Azimuthal quantum number
In addition, the spin quantum number ms can take two distinct values.
While originally used just for isolated atoms, atomic-like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids.
The quantum number ℓ plays an important role here via the connection to the angular dependence of the spherical harmonics for the different orbitals around each atom.
The term "azimuthal quantum number" was introduced by Arnold Sommerfeld in 1915[1]: II:132 as part of an ad hoc description of the energy structure of atomic spectra.
Only later with the quantum model of the atom was it understood that this number, ℓ, arises from quantization of orbital angular momentum.
Some textbooks[2]: 199 and the ISO standard 80000-10:2019[3] call ℓ the orbital angular momentum quantum number.
[4]: 240 The lowercase letter ℓ, is used to denote the orbital angular momentum of a single particle.
[3] There are four quantum numbers—n, ℓ, mℓ, ms— connected with the energy states of an isolated atom's electrons.
An electron's angular momentum, L, is related to its quantum number ℓ by the following equation:
where ħ is the reduced Planck constant, L is the orbital angular momentum operator and
The wavefunctions of these orbitals take the form of spherical harmonics, and so are described by Legendre polynomials.
The several orbitals relating to the different (integer) values of ℓ are sometimes called sub-shells—referred to by lowercase Latin letters chosen for historical reasons—as shown in the table "Quantum subshells for the azimuthal quantum number".
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.
Orbitals with higher ℓ than given in the table are perfectly permissible, but these values cover all atoms so far discovered.
A simplistic one-electron model results in energy levels depending on the principal number alone.
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.
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: Given a quantized total angular momentum
Due to the spin–orbit interaction in an atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin.
The quantum numbers describing the system, which are constant over time, are now j and mj, defined through the action of J on the wavefunction
So that j is related to the norm of the total angular momentum and mj to its projection along a specified axis.
As with any angular momentum in quantum mechanics, the projection of J along other axes cannot be co-defined with Jz, because they do not commute.
The angular momentum quantum numbers strictly refer to isolated atoms.
The ℓ m quantum number corresponds to specific spherical harmonics and are commonly used to describe features observed in spectroscopic methods such as X-ray photoelectron spectroscopy[6] and electron energy loss spectroscopy.
The angular momentum quantum numbers are also used when the electron states are described in methods such as Kohn–Sham density functional theory[8] or with gaussian orbitals.
[9] For instance, in silicon the electronic properties used in semiconductor device are due to the p-like states with
centered at each atom, while many properties of transition metals depend upon the d-like states with
[10] The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld.
[12] In three-dimensions the orbits become spherical without any nodes crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.
