Editing Azimuthal quantum number (section)

{{Short description|Quantum number denoting orbital angular momentum}}
[[File:HAtomOrbitals.png|thumb|The [[atomic orbital]] wavefunctions of a [[hydrogen atom]]: The azimuthal quantum number ({{mvar|ℓ}}) is denoted ''by letter'' at the top of each column. The principal quantum number ({{mvar|n}}) is shown at the right of each row.]]

In [[quantum mechanics]], the '''azimuthal quantum number''' {{mvar|ℓ}} is a [[quantum number]] for an [[atomic orbital]] that determines its [[angular momentum operator|orbital angular momentum]] and describes aspects of the angular shape of the orbital. The [[azimuth]]al quantum number is the second of a set of quantum numbers that describe the unique [[quantum state]] of an [[electron]] (the others being the [[principal quantum number]] {{mvar|n}}, the [[magnetic quantum number]] {{mvar|''m''{{sub|ℓ}}}}, and the [[spin quantum number]] {{mvar|''m''{{sub|s}}}}).

For a given value of the principal quantum number {{mvar|n}} (''[[electron shell]]''), the possible values of {{mvar|ℓ}} are the integers from 0 to {{math|1=''n'' − 1}}. For instance, the {{math|1=''n'' = 1}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and the {{math|1=''n'' = 2}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and <math>\ell=1</math>.

For a given value of the azimuthal quantum number {{mvar|ℓ}}, the possible values of the magnetic quantum number {{mvar|''m''{{sub|ℓ}}}} are the integers from {{math|1=''m''{{sub|ℓ}}=−ℓ}} to {{math|1=''m''{{sub|ℓ}}=+ℓ}}, including 0. In addition, the spin quantum number {{mvar|''m''{{sub|s}}}} can take two distinct values. The set of orbitals associated with a particular value of&nbsp;{{mvar|ℓ}} are sometimes collectively called a ''subshell''.

While originally used just for isolated atoms, atomic-like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids. The quantum number {{mvar|ℓ}} plays an important role here via the connection to the angular dependence of the [[spherical harmonics]] for the different orbitals around each atom.