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The '''quantum pendulum''' 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 non-linearity, the [[Schrödinger]] equation for the quantized system can be solved relatively easily.
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==Schrödinger Equation==
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Using Lagrangian theory from classical 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 is constant and there is no motion along the z axis). The kinetic energy and potential energy of the system can be found to be as follows:
 
:<math>T=\frac{1}{2} m l^2 \dot{\phi}^2 </math>
:<math>U=m g l (1-\cos(\phi)) </math>
 
This results in the Hamiltonian:
:<math>\hat{H} = \frac{\hat{p}^2}{2 m l^2} + m g l (1-\cos(\phi))</math>
 
The time-dependent [[Schrödinger]] equation for the system is as follows:
 
:<math>i \hbar \frac{d\Psi}{dt} = - \frac{\hbar^2}{2 m l^2} \frac {\mathrm{d}^2 \Psi} {\mathrm{d} \phi^2}+m g l (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>E \psi = - \frac{\hbar^2}{2 m l^2} \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+m g l (1+\cos(\eta)) \psi </math>
 
This is simply Mathieu's equation where the solutions are [[Mathieu functions]]
 
:<math>0 = \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+(\frac{2 m E^2} {\hbar^2}-\frac{2 m^2 g l^3} {\hbar^2}-\frac{2 m^2 g l^3} {\hbar^2} \cos(\eta)) \psi </math>
 
==Solutions==
 
===Energies===
Given <math>q</math>, for countably many special values of <math>a</math>, called ''characteristic values'', the Mathieu equation admits solutions which 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 n 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 L<sup>2</sup> norm equal <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 q:
 
:<math>0 = \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+(\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)) \psi </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 effective potential depth can be defined as follows:
 
<math>q=\frac{m^2 g l^3} {\hbar^2} </math>
 
A depth potential depth yields the dynamics of a particle in an independent potential. In contrast, a shallow potential depth, [[Bloch waves]] as well as [[quantum tunneling]] become of importance.
 
===General Solution===
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===
 
For positive values of q, the following is true:
:<math>C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}</math>
:<math>S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}.</math>
Here are the first few periodic Mathieu cosine functions for q=1:
[[Image:MathieuCE.gif|center]]
Note that, for example, <math>CE(1,1,x)</math> (green) resembles a cosine function, but with flatter hills and shallower valleys.
 
== 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}}
*{{cite book | last=Griffiths|first= David J.|title=Introduction to Quantum Mechanics |edition=2nd | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7}}
*Muhammad Ayub, ''Atom Optics Quantum Pendulum'', 2011, Islamabad, Pakistan., http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf
 
==External links==
* [http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf Atom Optics Quantum Pendulum]
 
{{DEFAULTSORT:Quantum Pendulum}}
[[Category:Quantum mechanics]]
[[Category:Quantum models]]

Latest revision as of 14:09, 16 September 2014

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Correct comfortable ups are crucial if you wish to construct muscles. Much stronger muscle tissues may well be more anxious and susceptible to injury. By warming up them up, damage may be eliminated. Don't do any hefty weightlifting until soon after you've done some light-weight training for five or ten moments, followed by several gentle to reasonable collections to get warmed up.

You need a great deal of proteins when muscle development. Health proteins offers the foundations that produce muscle groups. It will be tough for your body to increase its muscle tissue in case you are not consuming sufficient protein in your daily diet. Be sure that 2 or more of your own bigger foods, and also a couple of your day-to-day treats, have proteins.

While using creatine health supplements is effective, workout extreme caution in the course of use, particularly when utilizing for longer durations. Individuals with renal system troubles are typically encouraged to avoid creatine use. Other bad reactions could involve muscles area symptoms, cramping and arrhythmia. Adolescents employing this supplement tend to be at the greatest risk. Follow the advised levels for the safety when you use these supplements.

Though creatine remains safe and secure to use when trying to create muscle mass, you can find measures you should be aware. In case you have a preexisting renal system condition you need to stay away from this kind of dietary supplements. These dietary supplements have also been associated with muscle mass compartment symptoms, muscle tissue pains, and heart arrhythmia. Young people are especially afflicted. Consider to actually utilize the dietary supplements properly when you determine you are likely to integrate them in your workouts.

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Try to eat well around the days and nights that you simply decide to work towards your muscle building. You need to include unhealthy calories about sixty a few minutes before you decide to intend to elevate to present them time and energy to be ingested and available for the body to make use of. That doesn't indicate you are able to overindulge if you are timetabled for the exercise, but just be sure to try to eat a little bit more on those days and much less around the days and nights you aren't going to be weight training exercise.

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Make certain you use a healthy agenda for your exercises so that your muscle tissues can grow, and you will probably not end up with any type of injury. When you are dealing with your muscles initially, only do intense workouts two times per week. People that have a lot more practical experience and have already been conditioning their muscles can safely figure out this way three times in one week.

It is essential to remember that specific teams of muscle tissue will be more tough to develop than others. For those who have this challenge, objective a selected sets of muscle groups having a complete set. A load establish is a quick set up focusing on the trouble muscle tissues several days as soon as they were very last resolved hard.

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