Power transform - Revision history

en>DavidMCEddy: /* Definition */ error in Bickel & Doksum transform

2014-12-06T01:46:34Z

Definition: error in Bickel & Doksum transform

← Older revision		Revision as of 03:46, 6 December 2014
Line 1:		Line 1:
	The '''Transition dipole moment''' or '''Transition moment''', usually denoted <math>\scriptstyle{\mathbf{d}_{nm}}</math> for a transition between an initial state, <math>\scriptstyle{m}</math>, and a final state, <math>\scriptstyle{n}</math>, is ~~the [[electric dipole moment]] associated with the transition between the two states~~. ~~In general the transition dipole moment~~ is a ~~[[Complex number\|complex]] [[Vector (geometric)\|vector]] quantity that includes the phase factors associated~~ with the two states. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system. ~~The [[International System of Units\|SI unit]] of the transition dipole moment~~ is ~~the [[Coulomb]]-[[meter]] (Cm); a more conveniently sized unit is the [[Debye]] (D).~~		The author is called Irwin. What I adore doing is playing baseball but I haven't made a dime with it. South Dakota is where me and my spouse reside. Managing individuals is his profession.<br><br>My page: [http://www.pornextras.info/user/G49Z www.pornextras.info]

	~~== Definition ==~~

	The transition dipole moment for the <math>\scriptstyle{m, \rightarrow\, n}</math> transition is given by the relevant off-diagonal element of the dipole matrix, which can be calculated from an integral taken over the product of the wavefunctions of the initial and ~~final states of the transition, and the dipole moment operator,~~

	~~:<math>\mathbf{\hat{d}} = e\left(\sum_i x_i, \sum_i y_i, \sum_i z_i\right)</math>,~~

	~~where the [[summation]]s are over the positions of the [[electron]]s in the system~~. ~~Giving the transition dipole moment:~~

	~~:<math>\mathbf{d}_{nm} = \int \Psi^*_n \,\, \mathbf{\hat{d}} \,\, \Psi_m \, d^3r = \langle\Psi_n\|\,\mathbf{\hat{d}}\,\|\Psi_m\rangle</math>,~~

	~~where the integral~~ is in principle over all space, but it can be restricted to the region in which the initial and final state wavefunctions are non-negligible. If there are multiple electrons in the system, the wavefunctions depend on all their positions, and the integral must be over all the electron positions.

	~~== Analogy with a classical dipole ==~~
	~~{{main\|Electric dipole moment}}~~

	A basic, phenomenological understanding of the transition dipole moment can be obtained by analogy with a classical dipole. While the comparison can be very useful, care must be taken to ensure that one does not fall into the trap of assuming they are the same.

	~~In the case of two classical point charges, <math>\scriptstyle{+q}</math> and <math>\scriptstyle{-q}</math>, with a [[displacement (vector)\|displacement vector]],~~ <~~math~~>~~\scriptstyle{\mathbf{r}}~~<~~/math~~>~~, pointing from the negative charge to the positive charge, the electric dipole moment is given by~~

	:~~<math>\mathbf{p} = q\mathbf{r}</math>.~~

	~~In the presence of an~~ [[electric field]], such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net [[torque]] on the dipole. The magnitude of the torque is [[Proportionality (mathematics)\|proportional]] to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole:

	:~~<math>\|\mathbf{\tau}\| = \|q\mathbf{r}\|\|\mathbf{E}\|\sin\theta<~~/~~math>.~~

	~~Similarly, the coupling between an electromagnetic wave and an atomic transition with transition dipole moment <math>\scriptstyle{\mathbf{d}_{nm}}<~~/~~math> depends on the charge distribution within the atom, the strength of the electric field, and the relative polarizations of the field and the transition~~. ~~In addition, the transition dipole moment depends on the geometries and relative phases of the initial and final states~~.

