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{{Quantum book backlink|Mathematical structure and systems}}

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.
==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

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

This results in the Hamiltonian

:<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

:<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>

One must solve the time-independent Schrödinger equation to find the energy levels and corresponding eigenstates. This is best accomplished by changing the independent variable as follows:

:<math>\eta = \phi + \pi,</math>
:<math>\Psi = \psi e^{-iEt/\hbar},</math>
:<math>E \psi = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \psi}{d \eta^2} + mgl (1 + \cos\eta) \psi.</math>

This is simply Mathieu's differential equation

:<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]].

==Solutions==

===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>).

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>:

:<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>a_n(q), b_n(q) = \frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2}.</math>

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

:<math>q = \frac{m^2 g l^3}{\hbar^2}.</math>

A deep potential yields the dynamics of a particle in an independent potential. In contrast, in a shallow potential, Bloch waves, as well as quantum tunneling, become of importance.

===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>.

===Eigenstates===

For positive values of ''q'', the following is true:
:<math>C(a_n(q), q, x) = \frac{CE(n, q, x)}{CE(n, q, 0)},</math>
:<math>S(b_n(q), q, x) = \frac{SE(n, q, x)}{SE'(n, q, 0)}.</math>
Here are the first few periodic Mathieu cosine functions for <math>q = 1</math>.
center
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]]

==References==
{{Reflist}}

== Bibliography ==
*{{cite book | last1=Bransden | first1=B. H. | last2 = Joachain | first2 = C. J. | title = Quantum mechanics | edition = 2nd | publisher = Pearson Education|location=Essex| year = 2000|isbn=0-582-35691-1}}
*{{cite book | last=Davies|first= John H.|title=The Physics of Low-Dimensional Semiconductors: An Introduction | publisher=Cambridge University Press|year=2006|isbn=0-521-48491-X|edition=6th reprint}}
*Muhammad Ayub, ''Atom Optics Quantum Pendulum'', 2011, Islamabad, Pakistan., https://arxiv.org/abs/1012.6011

{{DEFAULTSORT:Quantum Pendulum}}
[[Category:Quantum models]]
[[Category:Pendulums]]

{{Sourceattribution|Quantum pendulum}}

Physics:Quantum pendulum - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template