Quantum mechanics obey very different laws than classical mechanics. These laws include the concept that a particle can be in more than one place at once, that a particle's location and momentum cannot be known at the same time and that a particle can act as both a particle and as a wave.

The Pauli exclusion principle is another law that seems to defy classical logic, but it is incredibly important for the electronic structure of atoms.

## Particle Classification

All elementary particles can be classified as **fermions or bosons**. Fermions have half-integer spin, meaning they can only have spin values of positive and negative 1/2, 3/2, 5/2 and so on; bosons have integer spin (this includes zero spin).

Spin is intrinsic angular momentum, or angular momentum that a particle simply has without it being created by any external force or influence. It is unique to quantum particles.

The Pauli exclusion principle **only applies to fermions**. Examples of fermions include electrons, quarks and neutrinos, as well as any combination of those particles in odd numbers. Protons and neutrons, which are made of three quarks, are therefore also fermions, as are atomic nuclei which have an odd number of protons and neutrons.

The most important application of the Pauli exclusion principle, electron configurations in atoms, involves electrons specifically. In order to understand their importance in atoms, it is first important to understand the foundational concept behind atomic structure: quantum numbers.

## Quantum Numbers in Atoms

The quantum state of an electron in an atom can be precisely defined by a set of four quantum numbers. These numbers are called the principal quantum number *n*, the azimuthal quantum number *l* (also called the orbital angular momentum quantum number), the magnetic quantum number *m _{l}* and the spin quantum number

*m*.

_{s}The set of quantum numbers provides the foundation for the shell, subshell and orbital structure of describing electrons in an atom. A shell contains a group of subshells with the same principal quantum number, *n*, and each subshell contains orbitals of the same orbital angular momentum quantum number, *l*. An s subshell contains electrons with *l*=0, a p subshell with *l*=1, a d subshell with *l*=2 and so on.

The value of *l* ranges from 0 to *n*-1. So the *n*=3 shell will have 3 subshells, with *l* values of 0, 1, and 2.

The magnetic quantum number, *m _{l}*, ranges from

*-l* to

*l* in increments of one, and defines the orbitals within a subshell. For example, there are three orbitals within a p (

*l*=1) subshell: one with

*m* =-1, one with

_{l}*m* =0 and one with

_{l}*m*=1.

_{l}The last quantum number, the spin quantum number *m _{s}*, ranges from

*-s* to

*s* in increments of one, where

*s* is the spin quantum number that is intrinsic to the particle. For electrons,

*s* is 1/2. This means

*all* electrons can only ever have spin equal to -1/2 or 1/2, and any two electrons with the same

*n*,

*l*, and

*m* quantum numbers must have antisymmetric or opposite spins.

_{l}As stated before, the *n*=3 shell will have 3 subshells, with *l* values of 0, 1 and 2 (s, p and d). The d subshell (*l*=2) of the *n*=3 shell will have five orbitals: *m _{l}*=-2, -1, 0, 1, 2. How many electrons will fit in this shell? The answer is determined by the Pauli exclusion principle.

## What Is the Pauli Exclusion Principle?

The Pauli principle is named for Austrian physicist **Wolfgang Pauli**, who wanted to explain why atoms with an even number of electrons were more chemically stable than those with an odd number.

He eventually came to the conclusion that there must be four quantum numbers, necessitating the invention of electron spin as the fourth, and, most importantly, no two electrons could have the same four quantum numbers in an atom. It was impossible for two electrons to be in the exact same state.

This is the Pauli exclusion principle: Identical fermions are not allowed to occupy the same quantum state at the same time.

We can now answer the previous question: How many electrons can fit in the d subshell of the *n*=3 subshell, given that it has five orbitals: *m _{l}*=-2, -1, 0, 1, 2? The question has already defined three of the four quantum numbers:

*n*=3,

*l*=2, and the five values of

*m*. So for each value of

_{l}*m* there are two possible values of

_{l},*m*: -1/2 and 1/2.

_{s}This means that ten electrons can fit in this subshell, two for each value of *m _{l}*. In each orbital, one electron will have

*m*=-1/2, and the other will have

_{s}*m*=1/2.

_{s}## Why Is the Pauli Exclusion Principle Important?

The Pauli exclusion principle informs electron configuration and the way atoms are classified in the periodic table of elements. Ground state, or lowest energy levels in an atom can fill up, forcing any additional electrons to higher energy levels. This is, fundamentally, the reason why ordinary matter in the solid or liquid phase occupies a **stable volume**.

Once the lower levels are filled, electrons cannot fall closer to the nucleus. Atoms therefore have a minimum volume and have a limit to how much they can be squeezed together.

Possibly the most dramatic example of the principle's importance can be seen in neutron stars and white dwarfs. The particles making up these small stars are under incredible gravitational pressure (with a bit more mass, these stellar remnants could have collapsed into black holes).

In normal stars, the heat energy produced at the center of the star by nuclear fusion creates enough outward pressure to oppose the gravity created by their incredible masses; but neither neutron stars nor white dwarfs undergo fusion in their cores.

What keeps these astronomical objects from collapsing under their own gravity is an internal pressure called degeneracy pressure, also known as Fermi pressure. In white dwarfs, the particles in the star are so crunched together, that to get any closer to each other, some of their electrons would have to occupy the same quantum state. But the Pauli exclusion principle says that they can't!

This also applies to neutron stars, because neutrons (which make up the entire star) are also fermions. But if they got too close together, they would be in the same quantum state.

Neutron degeneracy pressure is slightly stronger than electron degeneracy pressure, but both are directly caused by the Pauli exclusion principle. With their particles so impossibly close together, white dwarfs and neutron stars are the densest objects in the universe outside of black holes.

The white dwarf Sirius-B has a radius of just 4,200 km (Earth's radius is about 6,400 km) but is almost as massive as the Sun. Neutron stars are even more incredible: there is a neutron star in the constellation Taurus whose radius is only 13 km (just 6.2 miles), but it is *twice* as massive as the Sun! A *teaspoon* of neutron star material would weigh about a trillion pounds.

References

About the Author

Meredith is a science writer and physicist based in Seattle. She received her Bachelor of Science degree in physics from the University of Illinois at Urbana-Champaign and her Master of Science degree in physics from the University of Washington. She has written for Live Science, Physics, Symmetry, and WIRED, and was an AAAS Mass Media Fellow in 2019.