Integral equation - Revision history

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Reverted 1 edit by Anshul maitray identified as test/vandalism using STiki

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	~~{{Unreferenced\|date=July 2009}}~~		Luke Bryan is really a celebrity in the producing as well as profession progress to start with next to his 3rd casio recording, And , may be the proof. He broken on the scene in 2008 along with his special combination of straight down-house accessibility, motion [http://www.hotelsedinburgh.org luke bryan on sale dates] picture superstar excellent appears and lyrics, is scheduled t within a major way. The brand new re around the nation graph or chart and #2 about jason aldean and luke bryan tickets ([http://www.senatorwonderling.com www.senatorwonderling.com]) the burst maps, generating it [http://www.museodecarruajes.org luke bryan tickets indianapolis] the second maximum debut at that time of 2011 for a region musician. <br><br>The kid of a , is aware patience and perseverance are important elements with regards to a successful profession- . His very first album, Continue to be Me, created the best reaches “All My Friends “Country and Say” Guy,” when his hard work, Doin’ Point, identified the performer-about three direct No. 5 single men and women: Different Phoning Is usually a Good Issue.”<br><br>In the fall of 2015, Concerts: Luke & that had a remarkable set of , which includes Urban. “It’s almost like you are acquiring a endorsement to go to a higher level, states all those designers that have been a part of the Concert tourover right into a larger degree of musicians.” It twisted as one of the best excursions in its 10-season background.<br><br>Visit my web site: [http://lukebryantickets.neodga.com cheap luke bryan tickets]
	~~In [[quantum mechanics]], the case of~~ a ~~'''particle~~ in ~~a one-dimensional ring''' is similar~~ to ~~the [[particle in a box]]. The [[Schrödinger equation]] for a [[free particle]] which is restricted~~ to ~~a ring (technically~~, ~~whose [[configuration space]] is~~ the ~~[[circle]] <math>S^1</math>) is~~

	~~:<math> -\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi </math>~~

	~~== Wave function ==~~

	~~Using [[polar coordinates]]~~ on the ~~1 dimensional ring~~, ~~the~~ [~~[wave function]] depends only~~ on ~~the [[angle\|angular]] [[coordinate]~~], and so

	~~:<math> \nabla^~~2 ~~= \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} </math>~~

	~~Requiring that the wave function be~~ [~~[periodic function\|periodic]] in <math> \ \theta <~~/~~math> with a period <math> 2 \pi<~~/~~math> (from the demand that the wave functions be single-valued [[function (mathematics~~)~~\|function]]s on~~ the ~~[[circle]])~~, ~~and that they be~~ [~~[normalizing constant\|normalized]] leads to the conditions~~

	:~~<math> \int_{0}^{2 \pi} \left\| \psi ( \theta ) \right\|^2 \, d\theta = 1\ <~~/~~math>,~~

	~~and~~

	~~:<math> \ \psi (\theta) = \ \psi ( \theta + 2\pi)<~~/~~math>~~

	~~Under these conditions,~~ the ~~solution to the Schrödinger equation is given by~~

	:<~~math~~> ~~\psi_{\pm}(\theta) = \frac{1}{\sqrt{2 \pi}}\, e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta }~~ <~~/math~~>

	~~== Energy eigenvalues ==~~

	The ~~[[energy]] [[eigenvalue]]s <math> E </math> are [[quantization (physics)\|quantize]]d because~~ of ~~the periodic [[boundary condition]]s~~, and ~~they~~ are ~~required~~ to ~~satisfy~~

	~~:<math>~~ ~~e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } =~~ ~~e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} (\theta +2 \pi)}</math>~~, or
	~~:<math> e^{\pm i 2 \pi \frac{r}{\hbar} \sqrt{2 m E}~~ ~~} = 1 = e^{i 2 \pi n}</math>~~

	~~The eigenfunction~~ and ~~eigenenergies are~~
	~~:<math> \psi(\theta) = \frac{1}{\sqrt{2 \pi }} \~~, ~~e^{\pm i n \theta }</math>~~
	~~:<math> E_n = \frac{n^2 \hbar^2}{2 m r^2} </math> where <math>n = 0~~,~~\pm 1~~,~~\pm 2,\pm 3, \ldots~~<~~/math~~>

	~~Therefore, there are two degenerate [[quantum state]]s for every value of~~ <~~math~~> ~~n>0 </math> (corresponding to <math> \ e^{\pm i n \theta}</math>). Therefore there are 2n+1 states with energies up to an energy indexed by~~ the ~~number n.~~

	~~The case~~ of a ~~particle in a one-dimensional ring is an instructive example when studying the [[quantization (physics)\|quantization]]~~ of ~~[[angular momentum]] for~~, ~~say, an [[electron]] orbiting the [[Atomic nucleus\|nucleus]]~~. ~~The [[azimuth]]al wave functions in that case~~ are ~~identical~~ to ~~the energy [[eigenfunction]]s of the particle on~~ a ~~ring.~~

	~~The statement~~ that ~~any wavefunction for the particle on a ring can be written as~~ a ~~[[quantum superposition\|superposition]]~~ of ~~[[energy]] [[eigenfunction]]s is exactly identical to~~ the ~~[[Fourier theorem]] about the development of any periodic [[function (mathematics)\|function]] in~~ a ~~[[Fourier series]].~~

	~~This simple model can be used to find approximate energy levels~~ of ~~some ring molecules, such as benzene~~.

	~~==Application==~~

	~~In [[organic chemistry]], [[aromatic]] compounds contain atomic rings, such~~ as ~~[[benzene]] rings (~~the ~~[[Friedrich August Kekulé von Stradonitz\|Kekulé]] structure) consisting~~ ~~of five or six, usually [[carbon]], atoms. So does the surface of "[[buckyballs]]" (buckminsterfullerene). These molecules are exceptionally stable.~~

	~~The above explains why the ring behaves like a circular [[waveguide]], with the valence electrons orbiting~~ in ~~both directions~~.

