Physics:Quantum Time-dependent Schrödinger equation - Revision history

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	[[Category:Partial differential equations]]		[[Category:Partial differential equations]]

	{{Sourceattribution\|Quantum Time-dependent Schrödinger equation\|1}}		{{Sourceattribution\|Physics:Quantum Time-dependent Schrödinger equation\|1}}

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			{{Short description\|Quantum Collection topic on Quantum Time-dependent Schrödinger equation}}

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			{{continuum mechanics}}

	~~{{continuum mechanics}}~~		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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	A '''continuity equation''' or '''transport equation''' is an [[equation]] that describes the transport of some quantity. It is particularly simple and powerful when applied to a [[conserved quantity]], but it can be generalized to apply to any [[Intensive and extensive properties\|extensive quantity]]. Since [[mass]], [[energy]], [[momentum]], [[electric charge]] and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.		A '''continuity equation''' or '''transport equation''' is an [[equation]] that describes the transport of some quantity. It is particularly simple and powerful when applied to a [[conserved quantity]], but it can be generalized to apply to any [[Intensive and extensive properties\|extensive quantity]]. Since [[mass]], [[energy]], [[momentum]], [[electric charge]] and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

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	Flows governed by continuity equations can be visualized using a [[Sankey diagram]].		Flows governed by continuity equations can be visualized using a [[Sankey diagram]].
			s that can be differentiated into infinitesimal area elements).]]
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			[[File:Continuity_eqn_open_surface.png\|thumb\|280px\|Quantum Time-dependent Schrödinger equation.]]
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	[[File:Continuity_eqn_open_surface.png\|thumb\|Illustration of how the fluxes, or flux densities, {{math\|'''j'''<sub>1</sub>}} and {{math\|'''j'''<sub>2</sub>}} of a quantity {{math\|''q''}} pass through open surfaces {{math\|''S''<sub>1</sub>}} and {{math\|''S''<sub>2</sub>}}. (vectors {{math\|'''S'''<sub>1</sub>}} and {{math\|'''S'''<sub>2</~~sub~~>~~}} represent [[vector area]]s that can be differentiated into infinitesimal area elements).]]~~		</div>

	==General equation==		==General equation==

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	Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.		Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.

	Any continuity equation can be expressed in an "integral form" (in terms of a ~~[[Flux#Flux as a surface integral\|~~flux integral]]), which applies to any finite region, or in a "differential form" (in terms of the [[divergence]] operator) which applies at a point.		Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the [[divergence]] operator) which applies at a point.

	Continuity equations underlie more specific transport equations such as the [[convection–diffusion equation]], Boltzmann transport equation, and Navier–Stokes equations.		Continuity equations underlie more specific transport equations such as the [[convection–diffusion equation]], Boltzmann transport equation, and Navier–Stokes equations.
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	* {{math\|'''q'''}}, [[energy flux]] (transfer of energy per unit cross-sectional area per unit time) as a vector,		* {{math\|'''q'''}}, [[energy flux]] (transfer of energy per unit cross-sectional area per unit time) as a vector,

	An important practical example is the flow of heat. When heat flows inside a solid, the continuity equation can be combined with ~~[[Thermal conduction#~~Fourier's law~~\|Fourier's law]]~~ (heat flux is proportional to temperature gradient) to arrive at the [[heat equation]]. The equation of heat flow may also have source terms: Although ''energy'' cannot be created or destroyed, ''heat'' can be created from other types of energy, for example via [[friction]] or [[joule heating]].		An important practical example is the flow of heat. When heat flows inside a solid, the continuity equation can be combined with Fourier's law (heat flux is proportional to temperature gradient) to arrive at the [[heat equation]]. The equation of heat flow may also have source terms: Although ''energy'' cannot be created or destroyed, ''heat'' can be created from other types of energy, for example via [[friction]] or [[joule heating]].

	== Probability distributions ==		== Probability distributions ==
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	This is an important constraint on the form the Einstein field equations take in [[general relativity]].<ref>{{cite book \|title=Relativity DeMystified\|author=D. McMahon\|publisher=McGraw Hill (USA)\|year=2006\|isbn=0-07-145545-0}}</ref>		This is an important constraint on the form the Einstein field equations take in [[general relativity]].<ref>{{cite book \|title=Relativity DeMystified\|author=D. McMahon\|publisher=McGraw Hill (USA)\|year=2006\|isbn=0-07-145545-0}}</ref>

	However, the ''ordinary'' ~~[[Tensors in curvilinear coordinates#Second-order tensor field\|~~divergence]] of the stress–energy tensor does ''not'' necessarily vanish:<ref>{{cite book \|title=Gravitation \|author=C.W. Misner \|last2=K.S. Thorne \|last3=J.A. Wheeler \| publisher=W.H. Freeman & Co \|year=1973 \|isbn=0-7167-0344-0}}</ref>		However, the ''ordinary'' divergence of the stress–energy tensor does ''not'' necessarily vanish:<ref>{{cite book \|title=Gravitation \|author=C.W. Misner \|last2=K.S. Thorne \|last3=J.A. Wheeler \| publisher=W.H. Freeman & Co \|year=1973 \|isbn=0-7167-0344-0}}</ref>
	<math display="block">\partial_{\mu} T^{\mu\nu} = - \Gamma^{\mu}_{\mu \lambda} T^{\lambda \nu} - \Gamma^{\nu}_{\mu \lambda} T^{\mu \lambda},</math>		<math display="block">\partial_{\mu} T^{\mu\nu} = - \Gamma^{\mu}_{\mu \lambda} T^{\lambda \nu} - \Gamma^{\nu}_{\mu \lambda} T^{\mu \lambda},</math>

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	==Particle physics==		==Particle physics==

	Quarkss and [[gluon]]s have ''[[color charge]]'', which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at ~~[[Gluon field strength tensor#Equation of motion\|~~gluon field strength tensor]]).		Quarkss and [[gluon]]s have ''[[color charge]]'', which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at gluon field strength tensor).

	There are many other quantities in particle physics which are often or always conserved: [[baryon number]] (proportional to the number of quarks minus the number of antiquarks), [[lepton number\|electron number, mu number, tau number]], [[isospin]], and others.<ref>{{cite book \|title=Gravitation \|author=C.W. Misner \|last2=K.S. Thorne \|last3=J.A. Wheeler \| publisher=W.H. Freeman & Co \|year=1973 \|isbn=0-7167-0344-0\|pages=558–559}}</ref> Each of these has a corresponding continuity equation, possibly including source / sink terms.		There are many other quantities in particle physics which are often or always conserved: [[baryon number]] (proportional to the number of quarks minus the number of antiquarks), [[lepton number\|electron number, mu number, tau number]], [[isospin]], and others.<ref>{{cite book \|title=Gravitation \|author=C.W. Misner \|last2=K.S. Thorne \|last3=J.A. Wheeler \| publisher=W.H. Freeman & Co \|year=1973 \|isbn=0-7167-0344-0\|pages=558–559}}</ref> Each of these has a corresponding continuity equation, possibly including source / sink terms.