A quantum system has a very large matrix element |⟨f|H'|i⟩|² coupling initial and final states, yet it transitions extremely slowly. What is the most likely explanation?
AThe perturbation H' is too weak to drive transitions despite the large matrix element
BThe density of final states ρ(E_f) is very small — few states are available at the transition energy
CThe elapsed time is too short for the long-time limit of Fermi's Golden Rule to apply
DEnergy is not conserved in this transition, so the delta function suppresses the rate to zero
The Fermi Golden Rule rate Γ = (2π/ℏ)|⟨f|H'|i⟩|²ρ(E_f) has two independent factors: the matrix element measures coupling strength, and ρ(E_f) measures how many final states are energetically accessible. If ρ(E_f) is very small — as in a bandgap, a confined geometry, or a transition to a very narrow energy range — the rate is suppressed regardless of how strongly the perturbation couples the states. This is why spontaneous emission can be dramatically slowed by placing an atom in a photonic crystal that suppresses the photon density of states at the transition frequency.
Question 2 Multiple Choice
The delta function δ(E_f − E_i) that appears in Fermi's Golden Rule enforces:
AMomentum conservation — only final states with the same momentum as the initial state contribute
BEnergy conservation — only final states at exactly the initial energy are accessible
CNormalization of the final-state wavefunction to unity
DThe long-wavelength approximation used to simplify the matrix element
The delta function arises from the long-time behavior of the sinc-squared factor in first-order perturbation theory: as t → ∞, [sin(Δωt/2)/(Δω/2)]² → 2πt δ(E_f − E_i). This picks out only transitions where E_f = E_i — exact energy conservation. The perturbation mediates a transition between states at the same energy; it does not supply energy to the system. Momentum conservation, if required, must come separately from the structure of the matrix element.
Question 3 True / False
For transitions to a continuum of final states, Fermi's Golden Rule predicts a transition probability that grows linearly in time, corresponding to a constant transition rate.
TTrue
FFalse
Answer: True
When final states form a continuum, summing over them converts the oscillating sinc-squared factor into 2πt δ(E_f − E_i), making total transition probability proportional to t. Dividing by t gives a constant rate Γ independent of time — the system decays at a steady rate, as observed in radioactive decay, spontaneous emission, and scattering. This is in contrast to transitions between two discrete levels, where probability oscillates (Rabi oscillations).
Question 4 True / False
Fermi's Golden Rule applies equally well to transitions between two isolated discrete energy levels and to transitions into a continuum of final states.
TTrue
FFalse
Answer: False
Fermi's Golden Rule specifically requires a continuum of final states. For two discrete levels, transition probability oscillates periodically (Rabi oscillations) rather than growing linearly in time — there is no constant rate. The Golden Rule emerges only when final states are dense enough that the sinc-squared factor can be approximated as a delta function, which requires many closely spaced final states near the transition energy.
Question 5 Short Answer
Explain why a quantum system transitioning to a single discrete final state shows oscillatory probability over time, while a system transitioning to a continuum shows a constant transition rate.
Think about your answer, then reveal below.
Model answer: For a transition to a single discrete level, P(t) ∝ sin²(Δωt/2) oscillates as the system coherently tunnels back and forth between initial and final states — there is no irreversible decay. For a continuum, each final state individually oscillates with its own Δω. When you sum over all final states, the oscillations from states with different energies cancel by destructive interference, except near Δω = 0 (energy conservation). The result is incoherent, irreversible growth linear in t — a true constant rate.
The physics is analogous to why a coherently driven two-level system Rabi-flops while a system coupled to a large reservoir decays irreversibly. The continuum provides the reservoir: phase information leaks into the many final states and cannot be recovered, converting coherent oscillation into incoherent decay.