First-Order Transition Amplitudes and Probabilities

The sinc² lineshape is the workhorse of every spectroscopy: the finite-time width ΔE ~ h/t is the Fourier-transform limit that sets the ultimate resolution of a pulsed measurement, from femtosecond laser spectroscopy to Ramsey interferometry in atomic clocks, where longer interrogation times t buy narrower lines. The same first-order amplitude underlies the calculation of photoabsorption cross-sections, radiative decay rates, and — folded against a density of states — the scattering and reaction rates of particle and nuclear physics. The lineshape's collapse to a delta function is the formal justification for treating fast processes with constant rates in kinetics and laser physics, and its breakdown at long times or strong coupling is exactly where the coherent, oscillatory physics of qubits and driven two-level systems takes over.

Picture the system as a state vector pinned almost entirely on the initial eigenstate |i⟩, with a thin thread of amplitude leaking toward every other eigenstate |f⟩. The perturbation is the only thing that can move amplitude between the fixed energy shelves of H₀. To leading order — the "first order" of the title — we ignore the possibility that amplitude, once it reaches f, flows onward; we let the perturbation act exactly once. The amplitude that arrives at f is then the running sum of the perturbation's pushes, each carrying the interaction-picture phase e^{iω_fi t′} that the previous article isolated. The magic is in that phase. If the driving is steady (a constant V), the pushes at successive instants carry phases that rotate at the Bohr frequency ω_fi. When ω_fi is large, consecutive pushes point in different directions and cancel — little amplitude accumulates. When ω_fi → 0, the pushes align and the amplitude piles up. So the transition strongly favours final states at nearly the same energy as the initial one: energy conservation is not imposed, it emerges from the interference of the phases.

The height grows as t² and the width shrinks as 1/t. Multiply those together and the area under the peak grows only as t — a fixed rate of probability accumulation. That single observation, drawn on the plot above as the amber curve climbing while its shoulders pull in, is the whole reason transitions have well-defined rates. The width in energy, ΔE ~ 2πħ/t, is the time-energy uncertainty relation reappearing here as the finite-time fuzziness of energy conservation: to sharply conserve energy you must wait a long time, and Energy-Time Uncertainty makes exactly this trade quantitative. A constant perturbation is not the only case. If the perturbation oscillates at frequency ω, the same interference argument shifts the peak away from ω_fi = 0 to ω_fi = ±ω — the system can climb an energy ladder rung ħω by absorbing, or descend by stimulated emission. That shifted resonance is the subject of Harmonic Perturbations and Resonance; here we own the constant-V template that it specialises.

The starting point is the interaction-picture coefficient equation established in The Interaction Picture and Time-Dependent Hamiltonians: writing the state as |ψ_I(t)⟩ = Σ_n c_n(t)|n⟩, the coefficients obey ċ_m = −(i/ħ) Σ_n V_mn(t) e^{iω_mn t} c_n, where V_mn = ⟨m|V̂|n⟩ and ω_mn = (E_m − E_n)/ħ. This is exact. First-order perturbation theory, developed in Time-Dependent Perturbation Theory: General Formalism, keeps the leading solution: c_i ≈ 1, all other c_n ≈ 0, so a single final state f receives

where: c_f^{(1)} is the first-order amplitude for the transition i→f; V_{fi} = ⟨f|V̂|i⟩ is the matrix element of the perturbation between the two unperturbed eigenstates; ω_{fi} = (E_f − E_i)/ħ is the Bohr frequency of the pair. The physical reading is that the amplitude at f is the coherent sum of the perturbation's action at every earlier instant, each contribution stamped with the rotating phase e^{iω_{fi}t'}. Squaring gives the transition probability P_{i→f} = |c_f^{(1)}|². For the important case in which V_{fi} does not itself change while the perturbation is on (a constant coupling switched on at t=0), the integral is elementary and yields the lineshape below.

where: the prefactor |V_{fi}|²/ħ² sets the overall strength; the sinc² factor sin²(ω_{fi}t/2)/(ω_{fi}/2)² carries all the time and energy dependence. Physically: the probability is largest when ω_{fi} → 0 (final energy equal to initial), where the factor approaches t²; it oscillates and decays for ω_{fi} ≠ 0.

