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 Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...

). 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 In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...

(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 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 th ...

)

\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. 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 identi ...

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