Time-Dependent Perturbation Theory: General Formalism
Time-dependent perturbation theory is the systematic machinery for computing how a quantum system moves between the energy levels of a solvable Hamiltonian when a weak, time-varying influence is switched on. Where static perturbation theory asks how a fixed extra term shifts the energy levels, time-dependent theory asks a different question: given a system parked in one eigenstate, what is the probability that the perturbation has driven it into another by time t? The answer is organised as a power series in the strength of the perturbation, and this article owns that expansion — the order-by-order iteration that generates first-order, second-order, and all higher amplitudes. The setup is inherited from the interaction picture: the exact state, expanded in the fixed unperturbed basis, has coefficients obeying a coupled set of first-order equations driven entirely by the perturbation. We solve them iteratively. Zeroth order is free evolution (the system stays put); first order lets the perturbation act once, producing a single amplitude i→f; second order lets it act twice, i→m→f through an intermediate state, in a time-ordered double integral. We derive the recursion that generates the n-th order from the (n−1)-th, and identify that nested double integral as the seed of the Dyson series, resummed by name in a companion article. The one structural result proved here goes beyond cataloguing the orders. First-order theory is an approximation, and its domain of trust is not obvious. We prove — symbolically and numerically — that the first-order transition probability is exactly the weak-coupling, short-time tangent of the exact two-level solution: expanding the exact Rabi probability in the coupling reproduces the first-order sinc² term identically, and the two agree until the transferred population reaches about a quarter, where first order errs by 10%. That matched asymptotic is the honest statement of when you may stop at first order — and when the resonant, non-perturbative Rabi treatment must take over. We do not evaluate the first-order probability for specific perturbations here (that is the next article), nor resum the series (the Dyson article), nor solve the exact two-level problem (the Rabi article). We own the framework: the expansion, its structure, and the boundary of its validity.
- Split H = H₀ + H′(t) with H₀ solvable; expand the interaction-picture state in the H₀ basis; the coefficients obey ċ_m = −(i/ħ) Σ_n V_mn(t) e^{iω_mn t} c_n exactly.
- Solve iteratively: c_m^{(0)} = δ_mi; c_m^{(1)}(t) = −(i/ħ)∫₀ᵗ V_mi(t′) e^{iω_mi t′} dt′; c_f^{(2)} is a time-ordered double integral through an intermediate state m.
- First order = one action of the perturbation (i→f); second order = two actions through a virtual intermediate (i→m→f). Higher orders follow by the same recursion — the seed of the Dyson series.
- Time-dependent PT gives transition RATES and probabilities between levels; static PT gives energy SHIFTS of levels. They answer complementary questions and there are no energy denominators in the time-domain form.
- First order is valid only while Σ_{f≠i}|c_f^{(1)}|² ≪ 1 — the initial state stays mostly populated; it fails at resonance for long times.
- Structural result: first-order theory is the weak-coupling tangent of the exact two-level solution — the exact Rabi probability, expanded in the coupling, reproduces the first-order term exactly, and first order errs by 10% once ~26% of the population has transferred.
- Key idea: iterate the exact interaction-picture coefficient equation in powers of the perturbation — one action per order — and the leading term is trustworthy precisely until an appreciable fraction of the population has moved.
Expand the exact interaction-picture coefficient equation ċ_m = −(i/ħ)Σ_n V_mn e^{iω_mn t} c_n order by order in the perturbation. Each order adds one action of V: first order is a single time integral (i→f), second order a time-ordered double integral through an intermediate state (i→m→f). The leading term is the whole story only while little population has moved — quantitatively, until the transferred fraction reaches about a quarter.
Where this lives today
Time-dependent perturbation theory is the computational backbone of light–matter interaction: absorption and emission spectra, photoionisation and photodetachment cross-sections, and radiative decay rates are all first- or second-order transition calculations. In condensed matter it underlies linear-response theory and the Kubo formulas for conductivity and susceptibility, where the "perturbation" is an applied field and the response is a first-order transition rate. In particle and nuclear physics the same expansion, resummed into the Dyson series and made covariant, becomes the Feynman-diagram perturbation theory of quantum field theory. Wherever the honest boundary is respected — small transferred population — the framework is astonishingly accurate; wherever it is violated, as in strong-field and resonant-drive physics, its breakdown is itself the signpost pointing to the non-perturbative Rabi and Floquet methods.
