Azimuthal Quantum Number - Derivation

Derivation

Associated with the energy states of the electrons of an atom is a set of four quantum numbers: n, ℓ, m_ℓ, and m_s. These specify the complete and unique quantum state of a single electron in an atom, and make up its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to 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. To aid understanding of this concept of the azimuth, it may also prove helpful to review spherical coordinate systems, and/or other alternative mathematical coordinate systems besides the cartesian coordinate system. Generally, the spherical coordinate system works best with spherical models, the cylindrical system with cylinders, the cartesian with general volumes, etc.

An atomic electron's angular momentum, L, is related to its quantum number ℓ by the following equation:

where ħ is the reduced Planck's constant, L2 is the orbital angular momentum operator and is the wavefunction of the electron. The quantum number ℓ is always a nonnegative integer: 0,1,2,3, etc. (see angular momentum quantization). 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.

Atomic orbitals 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 sub-shells, 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 "sub-shells" is some poor dumb fool. Other mnemonics for the order of the "sub-shells" include silly professors dance funny and Scott picks dead flowers. The letters after the f sub-shell 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 z-axis) runs from −ℓ to ℓ in integer units, and so there are 2ℓ + 1 possible states. Each distinct n,ℓ,m_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 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 0. In a p orbital, one node traverses the nucleus and therefore ℓ has the value of 1. L has the value √2ħ.

Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The wavelengths listed are 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).