What is Fermi's golden rule used for?

It gives the constant rate of transitions from an initial state into a continuum of final states, and is used to compute spontaneous emission rates, absorption rates, decay lifetimes, and scattering rates whenever the final states form a dense band.

Why does resonance occur in time-dependent perturbation theory?

A harmonic perturbation contributes an oscillating phase that cancels over time unless its frequency matches the energy difference between initial and final states; at that resonance the contributions add coherently and the transition probability becomes large.

Time-Dependent Perturbation Theory

Time-dependent perturbation theory computes the probability that a quantum system makes a transition between states when acted on by a time-varying influence, and in the long-time limit it yields Fermi's golden rule for steady transition rates.

Definition

Time-dependent perturbation theory is the method of computing transition amplitudes and probabilities between unperturbed states under a time-dependent perturbation by expanding the evolution in powers of the perturbation, most often to first order.

Scope

The topic covers the interaction picture and the expansion of transition amplitudes in powers of a time-dependent perturbation, first-order transition probabilities, the response to harmonic and sudden perturbations, resonance when the driving frequency matches an energy gap, and Fermi's golden rule giving the transition rate into a continuum of final states.

Core questions

How is the probability of a transition between states computed under a time-varying perturbation?
Why does a harmonic perturbation drive transitions most strongly on resonance?
What is Fermi's golden rule and when does it apply?
How does the density of final states enter the transition rate?

Key concepts

interaction picture
transition amplitude
transition probability
resonance
Fermi's golden rule
density of final states

Key theories

First-order transition amplitude: In the interaction picture the leading transition amplitude is the time integral of the perturbation's matrix element times an oscillating phase, so a harmonic perturbation produces a large amplitude only when its frequency matches the energy gap between initial and final states.
Fermi's golden rule: For transitions into a dense set of final states the probability grows linearly in time, giving a constant rate proportional to the squared matrix element times the density of final states at the resonant energy, the standard formula for decay and absorption rates.

Clinical relevance

Time-dependent perturbation theory is the engine behind spectroscopy and decay: it gives the rates of absorption and emission of light by atoms, selection rules for transitions, the lifetimes of excited states, and scattering and decay rates throughout atomic, molecular, nuclear, and particle physics.

History

Dirac formulated time-dependent perturbation theory in 1927 and applied it to the emission and absorption of radiation, deriving Einstein's coefficients; Fermi's lectures made the transition-rate formula so widely used that it became known as the golden rule.

Key figures

Paul Dirac
Enrico Fermi
Albert Einstein

Seminal works

sakurai2017
cohentannoudji2019

Frequently asked questions

What is Fermi's golden rule used for?: It gives the constant rate of transitions from an initial state into a continuum of final states, and is used to compute spontaneous emission rates, absorption rates, decay lifetimes, and scattering rates whenever the final states form a dense band.
Why does resonance occur in time-dependent perturbation theory?: A harmonic perturbation contributes an oscillating phase that cancels over time unless its frequency matches the energy difference between initial and final states; at that resonance the contributions add coherently and the transition probability becomes large.