Physics:Quantum Time-dependent perturbation theory - Revision history

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	'''Time-dependent perturbation theory''' ~~is a Book I topic in the Quantum Collection. Time~~-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson, studies the effect of a time-dependent perturbation applied to a time-independent Hamiltonian .		'''Time-dependent perturbation theory''' time-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson, studies the effect of a time-dependent perturbation applied to a time-independent Hamiltonian .
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	'''Time-dependent perturbation theory''' is a method in ~~[[Physics:Quantum mechanics\|~~quantum mechanics]] used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the ~~[[Physics:~~Hamiltonian (quantum mechanics)\|Hamiltonian]] depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems.		'''Time-dependent perturbation theory''' is a Book I topic in the Quantum Collection. Time-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory is a method in quantum mechanics used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the Hamiltonian depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems. Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson, studies the effect of a time-dependent perturbation applied to a time-independent Hamiltonian .
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	== Introduction ==		== Introduction ==
	Time-dependent perturbation theory, initiated by [[Biography:Paul Dirac\|Paul Dirac]] and further developed by ~~[[Biography:~~John Archibald Wheeler~~\|John Archibald Wheeler]]~~, ~~[[Biography:Richard Feynman\|~~Richard Feynman]], and ~~[[Biography:Freeman Dyson\|~~Freeman Dyson]],<ref name=":0">{{Citation \|last=Dick \|first=Rainer \|title=Time-Dependent Perturbations in Quantum Mechanics \|date=2020 \|work=Advanced Quantum Mechanics: Materials and Photons \|pages=265–310 \|editor-last=Dick \|editor-first=Rainer \|series=Graduate Texts in Physics \|place=Cham \|publisher=Springer International Publishing \|language=en \|doi=10.1007/978-3-030-57870-1_13 \|isbn=978-3-030-57870-1}}</ref> studies the effect of a time-dependent perturbation {{math\|''V''(''t'')}} applied to a time-independent Hamiltonian {{math\|''H''<sub>0</sub>}}.<ref>~~[[Biography:Albert Messiah\|~~Albert Messiah]] (1966). ''Quantum Mechanics'', North Holland, John Wiley & Sons. {{ISBN\|0486409244}}; J. J. Sakurai (1994). ''Modern Quantum Mechanics'' (Addison-Wesley) {{ISBN\|9780201539295}}.</ref>		Time-dependent perturbation theory, initiated by [[Biography:Paul Dirac\|Paul Dirac]] and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson,<ref name=":0">{{Citation \|last=Dick \|first=Rainer \|title=Time-Dependent Perturbations in Quantum Mechanics \|date=2020 \|work=Advanced Quantum Mechanics: Materials and Photons \|pages=265–310 \|editor-last=Dick \|editor-first=Rainer \|series=Graduate Texts in Physics \|place=Cham \|publisher=Springer International Publishing \|language=en \|doi=10.1007/978-3-030-57870-1_13 \|isbn=978-3-030-57870-1}}</ref> studies the effect of a time-dependent perturbation {{math\|''V''(''t'')}} applied to a time-independent Hamiltonian {{math\|''H''<sub>0</sub>}}.<ref>Albert Messiah (1966). ''Quantum Mechanics'', North Holland, John Wiley & Sons. {{ISBN\|0486409244}}; J. J. Sakurai (1994). ''Modern Quantum Mechanics'' (Addison-Wesley) {{ISBN\|9780201539295}}.</ref>

	It is widely used to describe processes such as scattering, radiation, atomic transitions, and response of matter to external fields. Applications include proton scattering, photo-ionization, neutron interactions, and dielectric response of materials.<ref name=":0" />		It is widely used to describe processes such as scattering, radiation, atomic transitions, and response of matter to external fields. Applications include proton scattering, photo-ionization, neutron interactions, and dielectric response of materials.<ref name=":0" />

	Since the Hamiltonian is time-dependent, both the energy levels and eigenstates evolve in time. The main quantities of interest are:		Since the Hamiltonian is time-dependent, both the energy levels and eigenstates evolve in time. The main quantities of interest are:
	* The time-dependent ~~[[Physics:Expectation value (quantum mechanics)\|~~expectation value]] of an observable.		* The time-dependent expectation value of an observable.
	* The time-dependent probability amplitudes of energy eigenstates.		* The time-dependent probability amplitudes of energy eigenstates.

	These quantities are crucial for understanding phenomena such as ~~[[Physics:Spectral line\|~~spectral line]] broadening, ~~[[Physics:Particle decay\|~~particle decay]], and population dynamics in ~~[[Physics:Laser\|~~laser]] physics.		These quantities are crucial for understanding phenomena such as spectral line broadening, particle decay, and population dynamics in laser physics.

	== Method of variation of constants ==		== Method of variation of constants ==
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	<math display="block"> \|c_n(t)\|^2 = \|\langle n\|\psi(t)\rangle\|^2.</math>		<math display="block"> \|c_n(t)\|^2 = \|\langle n\|\psi(t)\rangle\|^2.</math>

	Substituting into the ~~[[Physics:~~Schrödinger equation~~\|Schrödinger equation]]~~ yields		Substituting into the Schrödinger equation yields
	<math display="block"> \frac{dc_n}{dt} = \frac{-i}{\hbar} \sum_k \langle n\|V(t)\|k\rangle \,c_k(t)\, e^{-i(E_k - E_n)t/\hbar}.</math>		<math display="block"> \frac{dc_n}{dt} = \frac{-i}{\hbar} \sum_k \langle n\|V(t)\|k\rangle \,c_k(t)\, e^{-i(E_k - E_n)t/\hbar}.</math>

	This exact system of coupled ~~[[Differential equation\|~~differential equations]] describes how amplitudes evolve in time.		This exact system of coupled differential equations describes how amplitudes evolve in time.

