Physics:Quantum harmonic oscillator - Revision history

WikiHarold: Remove imported red links from Quantum page

2026-05-23T23:47:31Z

Remove imported red links from Quantum page

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~{{Quantum mechanics}}~~
	{{redirect\|QHO\|text=It is also the IATA airport code for all airports in the Houston area}}		{{redirect\|QHO\|text=It is also the IATA airport code for all airports in the Houston area}}
	~~{{Inline citations\|date=November 2023}}~~

	according to Newton's laws of classical mechanics (A–B), and according to the Schrödinger equation of [[Physics:Quantum mechanics\|quantum mechanics]] (C–H). In A–B, the particle (represented as a ball attached to a 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 eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.]]		according to Newton's laws of classical mechanics (A–B), and according to the Schrödinger equation of [[Physics:Quantum mechanics\|quantum mechanics]] (C–H). In A–B, the particle (represented as a ball attached to a 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 eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.]]

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	\end{align}</math>		\end{align}</math>

	The operator {{mvar\|a}} is not Hermitian, since itself and its adjoint {{math\|''a''<sup>†</sup>}} are not equal. The energy eigenstates {{math\|~~{{ket\|''n''}}~~}}, when operated on by these ladder operators, give		The operator {{mvar\|a}} is not Hermitian, since itself and its adjoint {{math\|''a''<sup>†</sup>}} are not equal. The energy eigenstates {{math\|}}, when operated on by these ladder operators, give
	<math display="block">\begin{align}		<math display="block">\begin{align}
	\hat a^\dagger\|n\rangle &= \sqrt{n + 1} \| n + 1\rangle \\		\hat a^\dagger\|n\rangle &= \sqrt{n + 1} \| n + 1\rangle \\
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	<math display="block">\psi(x,t)=\int dy~ K(x,y;t) \psi(y,0) \,.</math>		<math display="block">\psi(x,t)=\int dy~ K(x,y;t) \psi(y,0) \,.</math>
	===Coherent states===		===Coherent states===

	~~{{main\|Physics:Coherent state}}~~

	[[File:Coherent state gif.gif\|thumb\|right\|450px\|Coherent state dynamics for <math>\alpha = \sqrt{10}</math>, in units of the harmonic oscillator length <math>x_0=\sqrt{\hbar/m\omega}</math>, showing the probability density <math>\|\psi(x,t)\|^2</math> and the quantum phase (color).]]

	The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty {{math\|1=''σ<sub>x</sub>'' ''σ<sub>p</sub>'' = {{frac\|''ℏ''\|2}}}}, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, ''not'' the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.{{sfnp\|Zwiebach\|2022\|pp=481-492}}		The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty {{math\|1=''σ<sub>x</sub>'' ''σ<sub>p</sub>'' = {{frac\|''ℏ''\|2}}}}, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, ''not'' the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.{{sfnp\|Zwiebach\|2022\|pp=481-492}}

	The coherent states are indexed by <math>\alpha \in \mathbb{C}</math> and expressed in the {{math\|~~{{braket\|ket\|''n''}}~~}} basis as		The coherent states are indexed by <math>\alpha \in \mathbb{C}</math> and expressed in the {{math\|}} basis as

	<math display="block">\|\alpha\rangle = \sum_{n=0}^\infty \|n\rangle \langle n \| \alpha \rangle = e^{-\frac{1}{2} \|\alpha\|^2} \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}} \|n\rangle = e^{-\frac{1}{2} \|\alpha\|^2} e^{\alpha a^\dagger} e^{-{\alpha^* a}} \|0\rangle.</math>		<math display="block">\|\alpha\rangle = \sum_{n=0}^\infty \|n\rangle \langle n \| \alpha \rangle = e^{-\frac{1}{2} \|\alpha\|^2} \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}} \|n\rangle = e^{-\frac{1}{2} \|\alpha\|^2} e^{\alpha a^\dagger} e^{-{\alpha^* a}} \|0\rangle.</math>
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	==Applications==		==Applications==
	===Harmonic oscillators lattice: phonons===		===Harmonic oscillators lattice: phonons===
	~~{{see also\|Physics:Canonical quantization}}~~

	The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical ''harmonic chain'' of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.		The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical ''harmonic chain'' of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.
	As in the previous section, we denote the positions of the masses by {{math\|''x''<sub>1</sub>, ''x''<sub>2</sub>, ...}}, as measured from their equilibrium positions (i.e. {{math\|1=''x<sub>i</sub>'' = 0}} if the particle {{mvar\|i}} is at its equilibrium position). In two or more dimensions, the {{math\|''x<sub>i</sub>''}} are vector quantities. The Hamiltonian for this system is		As in the previous section, we denote the positions of the masses by {{math\|''x''<sub>1</sub>, ''x''<sub>2</sub>, ...}}, as measured from their equilibrium positions (i.e. {{math\|1=''x<sub>i</sub>'' = 0}} if the particle {{mvar\|i}} is at its equilibrium position). In two or more dimensions, the {{math\|''x<sub>i</sub>''}} are vector quantities. The Hamiltonian for this system is
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	===Molecular vibrations===		===Molecular vibrations===
	~~{{main\|Physics:Molecular vibration}}~~
	* The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by <math display="block">\omega = \sqrt{\frac{k}{\mu}} </math> where <math>\mu = \frac{m_1 m_2}{m_1 + m_2}</math> is the reduced mass and <math>m_1</math> and <math>m_2</math> are the masses of the two atoms.<ref>{{Cite web \| title=Quantum Harmonic Oscillator \| website=Hyperphysics \| access-date=24 September 2009 \| url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html}}</ref>		* The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by <math display="block">\omega = \sqrt{\frac{k}{\mu}} </math> where <math>\mu = \frac{m_1 m_2}{m_1 + m_2}</math> is the reduced mass and <math>m_1</math> and <math>m_2</math> are the masses of the two atoms.<ref>{{Cite web \| title=Quantum Harmonic Oscillator \| website=Hyperphysics \| access-date=24 September 2009 \| url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html}}</ref>
	* Modelling phonons, as discussed above.		* Modelling phonons, as discussed above.
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	{{DEFAULTSORT:Quantum Harmonic Oscillator}}		{{DEFAULTSORT:Quantum Harmonic Oscillator}}
	~~[[Category:Quantum models]]~~
	~~[[Category:Oscillators]]~~