The physical picture
The picture is a system sitting on the ground floor of a building whose floors are the eigenstates of H₀. Left alone, it stays put and merely accumulates a phase. Switch on a weak time-dependent perturbation and it begins to tunnel between floors — but weakly, so at any moment the probability is still overwhelmingly on the starting floor, with thin threads of amplitude reaching the others. Perturbation theory counts how many times the perturbation has had to act to deliver a given thread. First order is the single-hop amplitude: the perturbation acts once, at some instant t′, carrying the system directly from i to f. Because it could have acted at any t′, the first-order amplitude is the coherent sum — an integral — over all those instants. Second order is the two-hop amplitude: the perturbation acts at t″, parking the system momentarily on an intermediate floor m, then acts again at a later t′ > t″ to complete the journey to f. The requirement t′ > t″ — you must arrive before you leave again — is time ordering, and the resulting nested double integral is the first appearance of the structure the Dyson series organises to all orders.
The contrast on this plot is the practical rule of thumb. When the dimensionless coupling V⁄ΔE is small, each order is smaller than the last by that factor, and the green bars have collapsed to nothing by second order — keeping only first order is not merely convenient, it is essentially exact. As the transition approaches resonance (ΔE → 0), the ratio V⁄ΔE grows toward and past one, the red bars stop shrinking, and no finite truncation converges. That is the analytic signature of the same breakdown the inset above showed graphically: perturbation theory is a statement about weak, off-resonant coupling, and resonance is where it must yield to the exact two-level solution. The inset's thesis in one frame. On resonance the exact two-level probability is sin²(Vt/ħ), a bounded oscillation that never exceeds one. First-order theory replaces it by the parabola (Vt/ħ)² — the tangent at the origin. The two curves hug each other while the transferred population is small and then peel apart; the marked point is where they differ by 10%, at a transferred fraction of about a quarter. Beyond it, first order predicts a probability climbing past one, which is nonsense, and the exact Rabi treatment must take over. This is the honest boundary of the whole first-order enterprise. It is worth being sharp about what time-dependent perturbation theory computes, because it is easy to confuse with static perturbation theory. Static theory takes a fixed extra term and asks how it shifts a stationary energy level — the answer carries energy denominators (E_i − E_m). Time-dependent theory takes a time-varying term and asks for the probability of a transition between unperturbed levels — the answer is a time integral of a phase, with no energy denominators at all. Shifts versus transitions; denominators versus time integrals. They are complementary tools, and Non-Degenerate Perturbation Theory owns the static, level-shifting side.
The core equations
The exact equation of motion for the expansion coefficients, derived in The Interaction Picture and Time-Dependent Hamiltonians, is the coupled first-order system
where: c_m(t) is the amplitude to find the system in unperturbed eigenstate |m⟩; V_mn = ⟨m|V̂|n⟩ is the perturbation matrix element; ω_mn = (E_m − E_n)/ħ. This is exact — no approximation yet. The perturbation is the only thing on the right-hand side, so if V = 0 the coefficients are frozen (free evolution in the interaction picture). To make progress we treat the perturbation as small, writing V → λV with a bookkeeping parameter λ, and expand each coefficient c_m = c_m^{(0)} + λ c_m^{(1)} + λ² c_m^{(2)} + ⋯. Substituting and matching powers of λ gives a hierarchy in which each order is driven by the one below it:
where: the N-th order coefficient is generated by feeding the (N−1)-th order into one more action of the perturbation. Physically: each order is one additional hop, and the recursion says "to reach order N, act once more on order N−1."
The goal is to convert the exact coefficient equation into an explicit hierarchy — c^{(0)}, c^{(1)}, c^{(2)} — so that "first order," "second order," and so on have precise meanings, and to expose the time-ordered double integral at second order that the Dyson series will resum. We start from a system prepared in |i⟩ at t = 0.
Step 1. Zeroth order. With λ = 0 the right-hand side vanishes, so c_m^{(0)} is constant. The initial condition c_m(0) = δ_mi gives c_m^{(0)}(t) = δ_mi for all t — the system sits in |i⟩. Why: with no perturbation the interaction-picture coefficients cannot change; only V moves amplitude between H₀ eigenstates.
Step 2. First order. Feed c_n^{(0)} = δ_ni into the recursion: ċ_m^{(1)} = −(i/ħ) V_mi(t) e^{iω_mi t}. Integrate from 0 to t: c_m^{(1)}(t) = −(i/ħ)∫₀ᵗ V_mi(t′) e^{iω_mi t′} dt′. This is a single action of the perturbation carrying the system directly from i to m. Why one integral: only the term n = i survives at this order, so the sum collapses to a single driving term.
