Physics:Quantum pendulum - Revision history

WikiHarold: Use Quantum See also index module

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Use Quantum See also index module

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	center		center
	Note that, for example, <math>CE(1, 1, x)</math> (green) resembles a cosine function, but with flatter hills and shallower valleys.		Note that, for example, <math>CE(1, 1, x)</math> (green) resembles a cosine function, but with flatter hills and shallower valleys.

	~~== See also ==~~

	* [[Physics:Quantum harmonic oscillator\|Quantum harmonic oscillator]]

	== See also ==		== See also ==

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	<div style="width:300px;">		<div style="width:300px;">
	[[File:Quantum_book1_pendulum_yellow.png]]		[[File:Quantum_book1_pendulum_yellow.png\|thumb\|280px\|Quantum pendulum.]]
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	'''pendulum''' ~~is a Book I topic in~~ the ~~Quantum Collection. The~~ quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily. The quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena.		'''pendulum''' the quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily. The quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Using Lagrangian mechanics, one can develop a Hamiltonian for the system.
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	~~The~~ '''~~quantum~~ pendulum''' is a theoretical model and experimental system that studies how a pendulum behaves under ~~[[Physics:Quantum mechanics\|~~quantum mechanics~~]].<ref>{{Cite journal \|date=2021-08-18 \|title=Exploring quantum gravity—for whom the pendulum swings. \|url=https://www.nist~~.~~gov/news-events/news/2021/08/exploring-quantum-gravity-whom-pendulum-swings \|journal=NIST \|language=en}}</ref>~~ It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena.<ref>{{Cite journal \|last=Ayub \|first=Muhammad \|last2=Naseer \|first2=Khalid \|last3=Ali \|first3=Manzoor \|last4=Saif \|first4=Farhan \|date=2009-05-01 \|title=Atom optics quantum pendulum \|url=https://doi.org/10.1007/s10946-009-9078-x \|journal=Journal of Russian Laser Research \|language=en \|volume=30 \|issue=3 \|pages=205–223 \|doi=10.1007/s10946-009-9078-x \|issn=1573-8760\|arxiv=1012.6011 }}</ref> Though a pendulum not subject to the ~~[[Small-angle approximation\|~~small-angle approximation]] has an inherent nonlinearity, the [[Schrödinger equation]] for the quantized system can be solved relatively easily.		'''pendulum''' is a Book I topic in the Quantum Collection. The quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily. The quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena.
	</div>		</div>

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	[[File:~~not found~~]]		[[File:Quantum_book1_pendulum_yellow.png]]
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	==Schrödinger equation==		==Schrödinger equation==

	Using [[Lagrangian mechanics]], one can develop a ~~[[Physics:~~Hamiltonian ~~(quantum mechanics)\|Hamiltonian]]~~ for the system. A simple pendulum has one ~~[[Generalized coordinates\|~~generalized coordinate]] (the angular displacement <math>\phi</math>) and two constraints (the length of the string and the plane of motion). The kinetic and ~~[[Physics:Potential energy\|~~potential energies]] of the system can be found to be		Using Lagrangian mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement <math>\phi</math>) and two constraints (the length of the string and the plane of motion). The kinetic and potential energies of the system can be found to be

	:<math>T = \frac{1}{2} m l^2 \dot{\phi}^2,</math>		:<math>T = \frac{1}{2} m l^2 \dot{\phi}^2,</math>
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	:<math>\hat{H} = \frac{\hat{p}^2}{2 m l^2} + mgl (1 - \cos\phi).</math>		:<math>\hat{H} = \frac{\hat{p}^2}{2 m l^2} + mgl (1 - \cos\phi).</math>

	The time-dependent [[Schrödinger equation]] for the system is		The time-dependent Schrödinger equation for the system is

	:<math>i \hbar \frac{d\Psi}{dt} = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \Psi}{d \phi^2} + mgl (1 - \cos\phi) \Psi.</math>		:<math>i \hbar \frac{d\Psi}{dt} = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \Psi}{d \phi^2} + mgl (1 - \cos\phi) \Psi.</math>
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	:<math>\frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,</math>		:<math>\frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,</math>

	whose solutions are [[Mathieu functions]].		whose solutions are Mathieu functions.