	~~== Origin ==~~

	~~When an atom or molecule interacts with an electromagnetic wave of frequency <math>\scriptstyle{\omega}<~~/~~math>, it can undergo a transition from an initial to a final state of energy difference <math>\scriptstyle{\hbar\omega}<~~/math> through the coupling of the electromagnetic field to the [[transition dipole moment]]. When this transition is from a lower energy state to a higher energy state, this results in the [[Absorption (electromagnetic radiation)\|absorption]] of a [[photon]]. ~~A transition from a higher energy state to a lower energy state results in the [[Emission (electromagnetic_radiation)\|emission]] of a photon~~. If the charge, <math>\scriptstyle{e}</math>, is omitted from the electric dipole operator during this calculation, one obtains <math>\scriptstyle{\mathbf{R}_\alpha}</math> as used in [[oscillator strength]].

	~~== Applications ==~~

	The transition dipole moment is useful for determining if transitions are allowed under the electric dipole interaction. For example, the transition from a bonding <math>\scriptstyle{\pi}</math> orbital to an antibonding <math>\scriptstyle{\pi^*}</math> orbital is allowed because the [[integral]] defining the transition dipole moment is nonzero. Such a transition occurs between an [[Even and odd functions#Even functions\|even]] and an [[Even and odd functions#Odd functions\|odd]] orbital; the dipole operator is an odd function of <math>\scriptstyle{\mathbf{r}}</math>, hence the [[Integral#Terminology and notation\|integrand]] is an even function. The integral of an odd function over symmetric limits returns a value of zero, while for an even function this is not ''necessarily'' the case. This result is reflected in the [[parity (physics)\|parity]] [[selection rule]] for [[electric dipole transition]]s.

	~~==References==~~

	~~{{cite web~~
	~~\| last =~~
	~~\| first =~~
	~~\| authorlink =~~
	~~\| coauthors =~~
	~~\| title = IUPAC compendium of Chemical Terminology~~
	~~\| work =~~
	~~\| publisher = IUPAC~~
	~~\| year = 1997~~
	~~\| url = http://goldbook.iupac.org/T06460.html~~
	~~\| format =~~
	~~\| doi =~~
	~~\| accessdate = 2007-01-15}}~~

	~~{{Physics operator}}~~

	~~[[Category:Atomic physics]~~]

82.128.188.151: /* Box–Cox transformation */

2013-10-17T14:48:23Z

Box–Cox transformation

← Older revision		Revision as of 16:48, 17 October 2013
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Irwin and I totally dig~~ that ~~name~~. ~~I am~~ a meter ~~reader~~ but ~~I strategy~~ on ~~altering it~~. ~~To gather coins~~ is 1 of [~~https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~9243~~ over the ~~counter std test~~] ~~things I love most~~. ~~North Dakota is where me and my husband reside~~.		The '''Transition dipole moment''' or '''Transition moment''', usually denoted <math>\scriptstyle{\mathbf{d}_{nm}}</math> for a transition between an initial state, <math>\scriptstyle{m}</math>, and a final state, <math>\scriptstyle{n}</math>, is the [[electric dipole moment]] associated with the transition between the two states. In general the transition dipole moment is a [[Complex number\|complex]] [[Vector (geometric)\|vector]] quantity that includes the phase factors associated with the two states. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system. The [[International System of Units\|SI unit]] of the transition dipole moment is the [[Coulomb]]-[[meter]] (Cm); a more conveniently sized unit is the [[Debye]] (D).

			== Definition ==

			The transition dipole moment for the <math>\scriptstyle{m, \rightarrow\, n}</math> transition is given by the relevant off-diagonal element of the dipole matrix, which can be calculated from an integral taken over the product of the wavefunctions of the initial and final states of the transition, and the dipole moment operator,

			:<math>\mathbf{\hat{d}} = e\left(\sum_i x_i, \sum_i y_i, \sum_i z_i\right)</math>,

			where the [[summation]]s are over the positions of the [[electron]]s in the system. Giving the transition dipole moment:

			:<math>\mathbf{d}_{nm} = \int \Psi^*_n \,\, \mathbf{\hat{d}} \,\, \Psi_m \, d^3r = \langle\Psi_n\|\,\mathbf{\hat{d}}\,\|\Psi_m\rangle</math>,

			where the integral is in principle over all space, but it can be restricted to the region in which the initial and final state wavefunctions are non-negligible. If there are multiple electrons in the system, the wavefunctions depend on all their positions, and the integral must be over all the electron positions.

