A weak drive accumulates a complex transition amplitude

Let H₀ be time independent and exactly understood, with a perturbation V(t). Define the interaction-picture state by multiplying the Schrödinger-picture state by exp(+iH₀t/ħ). Differentiating shows that it obeys a Schrödinger equation driven by V_I(t)=exp(+iH₀t/ħ)V(t)exp(−iH₀t/ħ). This is an exact change of representation; no approximation has yet been made. The initial state and measurement operators must be transformed consistently when comparing physical predictions.

The exact interaction-picture propagator is a time-ordered exponential. Expanding its integral equation gives a first integral, then nested integrals with later operators on the left. Operators at different times need not commute, so replacing the propagator by an ordinary exponential of the integral can erase physical effects. Truncating the expansion is the actual perturbative step. It is justified only when the relevant accumulated transition amplitudes and neglected corrections remain small over the specified protocol.

Prepare an unperturbed eigenstate i at time zero. At first order, the coefficient of a different eigenstate f is minus i over ħ times the time integral of the perturbation matrix element, multiplied by exp(iω_fi t), where ω_fi=(E_f−E_i)/ħ. This positive phase follows from the chosen interaction-picture definition. The integral preserves phase information: probabilities from different creation times cannot generally be added before their amplitudes are combined.

For a pulse whose relevant matrix element is v exp(−iωt), constant in magnitude between zero and T, define detuning δ=ω_fi−ω. The amplitude is proportional to T times sinc(δT/2), with a midpoint phase exp(iδT/2). At resonance the contributions align and amplitude grows linearly with duration, so the leading probability grows quadratically. Off resonance their phase winds and can suppress the result. A real cosine drive supplies Fourier components of half its peak matrix element; that amplitude convention must be included when assigning v.

The leading transition probability is the squared first-order amplitude when the initial population remains close to one. A result larger than one is not a prediction of impossible quantum behavior; it means the truncation has failed. On resonance an isolated driven two-level system instead develops coherent oscillations when the full evolution is retained. Detuning zeros in the first-order integral likewise establish cancellation at that order, not an automatic all-orders prohibition on transitions.

The waves lab isolates interference between two contributions. Reset and set both controls to phase zero degrees and ratio one, predicting a sum amplitude of two. Reset again and set phase 180 degrees and ratio one, predicting cancellation. These two same-frequency waves can represent two phase contributions to the integral only as an analogy. They are not initial and final energy levels, and the instrument has no pulse duration, detuning, matrix element, or time-ordered quantum propagator. Its playback changes the shared wave phase rather than applying a physical drive.