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.

Introduction

Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson,^[1] studies the effect of a time-dependent perturbation Template:Math applied to a time-independent Hamiltonian Template:Math.^[2]

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.^[1]

Since the Hamiltonian is time-dependent, both the energy levels and eigenstates evolve in time. The main quantities of interest are:

The time-dependent expectation value of an observable.
The time-dependent probability amplitudes of energy eigenstates.

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

Consider an unperturbed system with eigenstates $| n ⟩$ satisfying $H_{0} | n ⟩ = E_{n} | n ⟩ .$

If the system is initially in state $| j ⟩$ , its time evolution without perturbation is $| j (t) ⟩ = e^{- i E_{j} t / ℏ} | j ⟩ .$

Now introduce a time-dependent perturbation: $H = H_{0} + V (t) .$

The quantum state can be expanded as $| ψ (t) ⟩ = \sum_{n} c_{n} (t) e^{- i E_{n} t / ℏ} | n ⟩ .$

The coefficients Template:Math represent probability amplitudes, with $| c_{n} (t) |^{2} = | ⟨ n | ψ (t) ⟩ |^{2} .$

Substituting into the Schrödinger equation yields $\frac{d c_{n}}{d t} = \frac{- i}{ℏ} \sum_{k} ⟨ n | V (t) | k ⟩ c_{k} (t) e^{- i (E_{k} - E_{n}) t / ℏ} .$

This exact system of coupled differential equations describes how amplitudes evolve in time.

For weak perturbations, an iterative (perturbative) solution is used: $c_{n} (t) = c_{n}^{(0)} + c_{n}^{(1)} + c_{n}^{(2)} + \dots$

The first-order term is $c_{n}^{(1)} (t) = \frac{- i}{ℏ} \int_{0}^{t} d t^{'} ⟨ n | V (t^{'}) | k ⟩ e^{- i (E_{k} - E_{n}) t^{'} / ℏ} .$

This framework leads to important results such as:

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 $| ψ (t) ⟩ = T \exp [- \frac{i}{ℏ} \int_{t_{0}}^{t} d t^{'} H (t^{'})] | ψ (t_{0}) ⟩,$

where Template:Mvar is the time-ordering operator. Expanding the exponential yields the Dyson series: $| ψ (t) ⟩ = [1 - \frac{i}{ℏ} \int_{t_{0}}^{t} d t_{1} H (t_{1}) - \frac{1}{ℏ^{2}} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} H (t_{1}) H (t_{2}) + \dots] | ψ (t_{0}) ⟩ .$

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 $| α ⟩$ and $| β ⟩$ is $A_{α β} = - \frac{i λ}{ℏ} \int_{t_{0}}^{t} d t_{1} ⟨ β | V (t_{1}) | α ⟩ e^{- i (E_{α} - E_{β}) (t_{1} - t_{0}) / ℏ} .$

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.

Physics:Quantum Time-dependent perturbation theory

Introduction

Method of variation of constants

Method of Dyson series

Physical interpretation

Applications

See also

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