			== Analogy with a classical dipole ==
			{{main\|Electric dipole moment}}

			A basic, phenomenological understanding of the transition dipole moment can be obtained by analogy with a classical dipole. While the comparison can be very useful, care must be taken to ensure that one does not fall into the trap of assuming they are the same.

			In the case of two classical point charges, <math>\scriptstyle{+q}</math> and <math>\scriptstyle{-q}</math>, with a [[displacement (vector)\|displacement vector]], <math>\scriptstyle{\mathbf{r}}</math>, pointing from the negative charge to the positive charge, the electric dipole moment is given by

			:<math>\mathbf{p} = q\mathbf{r}</math>.

			In the presence of an [[electric field]], such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net [[torque]] on the dipole. The magnitude of the torque is [[Proportionality (mathematics)\|proportional]] to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole:

			:<math>\|\mathbf{\tau}\| = \|q\mathbf{r}\|\|\mathbf{E}\|\sin\theta</math>.

			Similarly, the coupling between an electromagnetic wave and an atomic transition with transition dipole moment <math>\scriptstyle{\mathbf{d}_{nm}}</math> depends on the charge distribution within the atom, the strength of the electric field, and the relative polarizations of the field and the transition. In addition, the transition dipole moment depends on the geometries and relative phases of the initial and final states.

			== Origin ==

			When an atom or molecule interacts with an electromagnetic wave of frequency <math>\scriptstyle{\omega}</math>, it can undergo a transition from an initial to a final state of energy difference <math>\scriptstyle{\hbar\omega}</math> through the coupling of the electromagnetic field to the [[transition dipole moment]]. When this transition is from a lower energy state to a higher energy state, this results in the [[Absorption (electromagnetic radiation)\|absorption]] of a [[photon]]. A transition from a higher energy state to a lower energy state results in the [[Emission (electromagnetic_radiation)\|emission]] of a photon. If the charge, <math>\scriptstyle{e}</math>, is omitted from the electric dipole operator during this calculation, one obtains <math>\scriptstyle{\mathbf{R}_\alpha}</math> as used in [[oscillator strength]].

			== Applications ==

			The transition dipole moment is useful for determining if transitions are allowed under the electric dipole interaction. For example, the transition from a bonding <math>\scriptstyle{\pi}</math> orbital to an antibonding <math>\scriptstyle{\pi^*}</math> orbital is allowed because the [[integral]] defining the transition dipole moment is nonzero. Such a transition occurs between an [[Even and odd functions#Even functions\|even]] and an [[Even and odd functions#Odd functions\|odd]] orbital; the dipole operator is an odd function of <math>\scriptstyle{\mathbf{r}}</math>, hence the [[Integral#Terminology and notation\|integrand]] is an even function. The integral of an odd function over symmetric limits returns a value of zero, while for an even function this is not ''necessarily'' the case. This result is reflected in the [[parity (physics)\|parity]] [[selection rule]] for [[electric dipole transition]]s.

			==References==

			{{cite web
			\| last =
			\| first =
			\| authorlink =
			\| coauthors =
			\| title = IUPAC compendium of Chemical Terminology
			\| work =
			\| publisher = IUPAC
			\| year = 1997
			\| url = http://goldbook.iupac.org/T06460.html
			\| format =
			\| doi =
			\| accessdate = 2007-01-15}}

			{{Physics operator}}

			[[Category:Atomic physics]]

en>Isheden at 08:24, 21 August 2012

2012-08-21T08:24:54Z

New page

Hello and welcome. My name is Irwin and I totally dig that name. I am a meter reader but I strategy on altering it. To gather coins is 1 of [https://www.machlitim.org.il/subdomain/megila/end/node/9243 over the counter std test] things I love most. North Dakota is where me and my husband reside.