	{{Sourceattribution\|Quantum harmonic oscillator}}		{{Sourceattribution\|Quantum harmonic oscillator}}

WikiHarold: Clean Quantum page image and red links

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	The consideration of the harmonic oscillator e.g. from the energy <math>E={\bold p}^2/2m + m\omega^2q^2/2</math> in analogy to a derivation of the Dirac equation – so-to-speak from the "square root" of the equation <math>p_{\mu}p^{\mu} + m^2=0</math> – has been explored by Lorella M. Jones.<ref>Lorella M. Jones, Another Dirac Oscillator, University of Illinois at Urbana-Champaign report ILLL-(TH)-91-24 (1991).</ref>		The consideration of the harmonic oscillator e.g. from the energy <math>E={\bold p}^2/2m + m\omega^2q^2/2</math> in analogy to a derivation of the Dirac equation – so-to-speak from the "square root" of the equation <math>p_{\mu}p^{\mu} + m^2=0</math> – has been explored by Lorella M. Jones.<ref>Lorella M. Jones, Another Dirac Oscillator, University of Illinois at Urbana-Champaign report ILLL-(TH)-91-24 (1991).</ref>

	==See also==		== See also ==
	*{{~~Annotated link\|Quantum pendulum}}~~		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	*{{Annotated link\|Quantum machine}}
	*{{Annotated link\|Gas in a harmonic trap}}
	*{{Annotated link\|Creation and annihilation operators}}
	*{{Annotated link\|Coherent state}}
	*{{Annotated link\|Morse potential}}
	*{{Annotated link\|~~Bertrand's theorem}}~~
	*{{Annotated link\|~~Mehler kernel}}~~
	*{{Annotated link\|Molecular vibration#Quantum ~~mechanics\|Molecular vibration~~}}

	==Notes==		==Notes==
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	*[http://www.brummerblogs.com/curvature/3d-harmonic-oscillator-eigenfunctions/ Live 3D intensity plots of quantum harmonic oscillator] {{Webarchive\|url=https://web.archive.org/web/20110712013635/http://www.brummerblogs.com/curvature/3d-harmonic-oscillator-eigenfunctions/ \|date=12 July 2011 }}		*[http://www.brummerblogs.com/curvature/3d-harmonic-oscillator-eigenfunctions/ Live 3D intensity plots of quantum harmonic oscillator] {{Webarchive\|url=https://web.archive.org/web/20110712013635/http://www.brummerblogs.com/curvature/3d-harmonic-oscillator-eigenfunctions/ \|date=12 July 2011 }}
	*[https://users.aalto.fi/~thunebe1/courses/monqo.pdf Driven and damped quantum harmonic oscillator (lecture notes of course "quantum optics in electric circuits")]		*[https://users.aalto.fi/~thunebe1/courses/monqo.pdf Driven and damped quantum harmonic oscillator (lecture notes of course "quantum optics in electric circuits")]


	{{DEFAULTSORT:Quantum Harmonic Oscillator}}		{{DEFAULTSORT:Quantum Harmonic Oscillator}}

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 {{Short description|Important, well-understood quantum mechanical model}}
 {{Quantum mechanics}}
 {{redirect|QHO|text=It is also the IATA airport code for 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 [[Physics:Newton's laws|Newton's laws]] of [[Physics:Classical mechanics|classical mechanics]] (A–B), and according to the [[Schrödinger equation]] of [[Physics:Quantum mechanics|quantum mechanics]] (C–H). In A–B, the particle (represented as a ball attached to a [[Physics: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 eigenstates. H is a [[Coherent states|coherent state]]—a quantum state that approximates the classical trajectory.]]
+{{Inline citations|date=November 2023}}
 The '''quantum harmonic oscillator''' is the [[Physics:Quantum mechanics|quantum-mechanical]] analog of the [[Physics:Harmonic oscillator|classical harmonic oscillator]]. Because an arbitrary smooth [[Physics:Potential energy|potential]] can usually be approximated as a [[Physics:Harmonic oscillator#Simple harmonic oscillator|harmonic potential]] at the vicinity of a stable [[Equilibrium point|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, [[Physics: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. |  title =Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian | website =M.A. Rashid – 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]]-[[Software:Microsoft PowerPoint|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><ref>Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (2012) World Scientific, ISBN 978-9810-4397-5. This reference includes in detail the method of operators, Hermite functions, contour integration, pp. 130-141.</ref>
 ==One-dimensional harmonic oscillator==

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