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Quantum number denoting orbital angular momentum

In quantum mechanics, the azimuthal quantum number ℓ is a quantum number for an atomic orbital that determines its orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number n, the magnetic quantum number , and the spin quantum number ). For a given value of the principal quantum number n (electron shell), the possible values of ℓ are the integers from 0 to 1=n − 1. For instance, the 1=n = 1 shell has only orbitals with , and the 1=n = 2 shell has only orbitals with , and
For a given value of the azimuthal quantum number ℓ, the possible values of the magnetic quantum number are the integers from to , including 0. In addition, the spin quantum number can take two distinct values. The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell. While originally used just for isolated atoms, atomic like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids. The quantum number ℓ plays an important role here via the connection to the angular dependence of the spherical harmonics for the different orbitals around each atom.

The term "azimuthal quantum number" was introduced by Arnold Sommerfeld in 1915 as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, ℓ, arises from quantization of orbital angular momentum. Some textbooks and the ISO standard 80000 10:2019 call ℓ the orbital angular momentum quantum number. The energy levels of an atom in an external magnetic field depend upon the value so it is sometimes called the magnetic quantum number. The lowercase letter ℓ, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter L is used. and electron energy loss spectroscopy. (The notation is slightly different, with X ray notation where K, L, M are used for excitations out of electron states with .) The angular momentum quantum numbers are also used when the electron states are described in methods such as Kohn–Sham density functional theory or with gaussian orbitals. For instance, in silicon the electronic properties used in semiconductor device are due to the p like states with centered at each atom, while many properties of transition metals depend upon the d like states with .

The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld. The Bohr model was derived from spectroscopic analysis of atoms in combination with the Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. Orbits with zero angular momentum were considered as oscillating charges in one dimension and so described as "pendulum" orbits, but were not found in nature. In three dimensions the orbits become spherical without any nodes crossing the nucleus, similar (in the lowest energy state) to a skipping rope that oscillates in one large circle.