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# Quantum number

A quantum number is a number used to parametrise certain properties of particles or other systems in quantum mechanics. Combinations of quantum numbers can be used to identify eigenstates of the system.

Each quantum number represents a specific degree of freedom that any particle can occupy. In this viewpoint, they can be seen as analogous to properties in classical systems. For example, the principal quantum number of an electron in an atom is roughly analogous to its orbital distance in classical mechanics. However, the defining characteristic of quantum mechanics is that these are quantised: that is, there is a specific discrete set of values that are allowed for each quantum number. This quantization property is the source of the word "quantum" in "quantum mechanics".

The nature of the sets of allowed values is quite fundamental. In the case of electrons in an atom, the restrictions on the allowed values of l and ml arise from the nature of the boundary condition imposed by the shape of the potential generated by the nucleus, and the restriction on ms is due to the electron itself.

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## Spatial quantum numbers

These are quantum numbers associated with bound electrons in a nucleus, and essentially they are related with electron's orbitals and energy levels.

Spatial quantum numbers arise in the solution to the Schödinger equation of the hydrogen atom, which originates from the application of the Hamiltonian to the electron wave function. The electron in an atom is under an electric potential of spherical symmetry. This justifies the use of polar coordinates in Schrödinger equation. After switching to polar coordinates (radius r, colatitude θ and azimuth φ) Schrödinger equation can be decomposed as a product of three parts:

$\Psi(r,\theta,\varphi)=R(r)\,\Theta(\theta)\,\Phi(\varphi)$

These parts are known as the radial R(r), colatitude Θ(θ) and azimuthal Φ(φ) parts of the Schrödinger equation. It is during the process of solving these three parts that the three spatial quantum numbers arise:

## Internal quantum numbers

These quantum numbers are intrinsic to elementary particles. They are also known as quantum charges. They are additive quantum numbers, i.e., the sum of them is usually preserved in a given interaction.

## Multiplicative quantum numbers

In some cases a quantum number product is preserved instead of addition. These is the case for some quantum numbers associated with discrete symmetries: