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, 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:

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

where ''ℓ'' is the azimuthal quantum number (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 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 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.

References

* * Albert Messiah, (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