In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the total angular momentum quantum number parametrises the total
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 ...
of a given
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
, by combining its
orbital angular momentum and its intrinsic angular momentum (i.e., its
spin).
If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is
The associated quantum number is the main total angular momentum quantum number ''j''. It can take the following range of values, jumping only in integer steps:
where ''ℓ'' is the
azimuthal quantum number
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 ...
(parameterizing the orbital angular momentum) and ''s'' is the
spin quantum number (parameterizing the spin).
The relation between the total angular momentum vector j and the total angular momentum quantum number ''j'' is given by the usual relation (see
angular momentum quantum number)
The vector's ''z''-projection is given by
where ''m
j'' is the secondary total angular momentum quantum number, and the
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 ...
. It ranges from −''j'' to +''j'' in steps of one. This generates 2''j'' + 1 different values of ''m''
''j''.
The total angular momentum corresponds to the
Casimir invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
so(3) of the three-dimensional
rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
.
See also
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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 ...
*
Orbital angular momentum quantum number
*
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 ...
*
Spin quantum number
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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 ...
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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 ...
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Angular momentum diagrams (quantum mechanics)
In quantum mechanics and its applications to quantum many-particle systems, notably quantum chemistry, angular momentum diagrams, or more accurately from a mathematical viewpoint angular momentum graphs, are a diagrammatic method for representing ...
*
Rotational spectroscopy
Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The spectra of polar molecules can be measured in absorption or emission by microwave ...
References
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Albert Messiah
Albert Messiah (23 September 1921, Nice – 17 April 2013, Paris) was a French physicist.
He studied at the Ecole Polytechnique.
He spent the Second World War in the Free France forces: he embarked on 22 June 1940 at Saint-Jean-de-Luz for Engla ...
, (1966). ''Quantum Mechanics'' (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
External links
Vector model of angular momentum
Angular momentum
Atomic physics
Quantum numbers
Rotation in three dimensions
Rotational symmetry
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