	==Solutions==		==Solutions==

	===Energies===		===Energies===
	Given <math>q</math>, for countably many special values of <math>a</math>, called ''characteristic values'', the Mathieu equation admits solutions that are periodic with period <math>2\pi</math>. The characteristic values of the Mathieu cosine, sine functions respectively are written <math>a_n(q), b_n(q)</math>, where <math>n</math> is a ~~[[Natural number\|~~natural number]]. The periodic special cases of the Mathieu cosine and sine functions are often written <math>CE(n,q,x), SE(n,q,x)</math> respectively, although they are traditionally given a different normalization (namely, that their <math>L^2</math>norm equals <math>\pi</math>).		Given <math>q</math>, for countably many special values of <math>a</math>, called ''characteristic values'', the Mathieu equation admits solutions that are periodic with period <math>2\pi</math>. The characteristic values of the Mathieu cosine, sine functions respectively are written <math>a_n(q), b_n(q)</math>, where <math>n</math> is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written <math>CE(n,q,x), SE(n,q,x)</math> respectively, although they are traditionally given a different normalization (namely, that their <math>L^2</math>norm equals <math>\pi</math>).

	The boundary conditions in the quantum pendulum imply that <math>a_n(q), b_n(q)</math> are as follows for a given <math>q</math>:		The boundary conditions in the quantum pendulum imply that <math>a_n(q), b_n(q)</math> are as follows for a given <math>q</math>:
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	The energies of the system, <math>E = m g l + \frac{\hbar^2 a_n(q), b_n(q)}{2 m l^2}</math> for even/odd solutions respectively, are quantized based on the characteristic values found by solving the Mathieu equation.		The energies of the system, <math>E = m g l + \frac{\hbar^2 a_n(q), b_n(q)}{2 m l^2}</math> for even/odd solutions respectively, are quantized based on the characteristic values found by solving the Mathieu equation.

	The ~~[[Physics:Effective potential\|~~effective potential]] depth can be defined as		The effective potential depth can be defined as

	:<math>q = \frac{m^2 g l^3}{\hbar^2}.</math>		:<math>q = \frac{m^2 g l^3}{\hbar^2}.</math>
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	===General solution===		===General solution===
	The general solution of the above ~~[[Differential equation\|~~differential equation]] for a given value of ''a'' and ''q'' is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively. In general, the Mathieu functions are aperiodic; however, for characteristic values of <math>a_n(q), b_n(q)</math>, the Mathieu cosine and sine become periodic with a period of <math>2\pi</math>.		The general solution of the above differential equation for a given value of ''a'' and ''q'' is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively. In general, the Mathieu functions are aperiodic; however, for characteristic values of <math>a_n(q), b_n(q)</math>, the Mathieu cosine and sine become periodic with a period of <math>2\pi</math>.

	===Eigenstates===		===Eigenstates===

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	* [[Physics:Quantum harmonic oscillator\|Quantum harmonic oscillator]]		* [[Physics:Quantum harmonic oscillator\|Quantum harmonic oscillator]]

			== See also ==
			{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	==References==		==References==
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	[[Category:Pendulums]]		[[Category:Pendulums]]

	{{Sourceattribution\|Quantum pendulum}}		{{Author\|Harold Foppele}}

			{{Sourceattribution\|Physics:Quantum pendulum\|1}}

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	{{Quantum article nav\|previous=Physics:Quantum Klein–Gordon equation\|previous label=Klein–Gordon equation\|next=Physics:Quantum configuration space\|next label=Configuration space}}		{{Quantum article nav\|previous=Physics:Quantum Klein–Gordon equation\|previous label=Klein–Gordon equation\|next=Physics:Quantum configuration space\|next label=Configuration space}}