orbital quantum number
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The azimuthal quantum number is a quantum number 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 an ...
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 i ...
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 the ...
, and the spin quantum number). 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 ...
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 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 ...
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 th ...
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 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 ...
. 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: \mathbf^2\Psi = \hbar^2 \ell(\ell + 1) \Psi, 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 ...
, L2 is the orbital angular momentum operator and \Psi 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 an ...
. 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 ...
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 applica ...
. 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(2''ℓ'' + 1) electrons. This is because the third quantum number ''m'' (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'' 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 t ...
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, entertai ...
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. L has the value \sqrt\hbar. 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 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 princip ...
(ultraviolet) : n = 2, L = \sqrt\hbar,
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) : n = 3, L = \sqrt\hbar, 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 arou ...
) : n = 4, L = 2\sqrt\hbar,
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) : n = 5, L = 2\sqrt\hbar, Pfund series ( mid-wavelength infrared).


Addition of quantized angular momenta

Given a quantized total angular momentum \vec which is the sum of two individual quantized angular momenta \vec and \vec, :\vec = \vec + \vec the quantum number j associated with its magnitude can range from , \ell_1 - \ell_2, to \ell_1 + \ell_2 in integer steps where \ell_1 and \ell_2 are quantum numbers corresponding to the magnitudes of the individual angular momenta.


Total angular momentum of an electron in the atom

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–or ...
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. 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 sp ...
J does commute with the one-electron Hamiltonian and so is constant. J is defined through :\vec = \vec + \vec L being the orbital angular momentum and S the spin. The total angular momentum satisfies the same commutation relations as orbital angular momentum, namely : _i, J_j = i \hbar \epsilon_ J_k from which follows :\left _i, J^2 \right= 0 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 \Psi :\mathbf^2\Psi = \hbar^2\Psi :\mathbf_z\Psi = \hbar\Psi 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 ...
, 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 i ...
, 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 k ...
''ℓ'', ''m''''ℓ'' and ''m''''s'' (the projection of the spin 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 ...
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 ...
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 ...
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 * 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 the ...
, 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 sp ...


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. 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 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 be ...
*
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




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