Editing Quantum harmonic oscillator (section)

{{Use American English|date = February 2019}}
{{Short description|Important, well-understood quantum mechanical model}}
{{Quantum mechanics}}
{{redirect|QHO|text=It is also the [[IATA airport code]] for [[Transportation in Houston#Airports|all airports in the Houston area]]}}{{Inline citations|date=November 2023}}[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|300px|right|Some trajectories of a [[harmonic oscillator]] according to [[Newton's laws]] of [[classical mechanics]] (A–B), and according to the [[Schrödinger equation]] of [[quantum mechanics]] (C–H). In A–B, the particle (represented as a ball attached to a [[Hooke's law|spring]]) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the [[wavefunction]]. C, D, E, F, but not G, H, are [[energy eigenstate]]s. H is a [[Coherent states|coherent state]]—a quantum state that approximates the classical trajectory.]]

The '''quantum harmonic oscillator''' is the [[quantum mechanics|quantum-mechanical]] analog of the [[harmonic oscillator|classical harmonic oscillator]]. Because an arbitrary smooth [[Potential energy|potential]] can usually be approximated as a [[Harmonic oscillator#Simple harmonic oscillator|harmonic potential]] at the vicinity of a stable [[equilibrium point]],  it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, [[List_of_quantum-mechanical_systems_with_analytical_solutions|analytical solution]] is known.{{sfn|Griffiths|2004}}{{sfn|Liboff|2002}}<ref>{{Cite web | last =Rashid | first =Muneer A. | author-link =Munir Ahmad Rashid | title =Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian | website =M.A. Rashid – [[National University of Sciences and Technology, Pakistan|Center for Advanced Mathematics and Physics]] | publisher =[[National Center for Physics]] | year =2006 | url =http://www.ncp.edu.pk/docs/12th_rgdocs/Munir-Rasheed.pdf | format =[[PDF]]-[[Microsoft PowerPoint]] | access-date =19 October 2010 | archive-date =3 March 2016 | archive-url =https://web.archive.org/web/20160303233341/http://www.ncp.edu.pk/docs/12th_rgdocs/Munir-Rasheed.pdf | url-status =dead }}</ref>