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The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L) is a quantum number
Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
 for an atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
 that determines its orbital angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
. The azimuthal quantum number is the second of a set of quantum numbers (the principal quantum number
Principal quantum number

In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
, following spectroscopic notation
Spectroscopic notation

Spectroscopic notation provides various ways to specify atomic ion, as well as atomic orbital and molecular orbitals....
, the azimuthal quantum number, the magnetic quantum number
Magnetic quantum number

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
, and the spin quantum number
Spin quantum number

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
) which describe the unique quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
 of an electron and is designated by the letter l.

e is a set of quantum numbers associated with the energy states of the electrons of an atom.






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The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L) is a quantum number
Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
 for an atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
 that determines its orbital angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
. The azimuthal quantum number is the second of a set of quantum numbers (the principal quantum number
Principal quantum number

In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
, following spectroscopic notation
Spectroscopic notation

Spectroscopic notation provides various ways to specify atomic ion, as well as atomic orbital and molecular orbitals....
, the azimuthal quantum number, the magnetic quantum number
Magnetic quantum number

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
, and the spin quantum number
Spin quantum number

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
) which describe the unique quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
 of an electron and is designated by the letter l.

Derivation

There is a set of quantum numbers associated with the energy states of the electrons of an atom. The four quantum numbers n, l, ml, and ms specify the complete and unique quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
 of a single electron in an atom called its wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 or orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
. The wavefunction of the Schrödinger wave equation reduces to the 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. In addition to understanding this concept of the azimuth, one may also find it necessary to review or learn more about spherical coordinate system
Spherical coordinate system

In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis to the point, and the azimuth angle from the positive x-axis to the orthogonal projection of the...
s, and/or other alternative mathematical coordinate systems other than the cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
. It is known that the spherical coordinate system works best with spherical models, the cylindrical system with cylinders, the cartesian with general areas, etc.. The concept of the azimuth and how it is used to explain electron orbital shapes may be more understandable after such a review.

An atomic electron's angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, L, which is related to its quantum number is described by the following equation:

where is the reduced Planck's constant, also called Dirac's constant, is the orbital angular momentum operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
 and is the wavefunction of the electron. While many introductory text books 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 .

The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.

This behavior manifests itself as the "shape" of the orbital.

Hatomorbitals
Electron shells have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
.

Their wavefunctions take the form of spherical harmonic
Spherical Harmonic

Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire series of books by Catherine Asaro which tells the story of Pharaoh Dyhianna Selei , ruler of the Skolian Empire, after the Radiance War fought by the Imperialate and their enemy Eubians....
s, and so are described by Legendre polynomials
Associated Legendre polynomials

In mathematics, the associated Legendre functions are the canonical solutions of the general Legendre equationorwhere the indices and m are referred to as the degree and order of the associated Legendre function respectively....
. 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:

Letter Max electrons Shape Name
0 s 2 sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
  		sharp
1 p 6 two dumbbell
Dumbbell

A dumbbell is a piece of equipment used in weight training, and is a type of weight. They can be used individually or in pairs ....
s 		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 "shells" is some poor dumb fool. Another mnemonic for the order of the "shells" is silly professors dance funny. The letters after the F subshell 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 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 nlml 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
Principal quantum number

In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
, 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 zero. In a p orbital, one node traverses the nucleus and therefore l has the value 1.

Depending on the value of n, the principal quantum number, 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
Lyman series

In physics, the Lyman series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n = 2 to n = 1 ....
 (ultraviolet)
n = 2, l = h, Balmer series
Balmer series

The Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom....
 (visible) Wavelength vary from 400 to 700 nm
n = 3, l = 2h, Ritz-Paschen series (short wave infrared)
n = 4, l = 3h, Pfund series (long wave 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
Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
  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


Due to the spin-orbit interaction
Spin-orbit interaction

In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's energy level , due to electromagnetic interaction between the electron's spin and the nucleus's electric field, through which it moves...
 in the atom, the orbital angular momentum no longer commutes
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
 with the Hamiltonian
Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* Hamiltonian ...
, nor does the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
. These therefore change over time. However the total angular momentum J does commute
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
 with the Hamiltonian
Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* 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 angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, namely from which follows where stand for , and .

The quantum number
Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
s describing the system, which are constant over time, are now j and , defined through the action of J on the wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 

So that j is related to the norm of the total angular momentum and to its projection along a specified axis.

As with any angular momentum in quantum mechanics
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, the projection of J along other axes cannot be co-defined with , because they do not commute
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
.

Relation between new and old quantum numbers
j and mj, together with the parity
Parity (physics)

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all spatial coordinates:...
 of the quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
, replace the three quantum numbers l, ml and ms (the projection of the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 along the specified axis). The former quantum numbers can be related to the latter.

Furthermore, the eigenvectors of j, mj and parity, which are also eigenvectors of the Hamiltonian
Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* Hamiltonian ...
, are linear combinations of the eigenvectors of l, ml and ms.

List of angular momentum quantum numbers

  • Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number
    Spin quantum number

    In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
  • orbital angular momentum quantum number (the subject of this article)
  • magnetic quantum number
    Magnetic quantum number

    In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
    , related to the orbital momentum quantum number
  • total angular momentum quantum number


History

The azimuthal quantum number was carried over from the Bohr model of the atom. The Bohr model was derived from spectroscopic analysis
Spectroscopy

Spectroscopy was originally the study of the interaction between radiation and matter as a function of wavelength . In fact, historically, spectroscopy referred to the use of visible light dispersed according to its wavelength, e.g....
 of the atom in combination with the Rutherford
Ernest Rutherford

Ernest Rutherford, 1st Baron Rutherford of Nelson, Order of Merit , Royal Society was a New Zealand-born British chemist who became known as the father of nuclear physics....
 atomic model. The lowest quantum level was found to have an angular momentum of zero. To simplify the mathematics, orbits were considered as oscillating charges in one dimension and so described as "pendulum" orbits. In three-dimensions the orbit becomes spherical without any nodes
Node (physics)

A node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes....
 crossing the nucleus, similar to a jump rope that oscillates in one large circle.

See also

  • Quantum number
    Quantum number

    Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
    • Principal quantum number
      Principal quantum number

      In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
    • Magnetic quantum number
      Magnetic quantum number

      In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
    • Spin quantum number
      Spin quantum number

      In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
    • Total angular momentum quantum number
  • Basic quantum mechanics
  • Angular momentum operator
    Angular momentum operator

    In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry....
  • Particle in a spherically symmetric potential
    Particle in a spherically symmetric potential

    In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a potential which has spherical symmetry....


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