In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number , the magnetic quantum number , and the spin quantum number ). It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as (pronounced ell).
Derivation
Connected with the energy states of the atom's electrons are four quantum numbers: n, ℓ, mℓ, and ms. These specify the complete, unique quantum state of a single electron in an atom, and make up its wavefunction or orbital. When solving to obtain the wave function, the Schrödinger equation reduces to three equations that 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 , reliant on the spherical coordinate system, which generally works best with models having some glimpse of spherical symmetry.
An atomic electron's angular momentum, , is related to its quantum number by the following equation:
where is the reduced Planck's constant, is the orbital angular momentum operator and is the wavefunction of the electron. The quantum number is always a non-negative integer: 0, 1, 2, 3, etc. has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is better to simply use the quantum number .
Atomic orbitals have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d (a convention originating in spectroscopy) 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 are referred to by lowercase Latin letters (chosen for historical reasons), as follows:
Each of the different angular momentum states can take 2(2ℓ + 1) electrons. This is because the third quantum number mℓ (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, mℓ orbital can be occupied by two electrons with opposing spins (given by the quantum number ms = ±), 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 ; therefore, the shell only possesses an s subshell and can only take 2 electrons, the shell possesses an s and a p subshell and can take 8 electrons overall, the shell possesses s, p, and d subshells and has a maximum of 18 electrons, and so on.
A simplistic one-electron model results in energy levels depending on the principal number alone. In more complex atoms these energy levels split for all , placing states of higher ℓ above states of lower ℓ. For example, the energy of 2p is higher than of 2s, 3d occurs higher than 3p, which in turn is above 3s, etc. This effect eventually forms the block structure of the periodic table. No known atom possesses an electron having ℓ higher than three (f) in its ground state.
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 magnitudes. 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. has the value .
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:
Addition of quantized angular momenta
Given a quantized total angular momentum which is the sum of two individual quantized angular momenta and ,
the quantum number 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.