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.
What a learner can do afterwards
- 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
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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)=<f|V(t)|i>. 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.
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.
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.
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.
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