Total angular momentum quantum number

In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum 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 {\ell }}~.

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:^[1]

\vert \ell -s\vert \leq j\leq \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 {j\,(j+1)}}\,\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. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of m_j.

The total angular momentum corresponds to the Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.

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References

^ 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.

External links

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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This page was last edited on 23 April 2024, at 19:01

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