The Azimuthal quantum number (or orbital angular momentum quantum number) l is a quantum number for an atomic orbital which determines its orbital angular momentum. Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials. The various orbitals relating to different values of l are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letter, as follows: A quantum number is a number used to parametrise certain properties of particles or other systems in quantum mechanics. ... An atomic orbital is the description of the behavior of an electron in an atom according to quantum mechanics. ... In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude... In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... The associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ...

Each of the different angular momentum states can take 2(2l+1) electrons. This is because the third quantum number m_l (which can be thought of [somewhat inaccurately] as the [quantised] projection of the angular momentum vector on the z-axis) runs from −l to l in integer units, and so there are 2l+1 possible states. Each distinct nlm_l orbital can be occupied by two electrons with opposing spins (given by the quantum number m_s), giving 2(2l+1) electrons overall. Orbitals with higher l than given in the table are perfectly permissible, but these values cover all atoms so far discovered.

For a given value of the Principal quantum number, n, the possible values of l range from 0 to n−1; therefore, the n=1 shell only possesses an s subshell and can only take 2 electrons, the n=2 shell possesses an s and a p subshell and can take 8 electrons overall, the n=3 shell possesses s, p and d subshells and has a maximum of 18 electrons, and so on (generally speaking, the maximum number of electrons in the nth energy level is 2n²). The Principal quantum number (usually written n) is the first quantum number of an atomic orbital. ...