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Total angular momentum quantum number

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Quantum number related to rotational symmetry

Inquantum mechanics, thetotal angular momentum quantum number parametrises the totalangular momentum of a givenparticle, by combining itsorbital angular momentum and its intrinsic angular momentum (i.e., itsspin).

Ifs is the particle's spin angular momentum andℓ its orbital angular momentum vector, the total angular momentumj is $\mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}~.$

The associated quantum number is themain total angular momentum quantum numberj. It can take the following range of values, jumping only in integer steps:^[1] $\vert \ell -s\vert \leq j\leq \ell +s$ whereℓ is theazimuthal quantum number (parameterizing the orbital angular momentum) ands is thespin quantum number (parameterizing the spin).

The relation between the total angular momentum vectorj and the total angular momentum quantum numberj is given by the usual relation (seeangular momentum quantum number) $\Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar$

The vector'sz-projection is given by $j_{z}=m_{j}\,\hbar$ wherem_j is thesecondary total angular momentum quantum number, and the $\hbar$ is thereduced Planck constant. It ranges from −j to +j in steps of one. This generates 2j + 1 different values ofm_j.

The total angular momentum corresponds to theCasimir invariant of theLie algebra so(3) of the three-dimensionalrotation group.

^Hollas, J. Michael (1996).Modern Spectroscopy (3rd ed.). John Wiley & Sons. p. 180.ISBN 0-471-96522-7.

Griffiths, David J. (2004).Introduction to Quantum Mechanics (2nd ed.). Prentice Hall.ISBN 0-13-805326-X.
Albert Messiah, (1966).Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.

v t e Electron configuration
Electron shell Atomic orbital Quantum mechanics Introduction to quantum mechanics
Quantum numbers	Principal quantum number (n) Azimuthal quantum number (ℓ) Magnetic quantum number (m) Spin quantum number (s)
Ground-state configurations	Periodic table (electron configurations) Electron configurations of the elements (data page)
Electron filling	Pauli exclusion principle Hund's rule Aufbau principle
Electron pairing	Electron pair Unpaired electron
Bonding participation	Valence electron Core electron
Electron counting rules	Octet rule 18-electron rule

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