![HAtomOrbitals](https://upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png)
The azimuthal quantum number is a
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
for an
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
that determines its
orbital angular momentum and describes the shape of the orbital. The
azimuthal
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematically, ...
quantum number is the second of a set of quantum numbers that describe the unique
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
of an electron (the others being the
principal quantum number
In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
A ...
, the
magnetic quantum number
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes th ...
, and the
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe th ...
). It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as ℓ (pronounced ''ell'').
Derivation
Connected with the energy states of the atom's electrons are four quantum numbers: ''n'', ''ℓ'', ''m''
''ℓ'', and ''m''
''s''. These specify the complete, unique quantum state of a single
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
in an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
, and make up its
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
or ''orbital''. When solving to obtain the wave function, the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, which generally works best with models having some glimpse of
spherical symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
.
![Vector model of orbital angular momentum](https://upload.wikimedia.org/wikipedia/commons/3/30/Vector_model_of_orbital_angular_momentum.svg)
An atomic electron's
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, ''L'', is related to its quantum number ''ℓ'' by the following equation:
where ''ħ'' is the
reduced Planck's constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, L
2 is the orbital angular momentum operator and
is the wavefunction of the electron. The quantum number ''ℓ'' is always a non-negative integer: 0, 1, 2, 3, etc. L has no real meaning except in its use as the
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
. When referring to angular momentum, it is better to simply use the quantum number ''ℓ''.
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
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
.
Their wavefunctions take the form of
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
s, and so are described by
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
. The various orbitals relating to different values of ''ℓ'' are sometimes called sub-shells, and are referred to by lowercase
Latin letters
The Latin script, also known as Roman script, is an alphabetic writing system based on the letters of the classical Latin alphabet, derived from a form of the Greek alphabet which was in use in the ancient Greek city of Cumae, in southern Italy ...
(chosen for historical reasons), as follows:
Each of the different angular momentum states can take 2(2''ℓ'' + 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 2''ℓ'' + 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(2''ℓ'' + 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
In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
A ...
''n'', the possible values of ''ℓ'' range from 0 to ; therefore, the
shell
Shell may refer to:
Architecture and design
* Shell (structure), a thin structure
** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses
** Thin-shell structure
Science Biology
* Seashell, a hard o ...
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 level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...
s depending on the principal number alone. In more complex atoms these energy levels
split
Split(s) or The Split may refer to:
Places
* Split, Croatia, the largest coastal city in Croatia
* Split Island, Canada, an island in the Hudson Bay
* Split Island, Falkland Islands
* Split Island, Fiji, better known as Hạfliua
Arts, enterta ...
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 ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the 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
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
:
:
,
Lyman series In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from ''n'' ≥ 2 to ''n'' = 1 (where ''n'' is the principa ...
(ultraviolet)
:
,
Balmer series
The Balmer series, or Balmer lines in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula, an empirical equation discovered b ...
(visible)
:
,
Ritz–Paschen series (
near infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
)
:
,
Brackett series
The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an ...
(
short-wavelength infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
)
:
,
Pfund series (
mid-wavelength infrared).
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
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
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.
Total angular momentum of an electron in the atom
![LS coupling](https://upload.wikimedia.org/wikipedia/commons/3/30/LS_coupling.svg)
Due to the
spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbi ...
in the atom, the orbital angular momentum no longer
commutes with the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, nor does the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
. These therefore change over time. However the
total angular momentum
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's s ...
J does commute with the one-electron Hamiltonian and so is constant. J is defined through
:
L being the
orbital angular momentum and S the spin. The total angular momentum satisfies the same
commutation relations as orbital angular momentum, namely
:
from which follows
:
where ''J''
i stand for ''J''
x, ''J''
y, and ''J''
z.
The quantum numbers describing the system, which are constant over time, are now ''j'' and ''m''
''j'', defined through the action of J on the wavefunction
:
:
So that ''j'' is related to the norm of the total angular momentum and ''m''
''j'' to its projection along a specified axis. The ''j'' number has a particular importance for
relativistic quantum chemistry
Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
, often featuring in subscript in
electron configuration of superheavy elements.
As with any
angular momentum in quantum mechanics, the projection of J along other axes cannot be co-defined with ''J''
z, because they do not commute.
Relation between new and old quantum numbers
''j'' and ''m''
''j'', together with the
parity of the
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
, replace the three
quantum numbers
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
''ℓ'', ''m''
''ℓ'' and ''m''
''s'' (the projection of the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
along the specified axis). The former quantum numbers can be related to the latter.
Furthermore, the
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of ''j'', ''s'', ''m''
''j'' and parity, which are also
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, are linear combinations of the
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of ''ℓ'', ''s'', ''m''
''ℓ'' and ''m''
''s''.
List of angular momentum quantum numbers
* Intrinsic (or spin) angular momentum quantum number, or simply
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe th ...
* orbital angular momentum quantum number (the subject of this article)
*
magnetic quantum number
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes th ...
, related to the orbital momentum quantum number
*
total angular momentum quantum number
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's s ...
History
The azimuthal quantum number was carried over from the
Bohr model of the atom
In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Sy ...
, and was posited by
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
. The Bohr model was derived from
spectroscopic analysis
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
of the atom in combination with the
Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. Orbits with zero angular momentum were considered as oscillating charges in one dimension and so described as "pendulum" orbits, but were not found in nature.
In three-dimensions the orbits become spherical without any
nodes
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.
See also
*
Angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
*
Introduction to quantum mechanics
Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the b ...
*
Particle in a spherically symmetric potential
In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potential ...
*
Angular momentum coupling
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact t ...
*
Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
References
External links
Development of the Bohr atom
{{electron configuration navbox
Angular momentum
Atomic physics
Quantum numbers
Rotational symmetry
de:Nebenquantenzahl