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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
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 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 azimuthal angular momentum is defined as = Define ... When the total angular momentum quantum number is a half-integer (1/2, 3/2, etc.), (^, ...
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.
s = spin quantum number; m s = spin magnetic quantum number; ℓ = Azimuthal quantum number; m ℓ = azimuthal magnetic quantum number; j = total angular momentum quantum number; m j = total angular momentum magnetic quantum number
The emitted particle carries away angular momentum, with quantum number λ, which for the photon must be at least 1, since it is a vector particle (i.e., it has J P = 1 − ). Thus, there is no radiation from E0 (electric monopoles) or M0 (magnetic monopoles, which do not seem to exist).
The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in [,): = (,) These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ().