The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as ℓ (lowercase L) is a quantum number for an atomic orbital that determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers (the principal quantum number, following spectroscopic notation, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) which describe the unique quantum state of an electron and is designated by the letter ℓ.
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There is a set of quantum numbers associated with the energy states of the electrons of an atom. The four quantum numbers n, ℓ, m_{ℓ}, and m_{s} specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation as shown below. In addition to understanding this concept of the azimuth, one may also find it necessary to review or learn more about spherical coordinate systems, and/or other alternative mathematical coordinate systems other than the cartesian coordinate system. It is known that the spherical coordinate system works best with spherical models, the cylindrical system with cylinders, the cartesian with general areas, etc. The concept of the azimuth and how it is used to explain electron orbital shapes may be more understandable after such a review.
An atomic electron's angular momentum, L, is related to its quantum number ℓ by the following equation:
where ħ=h/2π is the reduced Planck's constant, also called Dirac's constant, L^{2} is the orbital angular momentum operator and Ψ is the wavefunction of the electron. While many introductory textbooks on quantum mechanics will refer to L by itself, L has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is best to simply use the quantum number ℓ.
The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particlelike packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.
This behavior manifests itself as the "shape" of the orbital.
Electron shells have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the atomic orbital.
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 subshells, and (mainly for historical reasons) are referred to by letters, as follows:
ℓ  Letter  Max electrons  Shape  Name 

0  s  2  sphere  sharp 
1  p  6  two dumbbells  principal 
2  d  10  four dumbbells or unique shape one  diffuse 
3  f  14  eight dumbbells or unique shape two  fundamental 
4  g  18  
5  h  22  
6  i  26 
A mnemonic for the order of the "shells" is some poor dumb fool. Another mnemonic for the order of the "shells" is silly professors dance funny. The letters after the F subshell just follow F in alphabetical order.
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 loosely as the quantized projection of the angular momentum vector on the zaxis) 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 2n^{2}).
The angular momentum quantum number, ℓ, governs the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and trough, which has zero magnitude. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number ℓ takes the value of zero. In a p orbital, one node traverses the nucleus and therefore ℓ has the value 1.
Depending on the value of n, the principal quantum number, there is an angular momentum quantum number ℓ and the following series. The wavelengths listed are for a hydrogen atom:
Given a quantized total angular momentum which is the sum of two individual quantized angular momenta and ,
the quantum number j associated with its magnitude can range from to in integer steps where and are quantum numbers corresponding to the magnitudes of the individual angular momenta.
Due to the spinorbit interaction in the atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin. These therefore change over time. However the total angular momentum J does commute with the Hamiltonian and so is constant. J is defined through
L being the orbital angular momentum and S the spin. The total angular momentum satisfies the same commutation relations as angular momentum, namely
from which follows
where J_{i} stand for J_{x}, J_{y} and J_{z}.
The quantum numbers describing the system, which are constant over time, are now j and m_{j}, defined through the action of J on the wavefunction Ψ
So that j is related to the norm of the total angular momentum and m_{j} to its projection along a specified axis.
As with any angular momentum in quantum mechanics, the projection of J along other axes cannot be codefined with J_{z}, because they do not commute.
j and m_{j}, together with the parity of the quantum state, replace the three quantum numbers ℓ, m_{ℓ} and m_{s} (the projection of the spin along the specified axis). The former quantum numbers can be related to the latter.
Furthermore, the eigenvectors of j, m_{j} and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of ℓ, m_{ℓ} and m_{s}.
The azimuthal quantum number was carried over from the Bohr model of the atom. The Bohr model was derived from spectroscopic analysis of the atom in combination with the Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. To simplify the mathematics, orbits were considered as oscillating charges in one dimension and so described as "pendulum" orbits. In threedimensions the orbit becomes spherical without any nodes crossing the nucleus, similar to a jump rope that oscillates in one large circle.
