assumes in terms of energy, angular momentum, orientation of the angular momentum and orientation of the electron spin. For quantum particles, including electrons, all these physical quantities are abbreviated as so-called quantum numbers. Here, n stands for the energy, l for the angular momentum, m for the orientation of the angular momentum, s for the spin and sm for the orientation of the electron spin. A set of quantum numbers therefore consists of the set (n, l, m, s, sm).

The Pauli principle states that two electrons in a hydrogen atom must not match in all quantum numbers. More generally, according to the Pauli principle, all so-called 'fermions', which include electrons, must not match in all their properties.

The Pauli principle was discovered by the physicist Wolfgang Ernst Pauli, who was awarded the Nobel Prize in 1945 for this discovery, which he himself referred to as the 'exclusion principle'.

Consequences of the Pauli exclusion principle

The Pauli principle has many consequences. For example, it is used to explain the electron configurations of various elements. Even the stability of neutron stars is explained using the Pauli principle. Neutron stars are so heavy and compact that there is no known force that could withstand the gravitational pressure in a neutron star apart from the repulsion of the neutrons, which cannot occupy the same place due to the Pauli interdiction.

Without the enormous repulsive force of the neutrons in a neutron star, which comes from the Pauli interdiction, a neutron star would collapse to a point under its own weight. This also happens above a certain limit mass and a black hole remains. The Pauli principle is therefore valid within the framework of conventional forces but is not fundamentally insurmountable.

Even the exchange interaction of electrons in a solid, which is responsible for the phenomenon of ferromagnetism, can only be understood using the Pauli principle.

Exchange interaction

In today's formulation, the Pauli principle reads: "The total wave function of a system of N fermions is totally antisymmetric with respect to the interchange of two particles". This sounds very abstract at first but can be explained using the exchange interaction as an example.

Electrons are so-called fermions. All particles with a half-integer spin are fermions. The electron spin has the quantum number ½. Elementary particles can be described mathematically in general terms using a so-called wave function. Electrons can also be described using a product of wave functions, with each factor of the overall wave function representing a specific property. For example, the spatial wave function describes the location, the spin wave function describes the spin, etc...

According to the Pauli exclusion principle, the electrons cannot be in the same place if they do not differ in any other quantum number (such as the direction of the spin, for example). This stems from the formulation of "total antisymmetry with regard to the interchange of two particles". More precisely, the Pauli principle must be understood to mean that the wave functions of neighbouring electrons in a solid must be antisymmetrical to each other. This means that the electrons must differ in exactly one or three properties ('antisymmetric') if all other properties are the same, i.e. 'symmetric'. The electrons must also not differ in exactly two properties. The product of two antisymmetric wave functions is otherwise symmetric again. In general, the product of an even number of antisymmetric wave functions is always symmetric and the product of an odd number of antisymmetric wave functions is always antisymmetric. The symmetrical wave functions do not change the overall wave function.

An odd number of functions must therefore be antisymmetric if all other functions that describe the properties of the particles are symmetric. The neighbouring electrons in a solid are electrons with an antisymmetrical spatial wave function. All other functions are symmetrical. You can imagine this as meaning that the electrons differ in terms of their location but not in any other way. In the language of symmetry of wave functions, one would say: The spatial wave function of the electrons is antisymmetric, all wave functions except the spin wave function are symmetric, so the last remaining wave function, namely the wave function of the spin orientation, must also be symmetric so that the overall wave function is antisymmetric, as required by the Pauli principle.

The electrons must, therefore, not differ in spin.

For the same reason that the electrons within an atom cannot have the same spin orientation at the same location, the electrons of neighbouring atoms in a ferromagnetic solid must not have different electron spin orientations, as they would otherwise be symmetrical in terms of all properties.

This is why the electron spins in a ferromagnet stabilise each other based on the Pauli principle. This interaction is called the exchange interaction, just as the formulation of the Pauli interdiction speaks of a necessary antisymmetry in the 'interchange' or 'exchange' of the particles.