Azimuthal quantum number

Azimuthal quantum number is a quantum number or integer assigned to an atomic orbital. It is also denoted as “orbital angular momentum quantum number” or “second quantum number” and is designated by the symbol “ɩ”. It helps in the determination of the orbital angular momentum, which further helps in elucidating the shape of the electron’s orbital. Each electron has a set of four numbers, called as quantum numbers that is highly specific, and no two electrons in the same atom can have the same set of four quantum numbers. These constant numbers are essential to describe the position, spin, energy and orientation of an orbital in space.

Azimuthal quantum number is the second set of the quantum numbers which determines the quantum state of an electron, the other three being:

Principle quantum number (n)

It is the first quantum number that helps in determining the size of the orbital, or how far the electron extends from the nucleus. The higher the value of n, the further it is from the nucleus.

Magnetic quantum number (m)

It is the third quantum number that splits the subshells into individual orbitals. This orbital is oriented in a three dimensional form in space.

Spin quantum number (s)

The fourth and final quantum number gives the orientation of the two electrons in each orbital. Its values are considered to be +1/2 and -1/2. 

Azimuthal quantum number Picture

Picture – Azimuthal quantum number

Azimuthal quantum number History

It was first derived from the Bohr model of the atom and was postulated by Arnold Sommerfield. The Bohr model illustrates the properties of an atom, which was proposed by Niel Bohrs in 1915. It is quite familiar to a planetary model, where the neutrons and protons occupy a dense central region called the nucleus, and the electrons orbit around it like the planets revolving around the sun. The lowest quantum or energy level was found to have an angular momentum of zero. Initially, the orbits were considered as oscillating charges in one dimension like a “pendulum” as per the rule in mathematics. But later it was formulated that the orbit becomes spherical and oscillates in a circular motion in a three dimensional direction.

Azimuthal quantum number Derivation

The four quantum numbers n, ɩ, mɩ and ms signify the energy states of the electrons of an atom. These numbers denote the quantum state of a single electron which is usually represented as {n, ɩ, mɩ and ms}, which forms its wavefunction or orbital. The atomic orbital is basically a mathematical function that describes the wave-like nature of either one or a pair of electrons in an atom. Wave function on the other hand is represented as complex numbers, and is a function of space and time.

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur. The wave function of this particular wave equation gives rise to three equations that when solved; gives the first three quantum numbers that are found to be interrelated. The polar segment of the wave equation forms the azimuthal quantum number. The concept of the azimuth can be clearly understood from the spherical coordinate systems, which works well only with spherical models. It is the angle formed between the projected vector and a reference vector on the reference plane.

The angular momentum, L of an electron in an atom is related to its quantum number ɩ by the following equation:

L2ψ= ћ2 ɩ (ɩ+1) ψ

Where ћ is the reduced Planck’s constant, L2 is the angular momentum operator (cross product of a wave function’s position operator (r) and momentum operator (p)), and ψ is the wave function of the electron. Ɩ is always a positive integer like 0, 1, 2, 3, etc, according to angular momentum quantization, where the values do not vary continuously but have only certain quantized values. The angular momentum is usually referred to as L that does not have an exact meaning in quantum physics.

Wave is dissipated in small packets of energy called quanta because the energy of any wave is the frequency multiplied by Planck’s constant. Each of the quantum numbers is usually represented in the quantum state, as the formulae for each quantum number include Planck’s reduced constant which allows certain discrete values. The shape of the orbital characterizes this nature of the wave.

Atomic orbitals have unique shapes which are denoted by letters s, p and d which correspond to ɩ=0, ɩ=1, ɩ=2, and ɩ=3. The shapes of the various orbitals are analogous to the different values of ɩ, and are sometimes called as sub-shells.



Maximum  electrons



Number  of orbitals










Two dumbbells






Four dumbbells






Eight dumbbells



Each of the different angular momentum states can take 2(2ɩ+1) electrons. For a given value of the principal quantum number n, the possible values of ɩ range from 0 to n-1. Thus, n=1 shell possesses an s subshell with 2 electrons, n=2 shell possesses an s and a p subshell with 8 electrons, n=3 shell possesses s, p and d subshells with 18 electrons, and so on. In general, maximum number of electrons in the nth level is 2n2.

The angular momentum quantum number, ɩ, gives the number of planar nodes going through the nucleus. A planar node in an electromagnetic wave is the midpoint between crest and trough, having zero magnitude. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number ɩ takes the value of 0. In a p orbital, one node traverses the nucleus and therefore ɩ has the value of 1. L is assigned the value ħ

Based on the value of n, the angular momentum quantum number ɩ can be categorized. Below is the list of wavelengths for a hydrogen atom.

n = 1, L = 0, Lyman series (ultraviolet)

n = 2, L = √2ħ, Balmer series (visible)

n = 3, L = √6ħ, Ritz-Paschen series (short wave infrared)

n = 5, L = 2√5ħ, Pfund series (long wave infrared).

Addition of quantized angular momenta

The total angular momentum Symbol 1 Picture is the sum of two individual quantized angular momentaSymbol 2 Image and Symbol 3 Photo,

Equation 1 Picture

The quantum number j associated with its magnitude can range from Equation 2 Image to Equation 3 Photo where Symbol 4 Picture and Symbol 5 Image are quantum numbers corresponding to the magnitudes of the individual angular momenta.