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

\mathbf j = \mathbf s + \boldsymbol  ~.

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:

, \ell - s,  \le j \le \ell + s

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)

\Vert \mathbf j \Vert = \sqrt \, \hbar

The vector's ''z''-projection is given by

j_z = m_j \, \hbar

where ''m_j'' is the secondary total angular momentum quantum number, and the

\hbar

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. ...

References

* *

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 {{quantum-stub

See also

References

External links