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Azimuthal Quantum Number - Physics

Azimuthal Quantum Number is a quantum number for atomic orbitals which helps identify their orbital angular momentum. They define the shape of the orbital. Azimuthal quantum number is the second set of quantum numbers, and is very helpful in determining the unique set of an electron’s quantum state. The other quantum numbers are principal quantum number, magnetic quantum number, and spin quantum number. The other names of azimuthal quantum numbers are second quantum number, orbital quantum number, or orbital angular quantum number. The representation of this azimuthal quantum number is l and is pronounced as ell.

History

Azimuthal quantum number has been used from the time of Bohr’s atomic model and was realized by Arnold Sommerfeld. The model of Bohr was derived from a very specific analysis called spectroscopic analysis. Azimuthal quantum number has also been combined with Rutherford analysis on atomic models. The quantum number can be determined as lowest when it contains an angular momentum of zero. The orbits which contain these zero angular moments are found to be having one dimensional oscillating charge and can be defined as pendulum orbits. They were not found in nature. These orbits are supposed to become spherical in nature in a 3-dimensional view that does not contain any nodes crossing the nucleus.

Derivation

They consist of electrons that possess all four kinds of quantum numbers, n, ℓ, mℓ, and ms. They can specify the full and unique state of quantum numbers which contain a single electron in an atom. It helps make up its orbital. During the time of solving the wave function, the Schrödinger equation decreases to 3 equations. This is reduced up to the first 3 quantum numbers. So, it can be found out that the first three quantum numbers equations inter-relate each other. The azimuthal quantum number can arise in a solution with a polar part. It is highly reliant on the spherical coordinate system in which it works well along with the prototypes that have the same view as of spherical symmetry.

Given below is the equation for relating angular momentum of an atomic electron, L and its quantum number l,

L² Ψ=h² l l+1 Ψ

Where, h is considered as Planck’s constant, L2 can be termed as the operator for orbital angular momentum and Ψ is the electron’s wave function. The quantum number l can always be considered as a non-negative integer. It can either be 0 or positive integers such as 1, 2, 3, etc. The expression L has no original meaning. But it can only be used as an operator for angular momentum. Hence, it is much better to only use l as the illustration for quantum numbers.

Atomic orbitals contain appealing shapes that can be demonstrated by letters. Letters s, p and d are used to define the shape of the orbitals present in the atomic nature.

These wave functions are highly capable of taking the form of spherical harmonics. Hence, they are regarded as Legendre polynomials. The different orbitals equating to various values of l can sometimes be named as sub-sells.

Quantum subshells for the azimuthal quantum number
Azimuthal number (ℓ) Historical letter Maximum electrons Historical name Shape
0 s 2 sharp spherical
1 p 6 principal three dumbbell-shaped polar-aligned orbitals; one lobe on each pole of the x, y, and z (+ and − axes)
2 d 10 diffuse nine dumbbells and one doughnut
3 f 14 fundamental unique shape
4 g 18    
5 h 22    
6 i 26    

From the above table, the letters that come after f shell just follow the letter f in an alphabetical order.

It is considered that each of the various angular momentum state is capable of taking 2 (2 l + 1) electrons. The third quantum number mℓ can be used to understand the idea of loosely packed projections which are quantized in nature. Orbitals that contain higher l than that of the given table values, are used to cover all kinds of atoms so far discovered.

 

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