A transition rate emerges in a restricted continuum window

Switch on a weak time-independent perturbation at time zero and start in one unperturbed discrete state. The first-order probability for a specified final state is a sinc-squared function of its energy difference, multiplied by duration squared. The central peak has energy width of order ħ/T, not an infinitely sharp conservation law at finite T. The pulse onset and observation time supply this finite width. Its height grows as T² while its integrated area over final energy grows only as T.

A normalized version of this peak approaches a delta distribution as T increases. That statement means its integral against sufficiently smooth functions approaches their value at the resonant energy. It does not mean an individual finite-time probability becomes a literal delta function with an ordinary finite height. The replacement is useful only when the density of final states and squared matrix element vary slowly across the peak. A nearby continuum threshold, narrow resonance, or sharp selection feature can invalidate that simplification.

Replace the sum over closely spaced final states by an energy integral weighted by their density ρ(E). If both the density and matrix elements are smooth across the finite-time peak, evaluate their product at the initial energy and integrate the peak. The total leading probability becomes ΓT, with Γ equal to 2π over ħ times the squared coupling times the final-state density. For a static perturbation the resonant unperturbed final energy equals the initial one. A harmonic drive instead shifts that energy by its supplied or removed quantum.

The matrix-element convention must match the density convention. In a large normalization box, a final extended state often contributes a coupling proportional to inverse square root of volume, while the number of final states grows with volume. Their factors cancel in the physical rate. Counting the volume in the density but forgetting it in the matrix element yields a spurious size dependence. For a real cosine drive, use the relevant positive- or negative-frequency Fourier-component matrix element, rather than silently inserting the full peak amplitude.

Let Λ be an energy scale over which the product of density and coupling changes appreciably. The continuum peak should be narrow relative to Λ, requiring T much longer than ħ/Λ. For a finite dense spectrum, it should still cover many levels, requiring T much shorter than ħ divided by their typical spacing. Finally ΓT must remain small for the first-order calculation. These can be compatible when the coupling is weak; sending T to infinity at fixed coupling is not justified by the same derivation.

At sufficiently short times a finite-bandwidth system starts with quadratic transition probability, and at very long times depletion, recurrences, or more detailed spectral features matter. Exponential survival can follow from additional controlled resummation or reservoir assumptions, not merely by extending ΓT beyond one. The probability lab illustrates independent preparations: Reset and set both controls to p=0.05 and n=10, predicting mean count 0.5; Reset and set p=0.05 and n=20, predicting mean one. It computes exact binomial counts, not a continuum spectrum, transition trajectory, or golden-rule rate.