Physics:Quantum Stationary states - Revision history

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	{{Short description\|Quantum Collection topic on Quantum Stationary states}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Stationary states}}

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	'''Stationary states''' is a ~~Book I topic in the Quantum Collection. A~~ stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies). It is also called an energy eigenstate, energy eigenfunction, or energy eigenket. Stationary states are fundamental in quantum mechanics and are closely related to concepts such as atomic orbitals and molecular orbitals. in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]] A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies).		'''Stationary states''' a stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies). It is also called an energy eigenstate, energy eigenfunction, or energy eigenket. Stationary states are fundamental in quantum mechanics and are closely related to concepts such as atomic orbitals and molecular orbitals. in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]] A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies). A stationary state is called stationary because the system remains unchanged in every observable way as time elapses.
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	A stationary state is called ''stationary'' because the system remains unchanged in every observable way as time elapses. For a system with a time-independent Hamiltonian, this means that measurable quantities such as position probability, momentum, or spin remain constant in time.<ref>Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë. ''Quantum Mechanics: Volume One''. Hermann, 1977. p. 32.</ref>		A stationary state is called ''stationary'' because the system remains unchanged in every observable way as time elapses. For a system with a time-independent Hamiltonian, this means that measurable quantities such as position probability, momentum, or spin remain constant in time.<ref>Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë. ''Quantum Mechanics: Volume One''. Hermann, 1977. p. 32.</ref>

	The [[wavefunction]] itself is not constant: it evolves by a global complex [[phase factor]], forming a [[standing wave]]. The oscillation frequency of this phase, multiplied by the Planck constant, corresponds to the energy via the Planck–Einstein relation.		The wavefunction itself is not constant: it evolves by a global complex phase factor, forming a standing wave. The oscillation frequency of this phase, multiplied by the Planck constant, corresponds to the energy via the Planck–Einstein relation.

	== Definition ==		== Definition ==
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	== Superposition ==		== Superposition ==
	A general [[quantum state]] need not be stationary. A [[superposition]] of stationary states with different energies leads to time-dependent interference effects, producing a changing probability distribution.		A general quantum state need not be stationary. A superposition of stationary states with different energies leads to time-dependent interference effects, producing a changing probability distribution.

	== Spontaneous decay ==		== Spontaneous decay ==
	In ideal (nonrelativistic) quantum mechanics, systems such as the [[hydrogen atom]] have many stationary states. However, in reality, excited states are not perfectly stationary: an electron in a higher energy level can undergo [[spontaneous emission]], emitting a [[photon]] and decaying to a lower-energy state.<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, {{ISBN\|978-0-471-87373-0}}</ref>		In ideal (nonrelativistic) quantum mechanics, systems such as the hydrogen atom have many stationary states. However, in reality, excited states are not perfectly stationary: an electron in a higher energy level can undergo spontaneous emission, emitting a photon and decaying to a lower-energy state.<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, {{ISBN\|978-0-471-87373-0}}</ref>

	This occurs because the usual Hamiltonian is an approximation; more complete descriptions from [[quantum field theory]] include effects such as [[vacuum fluctuations]], which break exact stationarity for excited states.		This occurs because the usual Hamiltonian is an approximation; more complete descriptions from quantum field theory include effects such as vacuum fluctuations, which break exact stationarity for excited states.

	== Comparison to orbitals ==		== Comparison to orbitals ==
	~~{{main\|~~Atomic orbital\|Molecular orbital}}		''Related topic:'' Atomic orbital, Molecular orbital

	An [[atomic orbital]] or [[molecular orbital]] can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P. W. Atkins, Oxford University Press, 1978, {{ISBN\|0-19-855148-7}}.</ref>		An atomic orbital or molecular orbital can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P. W. Atkins, Oxford University Press, 1978, {{ISBN\|0-19-855148-7}}.</ref>

