quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, the total angular momentum quantum number parametrises the total

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

of a given

particle In the physical sciences, a particle (or corpuscle 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 s ...

, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its

spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...

). 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:

\vert \ell - s\vert \le j \le \ell + s

where ''ℓ'' is the

azimuthal quantum number In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its angular momentum operator, orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum ...

(parameterizing the orbital angular momentum) and ''s'' is the

spin quantum number In physics and chemistry, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply ''spin'') of an electron or other particle. It has the same value for all ...

(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 In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its angular momentum operator, orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum ...

)

\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 constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

. 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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

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