Algebraic vector bundle - Revision history

en>Bearcat: categorization/tagging using AWB

2014-12-28T16:06:22Z

categorization/tagging using AWB

← Older revision		Revision as of 18:06, 28 December 2014
Line 1:		Line 1:
	~~Much insight in [[quantum mechanics]] can be gained~~ from understanding the solutions to the time-dependent non-relativistic [[Schrödinger equation]] in an appropriate [[configuration space]]. In vector Cartesian coordinates <math>\mathbf{r}</math>, the equation takes the form		29 year-old Florist Colton Crochet from Spruce Grove, really likes comics, como ganhar dinheiro na internet and papercraft. Recently had a family voyage to Old City of Sana'a.<br><br>Look at my blog; [http://comoganhardinheironainternet.comoganhardinheiro101.com como conseguir dinheiro]

	~~:<math>~~
	~~H \psi\left(\mathbf{r}, t\right) = \left(T + V\right) \, \psi\left(\mathbf{r}, t\right) =~~
	~~\left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}, t\right) = i\hbar \frac{\partial\psi\left(\mathbf{r}, t\right)}{\partial t}~~
	~~</math>~~

	~~in which <math>\psi</math> is the [[wavefunction]] of the system~~, ~~H is the [[Hamiltonian operator]]~~, and ~~T and V are the operators for the [[kinetic energy]] and [[potential energy]], respectively~~. ~~(Common forms~~ of ~~these operators appear in the square brackets.) The quantity~~ '~~'t'' is the time~~. ~~[[Stationary state]]s of this equation are found by solving the [[eigenvalue]]-[[eigenfunction]] (time-independent) form of the Schrödinger equation,~~

	:<~~math~~>
	~~\left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}\right) = E \psi \left(\mathbf{r}\right)~~
	<~~/math~~>

	or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with [[spherical coordinates]]. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These '''quantum-mechanical systems with analytical solutions''' are listed below, and are quite useful for teaching and gaining intuition about quantum mechanics.

	~~== Solvable systems ==~~

	*The [[free particle]]
	*The [[delta potential]]
	*The [[particle in a box]] / [[infinite potential well]]
	*The [[finite potential well]]
	*The [[Airy function\|One-dimensional triangular potential]]
	*The [[particle in a ring]] or [[ring wave guide]]
	*The [[particle in a spherically symmetric potential]]
	*The [[quantum harmonic oscillator]]
	*The [[hydrogen atom]] or [[hydrogen-like atom]]
	*The [[particle in a one-dimensional lattice (periodic potential)]]
	*The [[Morse potential]]
	*The [[step potential]]
	*The [[Rigid_rotor#Quantum_mechanical_linear_rigid_rotor\|linear rigid rotor]]
	*The [[Rigid_rotor#Quantum_mechanical_rigid_rotor\|symmetric top]]
	*The [~~[Hooke's atom]]~~
	*The [[Spherium]]
	*Zero range interaction in a harmonic trap<ref>http://~~www~~.~~springerlink~~.com~~/content/k86t52r653522rk6/</ref>~~
	*The [[Quantum pendulum]]
	*The [[Rectangular potential barrier]]

	~~==References==~~
	~~{{reflist}}~~
	~~== See also ==~~
	* [[List of quantum-mechanical potentials]] – a list of physically relevant potentials without regard to analytic solubility
	* [[List of integrable models]]

	~~== Reading materials ==~~

	* {{cite book
	~~\| last = Mattis~~
	~~\| first = Daniel C.~~
	~~\| authorlink = Daniel C. Mattis~~
	~~\| title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension~~
	~~\| publisher = [[World Scientific]]~~
	~~\| date = 1993~~
	~~\| isbn = 981-02-0975-4}}~~