	For single-electron systems (such as [[hydrogen]]), orbitals correspond directly to stationary states. For many-electron systems, however, the full stationary state is a many-particle state, often approximated using methods such as Slater determinantss.<ref name=lowdin55_1>{{cite journal \|first1=Per-Olov \|last1=Löwdin \|journal=Physical Review \|title=Quantum theory of many-particle systems \|volume=97 \|issue=6 \|pages=1474–1489 \|year=1955}}</ref>		For single-electron systems (such as hydrogen), orbitals correspond directly to stationary states. For many-electron systems, however, the full stationary state is a many-particle state, often approximated using methods such as Slater determinantss.<ref name=lowdin55_1>{{cite journal \|first1=Per-Olov \|last1=Löwdin \|journal=Physical Review \|title=Quantum theory of many-particle systems \|volume=97 \|issue=6 \|pages=1474–1489 \|year=1955}}</ref>

	In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the Born–Oppenheimer approximation.		In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the Born–Oppenheimer approximation.

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	A '''~~stationary state~~''' is a [[quantum state]] with all ~~[[observable]]s~~ independent of time. It is an [[eigenvector]] of the [[energy operator]] (rather than a [[quantum superposition]] of different energies). It is also called an ~~'''~~energy eigenstate~~'''~~, ~~'''~~energy eigenfunction~~'''~~, or ~~'''~~energy ~~[[Bra-ket notation\|~~eigenket~~]]'''~~. Stationary states are fundamental in [[quantum mechanics]] and are closely related to concepts such as [[atomic ~~orbital]]s~~ and [[molecular ~~orbital]]s~~.		'''Stationary states''' is a Book I topic in the Quantum Collection. A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies). It is also called an energy eigenstate, energy eigenfunction, or energy eigenket. Stationary states are fundamental in quantum mechanics and are closely related to concepts such as atomic orbitals and molecular orbitals. in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]] A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies).

	in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]]
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	== Introduction ==		== Introduction ==
	A stationary state is called ''stationary'' because the system remains unchanged in every observable way as time elapses. For a system with a time-independent [[Hamiltonian ~~(quantum mechanics)\|Hamiltonian]]~~, this means that measurable quantities such as position probability, momentum, or ~~[[Spin (physics)\|~~spin]] remain constant in time.<ref>[[Claude Cohen-Tannoudji]], Bernard Diu, and [[Franck Laloë]]. ''Quantum Mechanics: Volume One''. Hermann, 1977. p. 32.</ref>		A stationary state is called ''stationary'' because the system remains unchanged in every observable way as time elapses. For a system with a time-independent Hamiltonian, this means that measurable quantities such as position probability, momentum, or spin remain constant in time.<ref>Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë. ''Quantum Mechanics: Volume One''. Hermann, 1977. p. 32.</ref>

	The [[wavefunction]] itself is not constant: it evolves by a global complex [[phase factor]], forming a [[standing wave]]. The oscillation frequency of this phase, multiplied by the [[Planck constant]], corresponds to the energy via the [[Planck–Einstein relation]].		The [[wavefunction]] itself is not constant: it evolves by a global complex [[phase factor]], forming a [[standing wave]]. The oscillation frequency of this phase, multiplied by the Planck constant, corresponds to the energy via the Planck–Einstein relation.

	== Definition ==		== Definition ==
	Stationary states are solutions to the time-independent [[Schrödinger equation]]:		Stationary states are solutions to the time-independent Schrödinger equation:
	<math display="block">\hat H \|\Psi\rangle = E \|\Psi\rangle</math>		<math display="block">\hat H \|\Psi\rangle = E \|\Psi\rangle</math>

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	* <math>E</math> is the energy eigenvalue.		* <math>E</math> is the energy eigenvalue.

	This is an ~~[[Eigenvalues and eigenvectors\|~~eigenvalue equation]]: the stationary states are eigenvectors of the Hamiltonian.		This is an eigenvalue equation: the stationary states are eigenvectors of the Hamiltonian.

	When inserted into the time-dependent Schrödinger equation, the evolution is:<ref>Quanta: A handbook of concepts, P. W. Atkins, Oxford University Press, 1974, {{ISBN\|0-19-855493-1}}.</ref>		When inserted into the time-dependent Schrödinger equation, the evolution is:<ref>Quanta: A handbook of concepts, P. W. Atkins, Oxford University Press, 1974, {{ISBN\|0-19-855493-1}}.</ref>
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	is time-independent.		is time-independent.

