--- title: "Time-Dependent Perturbation Theory and Fermi's Golden Rule" description: "Time-dependent perturbation theory computes transition amplitudes from weak couplings. In a suitable continuum and time regime, Fermi's golden rule gives a rate proportional to the squared matrix elem" canonical: https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC source: https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Time-Dependent Perturbation Theory and Fermi's Golden Rule Time-dependent perturbation theory computes transition amplitudes from weak couplings. In a suitable continuum and time regime, Fermi's golden rule gives a rate proportional to the squared matrix element and the density of energy-conserving final states. Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24 Page: https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC ## Ready when they can - Computes a first-order transition amplitude for a perturbation applied for a finite time - States the golden rule and identifies the matrix element and the density of states in a given problem - Explains why the rate depends on how many final states are available ## Lesson: How fast one state becomes another Under H0, state n gains phase exp(-iE_n t/hbar). The interaction picture removes these known phases. Start in i at t=0. For f different from i in an orthonormal H0 eigenbasis, a_f^(1)(T)=-(i/hbar) integral_0^T V_fi(t) exp[i(E_f-E_i)t/hbar] dt, where V_fi(t)=. The exponential is the states' relative-energy phase. Integrate this phased coupling over 0 to T. The amplitude can vanish; switching on V need not cause a transition. For weak coupling to a dense continuum, Gamma=(2pi/hbar)|M|²rho(E_res). M is the coupling matrix element; rho is final-state density per unit energy at the energy-conserving value. Choose T so hbar/T is small compared with the energy scale over which |M|²rho varies, but large compared with final-state level spacing. Initial-state depletion must stay small. This continuum rate is not an exact rule for an isolated pair. **Example.** Let V_fi=g be constant from 0 to T and omega=(E_f-E_i)/hbar. For omega not zero, the integral gives a=g(1-exp[i omega T])/(hbar omega). At resonance, omega=0, use a=-igT/hbar. With g=0.01, T=2 and hbar=1 in consistent units, a=-0.02i and |a|²=0.0004. This leading probability is useful while it remains small. Density is not the total number of states at all energies. Estimate rho near E_res by dividing the number of states in a narrow energy interval by that interval's width. For 50 states in 0.25 eV, rho is about 200 states/eV. This estimate assumes the density varies little across that interval. *(drawing: For Gamma_A>0, cancel common factors: Gamma_B/Gamma_A=rho_B/rho_A. Doubling |M| at fixed density makes the rate four times larger.)* The golden-rule contribution is zero if M=0 or rho(E_res)=0. Counting states at other energies cannot make this contribution nonzero. A zero first-order coupling does not rule out every higher-order process. At finite time, the amplitude has an energy width, so the sharp energy condition belongs to the continuum-rate approximation. For spontaneous atomic emission, the final state includes the lower-energy atom and an emitted photon. The atom-field interaction supplies M, and photon modes supply a continuum of final states. Neglecting recoil, photon energy is E_i-E_f. An atom dropping from 6 eV to 2 eV emits a 4 eV photon. Count only combined final states that conserve total energy. **Recap.** Integrate the phased coupling for a finite-time amplitude. In the weak-coupling continuum regime, Gamma=(2pi/hbar)|M|²rho(E_res), with only energy-conserving final states counted. *Original lesson, questions and flashcards written for this dataset about finite-time transition amplitudes and Fermi's golden rule.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Degenerate Perturbation Theory and the Variational Method](https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy) ## Optional connections - [Fourier Analysis and the Wave Packet](https://lightmysky.com/learn/science/fourier-analysis-and-the-wave-packet-mt_1ZaojWUZEw) ## Opens up - [Scattering: Cross-Sections and the Born Approximation](https://lightmysky.com/learn/science/scattering-cross-sections-and-the-born-approximation-mt_xmN9KDfSp7)