	For weak perturbations, an iterative (perturbative) solution is used:		For weak perturbations, an iterative (perturbative) solution is used:
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	This framework leads to important results such as:		This framework leads to important results such as:
	* ~~[[Physics:~~Fermi's golden rule~~\|Fermi's golden rule]]~~		* Fermi's golden rule
	* Transition probabilities between quantum states		* Transition probabilities between quantum states
	* Time-dependent expectation values		* Time-dependent expectation values
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	<math display="block">\|\psi(t)\rangle = T\exp{\left[-\frac{i}{\hbar}\int_{t_0}^t dt'H(t')\right]}\|\psi(t_0)\rangle,</math>		<math display="block">\|\psi(t)\rangle = T\exp{\left[-\frac{i}{\hbar}\int_{t_0}^t dt'H(t')\right]}\|\psi(t_0)\rangle,</math>

	where {{mvar\|T}} is the time-ordering operator. Expanding the exponential yields the ~~[[Physics:~~Dyson series~~\|Dyson series]]~~:		where {{mvar\|T}} is the time-ordering operator. Expanding the exponential yields the Dyson series:
	<math display="block">\|\psi(t)\rangle=\left[1-\frac{i}{\hbar}\int_{t_0}^t dt_1H(t_1)-\frac{1}{\hbar^2}\int_{t_0}^t dt_1\int_{t_0}^{t_1} dt_2H(t_1)H(t_2)+\ldots\right]\|\psi(t_0)\rangle.</math>		<math display="block">\|\psi(t)\rangle=\left[1-\frac{i}{\hbar}\int_{t_0}^t dt_1H(t_1)-\frac{1}{\hbar^2}\int_{t_0}^t dt_1\int_{t_0}^{t_1} dt_2H(t_1)H(t_2)+\ldots\right]\|\psi(t_0)\rangle.</math>

	In the ~~[[Physics:Interaction picture\|~~interaction picture]], the Schrödinger equation simplifies, and the Dyson series provides a systematic perturbative expansion in powers of the interaction.		In the interaction picture, the Schrödinger equation simplifies, and the Dyson series provides a systematic perturbative expansion in powers of the interaction.

	To first order, the transition amplitude between states <math>\|\alpha\rangle</math> and <math>\|\beta\rangle</math> is		To first order, the transition amplitude between states <math>\|\alpha\rangle</math> and <math>\|\beta\rangle</math> is
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	== Applications ==		== Applications ==
	Time-dependent perturbation theory is widely used in:		Time-dependent perturbation theory is widely used in:
	* ~~[[Physics:~~Fermi's golden rule~~\|Fermi's golden rule]]~~		* Fermi's golden rule
	* ~~[[Physics:Laser\|~~laser]] physics		* laser physics
	* ~~[[Physics:Spectral line\|~~spectral line]] broadening		* spectral line broadening
	* ~~[[Physics:Particle physics\|~~particle physics]] and ~~[[Physics:Nuclear physics\|~~nuclear physics]]		* particle physics and nuclear physics
	* ~~[[Physics:Permittivity\|~~dielectric response]] of materials		* dielectric response of materials

	=See also=		=See also=

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Quantum ~~time~~-dependent perturbation theory\|1}}		{{Sourceattribution\|Physics:Quantum Time-dependent perturbation theory\|1}}

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			{{Short description\|Quantum Collection topic on Quantum Time-dependent perturbation theory}}

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

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	'''Time-dependent perturbation theory''' is a method in [[Physics:Quantum mechanics\|quantum mechanics]] used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the [[Physics:Hamiltonian (quantum mechanics)\|Hamiltonian]] depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems.		'''Time-dependent perturbation theory''' is a method in [[Physics:Quantum mechanics\|quantum mechanics]] used to study systems subject to a time-dependent disturbance. It extends perturbation theory to cases where the [[Physics:Hamiltonian (quantum mechanics)\|Hamiltonian]] depends explicitly on time, allowing the calculation of transition probabilities, expectation values, and dynamical evolution of quantum systems.
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			[[File:Sinc-squared spectral lineshape perturbation theory.png\|thumb\|280px\|Quantum Time-dependent perturbation theory.]]
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	[[File:Sinc-squared spectral lineshape perturbation theory.png\|thumb\|400px\|Sinc-squared lineshape arising in time-dependent perturbation theory: the transition probability as a function of frequency shows a central peak at the Bohr frequency with width inversely proportional to the interaction time, and decaying side lobes characteristic of finite-duration signals.]]		</div>

	== Introduction ==		== Introduction ==

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	{{Quantum article nav\|previous=Physics:Quantum Perturbation theory\|previous label=Perturbation theory\|next=Physics:Quantum Adiabatic theorem\|next label=Adiabatic theorem}}		{{Quantum article nav\|previous=Physics:Quantum Perturbation theory\|previous label=Perturbation theory\|next=Physics:Quantum Adiabatic theorem\|next label=Adiabatic theorem}}