Step 3. Second order. Feed the first-order result back in: ċ_f^{(2)} = −(i/ħ) Σ_m V_fm(t′) e^{iω_fm t′} c_m^{(1)}(t′). Substituting c_m^{(1)} and integrating gives the time-ordered double integral c_f^{(2)}(t) = (−i/ħ)² Σ_m ∫₀ᵗ dt′ ∫₀^{t′} dt″ V_fm(t′) e^{iω_fm t′} V_mi(t″) e^{iω_mi t″}. The inner limit t″ < t′ enforces that the first hop (i→m) precedes the second (m→f). Why the nested limits: causality of the iteration — the system must reach m before the perturbation can send it onward to f.
Step 4. The general recursion and the Dyson seed. The pattern continues: c^{(N)} is an N-fold time-ordered integral summed over N−1 intermediate states, each factor a V weighted by its interaction-picture phase, with the times strictly ordered t > t₁ > t₂ > ⋯. This ordered multiple integral is exactly the structure the time-ordering operator organises; resumming it into the compact 𝒯-exponential is owned by Dyson Series and the Time-Ordered Evolution Operator. Why stop here: the recursion defines every order; the resummation and the 1/n! simplex counting are the Dyson article's, consumed by name. *Verified: verify_aq-qm-time-dependent-perturbation-theory.py confirms c_f^{(1)} = −(V/ħ)(e^{iωt}−1)/ω for constant V, its squared magnitude equals the sinc² lineshape, and the second-order nested integral ∫₀ᵗ∫₀^{t′} f f = ½(∫f)² for a concrete f (the time-ordering seed).*
The second derivation is the article's owned structural result: the precise sense in which first-order theory is the tangent to the exact solution, and the number at which it fails.
Go deeper · Derivation B · 2 minFirst-order theory as the tangent to the exact two-level solution
The goal is to pin down when first order can be trusted, by comparing it to a case that can be solved exactly: two levels coupled by a constant perturbation. We prove that the first-order probability is the leading term of the exact result in the coupling strength, and compute the transferred population at which the approximation errs by 10%.
Step 1. The exact two-level problem. For a constant coupling V between states split by ħδ, the interaction-frame Hamiltonian is the time-independent 2×2 matrix H = [[0, V],[V, ħδ]]. Diagonalising it gives the exact transition probability P_exact(t), a bounded oscillation. On resonance (δ = 0) the eigenstates are (|i⟩ ± |f⟩)/√2 with energies ±V, so P_exact(t) = sin²(Vt/ħ). Why exactly solvable: a time-independent 2×2 Hamiltonian is diagonalised in closed form — no perturbation needed.
Step 2. The weak-coupling expansion. Expand the exact off-resonant probability in powers of V. To leading order the generalised frequency Ω = √((V/ħ)² + δ²) → |δ|, and P_exact → (V²/ħ²) sin²(δt/2)/(δ/2)². This is identical to the first-order sinc² lineshape with ω_fi = δ. So first order is the O(V²) term of the exact result. Why it must match: first order is by construction the leading power of V, so it has to coincide with the exact solution's leading power.
Step 3. Quantify the breakdown on resonance. On resonance the exact probability is sin²(φ) with φ = Vt/ħ, while first order gives the parabola φ² (the δ→0 limit of the sinc²). Their relative error (φ² − sin²φ)/sin²φ reaches 0.10 at φ ≈ 0.532. Why on resonance: this is the worst case, where the two curves diverge fastest and the approximation is stressed hardest.
Step 4. Read the transferred population. At φ = 0.532 the exact transferred probability is sin²(0.532) = 0.257. So first order is good to 10% until about a quarter of the population has moved — a clean, checkable boundary. Beyond it the parabola climbs past one (unphysical) and the exact Rabi oscillation, owned by the two-level article, must be used. Why a population number, not a time: 26% transfer is the model-independent statement; the corresponding time depends on V and δ. *Verified: verify_...py diagonalises H = [[0,V],[V,ħδ]] and confirms P_exact/P_first-order → 1 as V → 0 (ratios 1.0000 at V/ħδ = 0.01), and solves (φ² − sin²φ)/sin²φ = 0.10 to get φ = 0.532, transferred population 0.257.*
Limiting cases
| regime | behaviour | reading | ||
|---|---|---|---|---|
| V → 0 (weak coupling) | c^{(1)} dominates; c^{(2)}/c^{(1)} ~ V/ΔE | first order suffices; higher orders negligible | ||
| transferred fraction ≪ 1 | first order accurate | the trustworthy regime, Σ_{f≠i} | c_f^{(1)} | ² ≪ 1 |
| transferred fraction ≈ 26% | first order errs by 10% | the honest boundary; resonance stresses it hardest | ||
| exact resonance, long t | first-order P climbs past 1 (unphysical) | must switch to exact Rabi flopping | ||
| static (time-independent) limit | asks for level SHIFTS not transitions | use non-degenerate PT (energy denominators) |
Worked examples
Example 1 — first-order amplitude and probability for a two-level pair. A constant perturbation with matrix element V = 0.05 eV couples two states split by ΔE = 1.5 eV. Using ħ = 6.582 × 10⁻¹⁶ eV·s:
Step 1. Bohr frequency: ω_fi = ΔE/ħ = 1.5/(6.582 × 10⁻¹⁶) = 2.28 × 10¹⁵ rad/s.
