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

'''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.

[[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.]]

== 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>

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:
* The time-dependent [[Physics:Expectation value (quantum mechanics)|expectation value]] of an observable.
* 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.

== Method of variation of constants ==
Consider an unperturbed system with eigenstates <math>|n\rangle</math> satisfying
<math display="block">H_0 |n\rangle = E_n |n\rangle.</math>

If the system is initially in state <math>|j\rangle</math>, its time evolution without perturbation is
<math display="block"> |j(t)\rangle = e^{-iE_j t /\hbar} |j\rangle.</math>

Now introduce a time-dependent perturbation:
<math display="block"> H = H_0 + V(t).</math>

The quantum state can be expanded as
<math display="block"> |\psi(t)\rangle = \sum_n c_n(t) e^{- i E_n t / \hbar} |n\rangle.</math>

The coefficients {{math|''c''<sub>''n''</sub>(''t'')}} represent probability amplitudes, with
<math display="block"> |c_n(t)|^2 = |\langle n|\psi(t)\rangle|^2.</math>

Substituting into the [[Physics:Schrödinger equation|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>

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

For weak perturbations, an iterative (perturbative) solution is used:
<math display="block">c_n(t) = c_n^{(0)} + c_n^{(1)} + c_n^{(2)} + \cdots</math>

The first-order term is
<math display="block">c_n^{(1)}(t) = \frac{-i}{\hbar} \int_0^t dt' \;\langle n|V(t')|k\rangle \, e^{-i(E_k - E_n)t'/\hbar}.</math>

This framework leads to important results such as:
* [[Physics:Fermi's golden rule|Fermi's golden rule]]
* Transition probabilities between quantum states
* Time-dependent expectation values

== Method of Dyson series ==
The time evolution operator can be written formally as
<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]]:
<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.

To first order, the transition amplitude between states <math>|\alpha\rangle</math> and <math>|\beta\rangle</math> is
<math display="block">A_{\alpha\beta}=-\frac{i\lambda}{\hbar}\int_{t_0}^t dt_1\langle\beta|V(t_1)|\alpha\rangle e^{-i(E_\alpha-E_\beta)(t_1-t_0)/\hbar}.</math>

This leads directly to transition probabilities and scattering rates.

== Physical interpretation ==
Time-dependent perturbation theory describes how external influences cause transitions between quantum states. The perturbation couples different eigenstates, and the time dependence determines how probability amplitudes evolve.

Oscillatory behavior arises due to phase factors, and resonance effects occur when the perturbation frequency matches energy differences between states. These mechanisms underlie many physical phenomena, including absorption, emission, and scattering processes.

== Applications ==
Time-dependent perturbation theory is widely used in:
* [[Physics:Fermi's golden rule|Fermi's golden rule]]
* [[Physics:Laser|laser]] physics
* [[Physics:Spectral line|spectral line]] broadening
* [[Physics:Particle physics|particle physics]] and [[Physics:Nuclear physics|nuclear physics]]
* [[Physics:Permittivity|dielectric response]] of materials

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum time-dependent perturbation theory|1}}

Physics:Quantum Time-dependent perturbation theory - Revision history

imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template

imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template