	~~[[Category:Quantum mechanics]]~~
	~~[[Category:Quantum models]]~~
	~~[[Category:Physics-related lists\|Quantum-mechanical systems with analytical solutions]~~]

en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

2013-01-09T00:29:16Z

Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

← Older revision		Revision as of 02:29, 9 January 2013
Line 1:		Line 1:
	~~Hi there~~, ~~I am Yoshiko Villareal but I never truly liked that name~~. The ~~thing she adores most~~ is ~~flower arranging and she is attempting~~ to ~~make it~~ a ~~occupation~~. ~~I am~~ a ~~manufacturing~~ and ~~distribution officer~~. ~~Arizona has always been my living location but my spouse wants us to transfer~~.<br>~~<br>Feel free to surf to my homepage [~~http://~~Www~~.~~dubaitoursrus~~.com/~~UserProfile~~/~~tabid~~/61/~~userId/122235/Default~~.~~aspx car warranty~~]		Much insight in [[quantum mechanics]] can be gained from understanding the solutions to the time-dependent non-relativistic [[Schrödinger equation]] in an appropriate [[configuration space]]. In vector Cartesian coordinates <math>\mathbf{r}</math>, the equation takes the form

			:<math>
			H \psi\left(\mathbf{r}, t\right) = \left(T + V\right) \, \psi\left(\mathbf{r}, t\right) =
			\left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}, t\right) = i\hbar \frac{\partial\psi\left(\mathbf{r}, t\right)}{\partial t}
			</math>

			in which <math>\psi</math> is the [[wavefunction]] of the system, H is the [[Hamiltonian operator]], and T and V are the operators for the [[kinetic energy]] and [[potential energy]], respectively. (Common forms of these operators appear in the square brackets.) The quantity ''t'' is the time. [[Stationary state]]s of this equation are found by solving the [[eigenvalue]]-[[eigenfunction]] (time-independent) form of the Schrödinger equation,

			:<math>
			\left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}\right) = E \psi \left(\mathbf{r}\right)
			</math>

			or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with [[spherical coordinates]]. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These '''quantum-mechanical systems with analytical solutions''' are listed below, and are quite useful for teaching and gaining intuition about quantum mechanics.

			== Solvable systems ==

			*The [[free particle]]
			*The [[delta potential]]
			*The [[particle in a box]] / [[infinite potential well]]
			*The [[finite potential well]]
			*The [[Airy function\|One-dimensional triangular potential]]
			*The [[particle in a ring]] or [[ring wave guide]]
			*The [[particle in a spherically symmetric potential]]
			*The [[quantum harmonic oscillator]]
			*The [[hydrogen atom]] or [[hydrogen-like atom]]
			*The [[particle in a one-dimensional lattice (periodic potential)]]
			*The [[Morse potential]]
			*The [[step potential]]
			*The [[Rigid_rotor#Quantum_mechanical_linear_rigid_rotor\|linear rigid rotor]]
			*The [[Rigid_rotor#Quantum_mechanical_rigid_rotor\|symmetric top]]
			*The [[Hooke's atom]]
			*The [[Spherium]]
			*Zero range interaction in a harmonic trap<ref>http://www.springerlink.com/content/k86t52r653522rk6/</ref>
			*The [[Quantum pendulum]]
			*The [[Rectangular potential barrier]]

			==References==
			{{reflist}}
			== See also ==
			* [[List of quantum-mechanical potentials]] – a list of physically relevant potentials without regard to analytic solubility
			* [[List of integrable models]]

			== Reading materials ==

			* {{cite book
			\| last = Mattis
			\| first = Daniel C.
			\| authorlink = Daniel C. Mattis
			\| title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension
			\| publisher = [[World Scientific]]
			\| date = 1993
			\| isbn = 981-02-0975-4}}

			[[Category:Quantum mechanics]]
			[[Category:Quantum models]]
			[[Category:Physics-related lists\|Quantum-mechanical systems with analytical solutions]]

en>Spaetzle: started 'see also'-section with a wikilink

2012-06-28T08:20:20Z

started 'see also'-section with a wikilink

New page

Hi there, I am Yoshiko Villareal but I never truly liked that name. The thing she adores most is flower arranging and she is attempting to make it a occupation. I am a manufacturing and distribution officer. Arizona has always been my living location but my spouse wants us to transfer.<br><br>Feel free to surf to my homepage [http://Www.dubaitoursrus.com/UserProfile/tabid/61/userId/122235/Default.aspx car warranty]