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The azimuthal quantum number, also known as the orbital angular momentum quantum number, describes the subshell, and gives the magnitude of the orbital angular momentum through the relation L 2 = ℏ 2 ℓ ( ℓ + 1 ) . {\displaystyle L^{2}=\hbar ^{2}\ell (\ell +1).}
The quantum state vector of a single particle with total angular momentum quantum number j and total magnetic quantum number m = j, j − 1, ..., −j + 1, −j, is denoted as a ket |j, m . As a diagram this is a singleheaded arrow. Symmetrically, the corresponding bra is j, m|.
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 ] | ℓ − s | ≤ j ≤ ℓ + s {\displaystyle \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 term "azimuthal quantum number" was introduced by Arnold Sommerfeld in 1915 [1]: II:132 as part of an ad hoc description of the energy structure of atomic spectra. . Only later with the quantum model of the atom was it understood that this number, ℓ, arises from quantization of orbital angular moment
Here L is the total orbital angular momentum quantum number. [18] For atoms with a well-defined S, the multiplicity of a state is defined as 2 S + 1. This is equal to the number of different possible values of the total (orbital plus spin) angular momentum J for a given (L, S) combination, provided that S ≤ L (the typical case).
The spin magnetic quantum number m s specifies the z-axis component of the spin angular momentum for a particle having spin quantum number s. For an electron, s is 1 ⁄ 2 , and m s is either + 1 ⁄ 2 or − 1 ⁄ 2 , often called "spin-up" and "spin-down", or α and β.
In atomic physics, a term symbol is an abbreviated description of the total spin and orbital angular momentum quantum numbers of the electrons in a multi-electron atom.So while the word symbol suggests otherwise, it represents an actual value of a physical quantity.
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics.The overall sign of the coefficients for each set of constant , , is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign convention as discussed by Baird and Biedenharn. [1]