	In the [[Heisenberg picture]], stationary states are mathematically constant in time.		In the Heisenberg picture, stationary states are mathematically constant in time.

	These results assume a time-independent Hamiltonian; if the system changes, the state will generally no longer be stationary.		These results assume a time-independent Hamiltonian; if the system changes, the state will generally no longer be stationary.
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	An [[atomic orbital]] or [[molecular orbital]] can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P. W. Atkins, Oxford University Press, 1978, {{ISBN\|0-19-855148-7}}.</ref>		An [[atomic orbital]] or [[molecular orbital]] can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P. W. Atkins, Oxford University Press, 1978, {{ISBN\|0-19-855148-7}}.</ref>

	For single-electron systems (such as [[hydrogen]]), orbitals correspond directly to stationary states. For many-electron systems, however, the full stationary state is a many-particle state, often approximated using methods such as [[Slater ~~determinant]]s~~.<ref name=lowdin55_1>{{cite journal \|first1=Per-Olov \|last1=Löwdin \|journal=[[Physical Review]] \|title=Quantum theory of many-particle systems \|volume=97 \|issue=6 \|pages=1474–1489 \|year=1955}}</ref>		For single-electron systems (such as [[hydrogen]]), orbitals correspond directly to stationary states. For many-electron systems, however, the full stationary state is a many-particle state, often approximated using methods such as Slater determinantss.<ref name=lowdin55_1>{{cite journal \|first1=Per-Olov \|last1=Löwdin \|journal=Physical Review \|title=Quantum theory of many-particle systems \|volume=97 \|issue=6 \|pages=1474–1489 \|year=1955}}</ref>

	In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the [[Born–Oppenheimer approximation]].		In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the Born–Oppenheimer approximation.

	=See also=		=See also=

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	{{Sourceattribution\|Physics:Stationary ~~state~~\|1}}		{{Sourceattribution\|Physics:Quantum Stationary states\|1}}

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			{{Short description\|Quantum Collection topic on Quantum Stationary states}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}

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

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			__TOC__
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	A '''stationary state''' is a [[quantum state]] with all [[observable]]s independent of time. It is an [[eigenvector]] of the [[energy operator]] (rather than a [[quantum superposition]] of different energies). It is also called an '''energy eigenstate''', '''energy eigenfunction''', or '''energy [[Bra-ket notation\|eigenket]]'''. Stationary states are fundamental in [[quantum mechanics]] and are closely related to concepts such as [[atomic orbital]]s and [[molecular orbital]]s.		A '''stationary state''' is a [[quantum state]] with all [[observable]]s independent of time. It is an [[eigenvector]] of the [[energy operator]] (rather than a [[quantum superposition]] of different energies). It is also called an '''energy eigenstate''', '''energy eigenfunction''', or '''energy [[Bra-ket notation\|eigenket]]'''. Stationary states are fundamental in [[quantum mechanics]] and are closely related to concepts such as [[atomic orbital]]s and [[molecular orbital]]s.

	~~[[File:Quantum_stationary_states_energy_levels_yellow.jpg\|thumb\|400px\|Stationary states as energy eigenstates: discrete energy levels with time-independent probability densities.]]~~		in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]]
	~~[[File:QuantumHarmonicOscillatorAnimation.gif\|thumb\|400px\|A [[harmonic oscillator]]~~ in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]]		</div>

			<div style="width:300px;">
			[[File:Quantum_stationary_states_energy_levels_yellow.jpg\|thumb\|280px\|Quantum Stationary states.]]
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			</div>

	== Introduction ==		== Introduction ==

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	{{Quantum article nav\|previous=Physics:Quantum Time-dependent Schrödinger equation\|previous label=Time-dependent Schrödinger equation\|next=Physics:Quantum Perturbation theory\|next label=Perturbation theory}}		{{Quantum article nav\|previous=Physics:Quantum Time-dependent Schrödinger equation\|previous label=Time-dependent Schrödinger equation\|next=Physics:Quantum Perturbation theory\|next label=Perturbation theory}}