Regularized canonical correlation analysis: Difference between revisions

Revision as of 01:20, 25 July 2013

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.

Schrödinger Equation

Using Lagrangian theory from classical mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement $ϕ$ ) 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:

T = \frac{1}{2} m l^{2} {\dot{ϕ}}^{2}

U = m g l (1 - \cos (ϕ))

This results in the Hamiltonian:

\hat{H} = \frac{{\hat{p}}^{2}}{2 m l^{2}} + m g l (1 - \cos (ϕ))

The time-dependent Schrödinger equation for the system is as follows:

i ℏ \frac{d Ψ}{d t} = - \frac{ℏ^{2}}{2 m l^{2}} \frac{d^{2} Ψ}{d ϕ^{2}} + m g l (1 - \cos (ϕ)) Ψ

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:

η = ϕ + π

E ψ = - \frac{ℏ^{2}}{2 m l^{2}} \frac{d^{2} ψ}{d η^{2}} + m g l (1 + \cos (η)) ψ

This is simply Mathieu's equation where the solutions are Mathieu functions

0 = \frac{d^{2} ψ}{d η^{2}} + (\frac{2 m E^{2}}{ℏ^{2}} - \frac{2 m^{2} g l^{3}}{ℏ^{2}} - \frac{2 m^{2} g l^{3}}{ℏ^{2}} \cos (η)) ψ

Solutions

Energies

Given $q$ , for countably many special values of $a$ , called characteristic values, the Mathieu equation admits solutions which are periodic with period $2 π$ . The characteristic values of the Mathieu cosine, sine functions respectively are written $a_{n} (q), b_{n} (q)$ , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written $C E (n, q, x), S E (n, q, x)$ respectively, although they are traditionally given a different normalization (namely, that their L² norm equal $π$ ).

The boundary conditions in the quantum pendulum imply that $a_{n} (q), b_{n} (q)$ are as follows for a given q:

0 = \frac{d^{2} ψ}{d η^{2}} + (\frac{2 m E l^{2}}{ℏ^{2}} - \frac{2 m^{2} g l^{3}}{ℏ^{2}} - \frac{2 m^{2} g l^{3}}{ℏ^{2}} \cos (η)) ψ

a_{n} (q), b_{n} (q) = \frac{2 m E l^{2}}{ℏ^{2}} - \frac{2 m^{2} g l^{3}}{ℏ^{2}}

The energies of the system, $E = m g l + \frac{ℏ^{2} a_{n} (q), b_{n} (q)}{2 m l^{2}}$ 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:

$q = \frac{m^{2} g l^{3}}{ℏ^{2}}$

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 $a_{n} (q), b_{n} (q)$ , the Mathieu cosine and sine become periodic with a period of $2 π$ .

Eigenstates

For positive values of q, the following is true:

C (a_{n} (q), q, x) = \frac{C E (n, q, x)}{C E (n, q, 0)}

S (b_{n} (q), q, x) = \frac{S E (n, q, x)}{S E^{'} (n, q, 0)} .

Here are the first few periodic Mathieu cosine functions for q=1:

File:MathieuCE.gif

Note that, for example, $C E (1, 1, x)$ (green) resembles a cosine function, but with flatter hills and shallower valleys.

Bibliography

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Muhammad Ayub, Atom Optics Quantum Pendulum, 2011, Islamabad, Pakistan., http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf

External links

Atom Optics Quantum Pendulum

@@ Line 1: / Line 1: @@
+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.
+==Schrödinger Equation==
-uggs suitable now are very considerably repeated in vogue and growth simply due to the fact of their distinctive beauty as properly as grace.They search pleasurable about everybody precisely in women and girls. Their leeway along with loveliness raises when they have for the rationale that alongside with little attire, restricted jeans along with kaftans. This type of produced his or her beauty a ton much more perfectly regarded alongside with desirable.
+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:
- You can get coordinating sneakers or boots with just about every regarding your have dwelling clothing to be ready to designed your very own character thus wonderful and also refined. On top of that, outfits alongside one another with distinct coloredgives that you merely invigorate alongside with carry back seem. Probably the most critical features relating to lower-price tag uggs is that this they are therefore soft and snug. They could sometimes use with out hosiery just put on them without selling.
+:<math>T=\frac{1}{2} m l^2 \dot{\phi}^2 </math>
+:<math>U=m g l (1-\cos(\phi)) </math>
- This is because of to their possess warmness and their truly feel. These are composed of wonderful lamb pores and pores and skin consequently that is why they are so simple to carry. For ladies very low-price ugg boot can be observed in each and every and every single shade in purchase to deliver them with every single of their garments. Moreover these come in both equally styles of high heel sandals and without having heels in [http://www.ehow.com/search.html?s=toned+type toned type].
+This results in the Hamiltonian:
+:<math>\hat{H} = \frac{\hat{p}^2}{2 m l^2} + m g l (1-\cos(\phi))</math>
-pe. These are therefore cozy and cozy that a single could don them even in nearly all cold temperature. It is probable to convey all of them and you'll step out by placing on these individuals during chilly. Moreover, ft far too keep on to be defensive and under refuge by donning all of them. uggs can be procured in huge amount as perfectly as assortment. They are offered in the two limited and prolonged variations.
+The time-dependent [[Schrödinger]] equation for the system is as follows:
- Variations are their individual within the amount of gals and adult males. Males, ugg boots can be located in hence fairly neutral colors that are for that reason awesome tasteful. Not just for females you will uncover there's extensive choice offor men as well. Boys and gentlemen could get pretty elegant and comfortable ugg boot sale built dependent on their alternative and may possibly. Financial markets are total with all the sneakers stores so that everyone is capable to simply go and acquire a pretty pair for them selves.
+:<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>
- ugg boot appears to be to be great any time males have these with presently there unwell fitting denims. They surface so wonderful even with trousers. Young young children will also get an array of sweet along with attractive ugg boot. For small children,uggs can be obtained in extremely distinctive as properly as excellent shades. [http://search.Huffingtonpost.com/search?q=Ugg+sheepskin&s_it=header_form_v1 Ugg sheepskin] boots boos are extremely repeated throughout showbiz and method market place as well.
+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>
-  A lot of Television superstars and movie stars have grow to be a lot susceptible in the path of placing on these forms of beautiful sneakers or boots. These footwear are generally unique and also special within their kind and stylish. Consistently consider and have almost certainly the most suited and cozy footwear and will be truly none other than uggs.<br><br>If you liked this write-up and you would certainly such as to get additional info relating to [http://tinyurl.com/k3zh4ok http://tinyurl.com/pkup9lo] kindly browse through our web-site.
+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]]

Regularized canonical correlation analysis: Difference between revisions

Revision as of 01:20, 25 July 2013

Contents

Schrödinger Equation

Solutions

Energies

General Solution

Eigenstates

Bibliography

External links

Navigation menu

Regularized canonical correlation analysis: Difference between revisions

Revision as of 01:20, 25 July 2013

Schrödinger Equation

Solutions

Energies

General Solution

Eigenstates

Bibliography

External links

Navigation menu

Search