	~~To fill all energy levels up to n requires~~ <~~math~~>~~2\times(2n+1)~~</~~math> electrons, as electrons have additionally two possible orientations of their spins.~~

	~~The rule that <math>4n+2<~~/~~math> excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the [[Hückel's rule]]~~.

	~~Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels~~.

	~~== See also ==~~

	* [[Angular momentum]]
	* [[Harmonic analysis]]
	* [[One-dimensional periodic case]]

	~~{{DEFAULTSORT:Particle In A Ring}}~~
	~~[[Category:Quantum models]~~]

en>Policron: Added an external link (EOM).

2013-10-31T02:23:59Z

Added an external link (EOM).

← Older revision		Revision as of 04:23, 31 October 2013
Line 1:		Line 1:
	The ~~person who wrote~~ the ~~post~~ is ~~known as Jayson Hirano~~ and ~~he completely digs~~ that ~~title. One of~~ the ~~very best issues~~ in the ~~globe for him~~ is ~~performing ballet~~ and ~~he'll be starting something else along~~ with it. ~~My working day occupation~~ is an ~~invoicing officer but I've already utilized~~ for an ~~additional one~~. ~~North Carolina~~ is exactly ~~where we've been residing for years and will by no means move~~.<br><br>~~Feel free to visit my page; best psychics~~ ([~~http://www~~.~~thisisviral~~.~~com/videos/profile.php?u~~=~~LyWynn http~~:~~//www.thisisviral.com/~~])		{{Unreferenced\|date=July 2009}}
			In [[quantum mechanics]], the case of a '''particle in a one-dimensional ring''' is similar to the [[particle in a box]]. The [[Schrödinger equation]] for a [[free particle]] which is restricted to a ring (technically, whose [[configuration space]] is the [[circle]] <math>S^1</math>) is

			:<math> -\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi </math>

			== Wave function ==

			Using [[polar coordinates]] on the 1 dimensional ring, the [[wave function]] depends only on the [[angle\|angular]] [[coordinate]], and so

			:<math> \nabla^2 = \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} </math>

			Requiring that the wave function be [[periodic function\|periodic]] in <math> \ \theta </math> with a period <math> 2 \pi</math> (from the demand that the wave functions be single-valued [[function (mathematics)\|function]]s on the [[circle]]), and that they be [[normalizing constant\|normalized]] leads to the conditions

			:<math> \int_{0}^{2 \pi} \left\| \psi ( \theta ) \right\|^2 \, d\theta = 1\ </math>,

			and

			:<math> \ \psi (\theta) = \ \psi ( \theta + 2\pi)</math>

			Under these conditions, the solution to the Schrödinger equation is given by

			:<math> \psi_{\pm}(\theta) = \frac{1}{\sqrt{2 \pi}}\, e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } </math>

			== Energy eigenvalues ==

			The [[energy]] [[eigenvalue]]s <math> E </math> are [[quantization (physics)\|quantize]]d because of the periodic [[boundary condition]]s, and they are required to satisfy

			:<math> e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} \theta } = e^{\pm i \frac{r}{\hbar} \sqrt{2 m E} (\theta +2 \pi)}</math>, or
			:<math> e^{\pm i 2 \pi \frac{r}{\hbar} \sqrt{2 m E} } = 1 = e^{i 2 \pi n}</math>

			The eigenfunction and eigenenergies are
			:<math> \psi(\theta) = \frac{1}{\sqrt{2 \pi }} \, e^{\pm i n \theta }</math>
			:<math> E_n = \frac{n^2 \hbar^2}{2 m r^2} </math> where <math>n = 0,\pm 1,\pm 2,\pm 3, \ldots</math>

			Therefore, there are two degenerate [[quantum state]]s for every value of <math> n>0 </math> (corresponding to <math> \ e^{\pm i n \theta}</math>). Therefore there are 2n+1 states with energies up to an energy indexed by the number n.

			The case of a particle in a one-dimensional ring is an instructive example when studying the [[quantization (physics)\|quantization]] of [[angular momentum]] for, say, an [[electron]] orbiting the [[Atomic nucleus\|nucleus]]. The [[azimuth]]al wave functions in that case are identical to the energy [[eigenfunction]]s of the particle on a ring.

			The statement that any wavefunction for the particle on a ring can be written as a [[quantum superposition\|superposition]] of [[energy]] [[eigenfunction]]s is exactly identical to the [[Fourier theorem]] about the development of any periodic [[function (mathematics)\|function]] in a [[Fourier series]].

			This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.

			==Application==

			In [[organic chemistry]], [[aromatic]] compounds contain atomic rings, such as [[benzene]] rings (the [[Friedrich August Kekulé von Stradonitz\|Kekulé]] structure) consisting of five or six, usually [[carbon]], atoms. So does the surface of "[[buckyballs]]" (buckminsterfullerene). These molecules are exceptionally stable.

			The above explains why the ring behaves like a circular [[waveguide]], with the valence electrons orbiting in both directions.

			To fill all energy levels up to n requires <math>2\times(2n+1)</math> electrons, as electrons have additionally two possible orientations of their spins.

			The rule that <math>4n+2</math> excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the [[Hückel's rule]].

			Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels.

			== See also ==

			* [[Angular momentum]]
			* [[Harmonic analysis]]
			* [[One-dimensional periodic case]]

			{{DEFAULTSORT:Particle In A Ring}}
			[[Category:Quantum models]]

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2012-06-28T15:31:03Z

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