Editing Azimuthal quantum number (section)

== Beyond isolated atoms ==
{{See also|Density functional theory|History of spectroscopy}}
[[File:Electron_energy_loss_spectroscopy_coreloss_lsmo.svg|thumb|Example of inner shell ionization edge (core loss) EELS data from La<sub>0.7</sub>Sr<sub>0.3</sub>MnO<sub>3</sub>, acquired in a [[Scanning transmission electron microscopy|scanning transmission electron microscope]].]]
The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The {{math|''ℓ m''}} quantum number corresponds to specific [[spherical harmonics]] and are commonly used to describe features observed in spectroscopic methods such as [[X-ray photoelectron spectroscopy]]<ref>{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}}</ref> and [[electron energy loss spectroscopy]].<ref>{{Cite book |last=Egerton |first=R.F. |url=https://link.springer.com/10.1007/978-1-4419-9583-4 |title=Electron Energy-Loss Spectroscopy in the Electron Microscope |date=2011 |publisher=Springer US |isbn=978-1-4419-9582-7 |location=Boston, MA |language=en |doi=10.1007/978-1-4419-9583-4}}</ref> (The notation is slightly different, with [[X-ray notation]] where K, L, M are used for excitations out of electron states with <math>n=0, 1, 2</math>.)

The angular momentum quantum numbers are also used when the electron states are described in methods such as [[Kohn–Sham equations|Kohn–Sham density functional theory]]<ref>{{Cite journal |last=Kohn |first=W. |last2=Sham |first2=L. J. |date=1965 |title=Self-Consistent Equations Including Exchange and Correlation Effects |url=https://link.aps.org/doi/10.1103/PhysRev.140.A1133 |journal=Physical Review |language=en |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |issn=0031-899X}}</ref> or with [[gaussian orbital]]s.<ref>{{Citation |last=Gill |first=Peter M.W. |title=Molecular integrals Over Gaussian Basis Functions |date=1994 |work=Advances in Quantum Chemistry |volume=25 |pages=141–205 |url=https://linkinghub.elsevier.com/retrieve/pii/S0065327608600192 |access-date=2024-02-20 |publisher=Elsevier |language=en |doi=10.1016/s0065-3276(08)60019-2 |isbn=978-0-12-034825-1|url-access=subscription }}</ref> For instance, in [[silicon]] the electronic properties used in [[semiconductor device]] are due to the p-like states with <math>l=1</math> centered at each atom, while many properties of [[transition metal]]s depend upon the d-like states with <math>l=2</math>.<ref>{{Cite book |last=Pettifor |first=D. G. |title=Bonding and structure of molecules and solids |date=1996 |publisher=Clarendon Press |isbn=978-0-19-851786-3 |edition=Reprint with corr |series=Oxford science publications |location=Oxford}}</ref>