The next card is the article's owned structural result: the exact-area identity that makes the lineshape a delta function, and hence the transition into a continuum a constant rate. This is the piece Fermi's Golden Rule consumes by name.

Example 1 — the sinc² lineshape for a concrete two-level pair. A perturbation with constant matrix element V_{fi} = 0.10 eV couples two states split by ΔE = E_f − E_i = 2.0 eV. Using ħ = 6.582 × 10⁻¹⁶ eV·s:

Step 1. The Bohr frequency: ω_{fi} = ΔE/ħ = 2.0 / (6.582 × 10⁻¹⁶) = 3.04 × 10¹⁵ rad/s.

Step 2. The probability at t = 1 fs. The dimensionless form is convenient: (|V_{fi}|²/ħ²)/(ω_{fi}/2)² = (2V_{fi}/ħω_{fi})² = (2V_{fi}/ΔE)² = (2·0.10/2.0)² = 0.010. Then P^{(1)}(1 fs) = 0.010 × sin²(ω_{fi}·10⁻¹⁵/2) = 0.010 × sin²(1.519) = 0.010 × 0.997 = 9.97 × 10⁻³.

Step 3. The first zero. The lineshape vanishes when ω_{fi}t/2 = π, i.e. t = 2π/ω_{fi} = 6.283/(3.04 × 10¹⁵) = 2.07 fs. At any multiple of this time the far-off-resonant transition amplitude returns to zero — the probability into this single non-resonant state simply oscillates and never accumulates.

Reading: even with a coupling as large as 0.10 eV, an off-resonant transition (ΔE = 2 eV ≫ V_{fi}) transfers only ~1% of the population, and that 1% sloshes back and forth. Real net transitions require either resonance (ω_{fi} → 0) or a continuum of final states.

Example 2 — the area is fixed while the peak climbs. Take V = ħ = 1 (natural units) so the lineshape is sin²(ω t/2)/(ω/2)².

Step 1. Central peak at three times: P(0, t) = t², so P(0, 1) = 1, P(0, 2) = 4, P(0, 4) = 16 — the peak climbs as t².

Step 2. Central width at the same times: first zero at ω = 2π/t, so widths 6.28, 3.14, 1.57 — the peak narrows as 1/t.

Step 3. The area, height × width up to a shape constant: ∫ sin²(ωt/2)/(ω/2)² dω = 2πt = 6.28, 12.57, 25.13. The area grows only linearly — height × width = t² × (1/t) = t. This is why probability into a continuum accumulates at a constant rate while probability into one state merely oscillates.

Reading: the plot's three curves make the point visually — the amber (t = 4) curve is sixteen times taller than the green (t = 1) but only a quarter as wide, and the balance is a linear-in-t area, the mathematical origin of every transition rate in quantum mechanics.

For the exact statements and derivations behind this article, the standard references are Griffiths (Introduction to Quantum Mechanics, 3rd ed., Ch. 11), Sakurai & Napolitano (Modern Quantum Mechanics, 2nd ed., §5.6–5.7), Shankar (Ch. 18), Cohen-Tannoudji (Vol. II, Ch. XIII), and Townsend (Ch. 14). The first-order amplitude here is set up by The Interaction Picture and Time-Dependent Hamiltonians and the perturbation expansion by Time-Dependent Perturbation Theory: General Formalism; the sinc²-to-delta limit is consumed by Fermi's Golden Rule together with the Density of Final States, the peak-shift to ±ω by Harmonic Perturbations and Resonance, the switched-on detail by Constant Perturbation Switched On, and the exact non-perturbative treatment by the two-level Rabi article. Provenance note (verbatim from the source export): Equations derived from the interaction-picture Schrödinger equation; verifier should reproduce the first-order amplitude (Griffiths 3e Eq. 11.17 / Sakurai 2e §5.6), the constant-perturbation sinc² (Griffiths Eq. 11.34), the harmonic two-term result (Griffiths Eq. 11.35), and Fermi's Golden Rule (Sakurai §5.7) including the $\frac{4\sin^2(\omega t/2)}{\omega^2}\to 2\pi t\,\delta(\omega)$ limit. Numerical constants ($2\pi/\hbar$, factors of 4) are standard textbook values; no nonstandard numbers used.