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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.
Spatial quantum numbers
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:
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:
- Principal quantum number (n = 1, 2, 3,...) appears on the radial part;
- Azimuthal quantum number (l = 0, 1 ... n−1) (also known as the angular quantum number or orbital quantum number) is linked to the azimuthal part;
- Magnetic quantum number (ml = −l, −l+1 ... 0 ... l−1, l) is linked to the colatitudinal part;
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.
- angular-momentum related:
- Baryon number
- Lepton number
- Electric charge
- Color charge
- Weak isospin
- Weak hypercharge
- Quantum numbers associated with quark's flavor:
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:
- Quantum Numbers and Electron Configurations
- Quantum numbers for the hydrogen atom
- Lecture notes on quantum numbers
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