Step 2. First-order amplitude magnitude: |c_f^{(1)}(t)| = (2V/ħω_fi)|sin(ω_fi t/2)| = (2V/ΔE)|sin(ω_fi t/2)|, with 2V/ΔE = 2(0.05)/1.5 = 0.0667.
Step 3. Probability at t = 0.5 fs: P^{(1)} = (2V/ΔE)² sin²(ω_fi t/2) = (0.0667)² sin²(0.570) = 4.44 × 10⁻³ × 0.291 = 1.29 × 10⁻³.
Step 4. The bound. Because ΔE ≫ V (far off resonance), the transferred probability can never exceed (2V/ΔE)² = 4.44 × 10⁻³ — it oscillates below half a percent forever. First order is superb here precisely because so little population moves.
Reading: the transition is real but tiny and non-accumulating; the smallness of 2V/ΔE both guarantees first order and caps the transfer. This is the generic off-resonant story.
Example 2 — where first order breaks: the 10% boundary on resonance. Now put the two states on resonance (ΔE = 0) with the same coupling, and use natural units V/ħ = 1, so the Rabi angle is φ = Vt/ħ.
Step 1. Exact resonant probability: P_exact = sin²φ. First-order estimate: P_first = φ².
Step 2. They agree at small φ: at φ = 0.2, P_exact = sin²(0.2) = 0.0395 and P_first = 0.04 — a 1.3% error.
Step 3. They diverge as population transfers: solving (φ² − sin²φ)/sin²φ = 0.10 gives φ = 0.532, where P_exact = sin²(0.532) = 0.257.
Step 4. Beyond it, first order is unusable: at φ = π/2 (a full π/2-pulse) the exact probability is 1 but first order predicts (π/2)² = 2.47 — nonsense. The exact two-level solution takes over.
Reading: first-order time-dependent perturbation theory is trustworthy to 10% until about a quarter of the population has transferred; past that, resonant physics is inherently non-perturbative and belongs to the Rabi article.
Physicist
Origins · 1 minTime-Dependent Perturbation Theory and how this was discovered
The order-by-order expansion of transition amplitudes in a time-dependent perturbation was developed by Paul Dirac in 1926–27, culminating in his 1927 paper on the quantum theory of the emission and absorption of radiation — the work that also produced the first-order transition rate later named for Fermi. Dirac's move was to expand the state in the fixed unperturbed basis and track the coefficients, exactly the framework used here. The perturbative idea itself descends from Schrödinger's 1926 treatment of stationary perturbations; the distinctively time-dependent version — transitions and rates rather than level shifts — is Dirac's, and it became the standard language of radiative processes, scattering, and decay throughout twentieth-century physics.
Going further
The canonical treatments are Griffiths (Introduction to Quantum Mechanics, 3rd ed., Ch. 11), Sakurai & Napolitano (Modern Quantum Mechanics, 2nd ed., Ch. 5), Shankar (Ch. 18), Cohen-Tannoudji (Vol. II, Ch. XIII), and Townsend (Ch. 14). The coefficient equation used here is established in The Interaction Picture and Time-Dependent Hamiltonians; the first-order probability is evaluated in First-Order Transition Amplitudes and Probabilities; the resummation of the time-ordered series is Dyson Series and the Time-Ordered Evolution Operator; the resonant, non-perturbative limit is the two-level Rabi article; and the static, level-shifting counterpart is Non-Degenerate Perturbation Theory. Provenance note (verbatim from the source export): Coefficient equation and first/second-order amplitudes follow Sakurai & Napolitano Ch. 5 and Griffiths Ch. 11. A verifier should: (1) re-derive by projecting the Schrödinger equation; (2) confirm the constant-perturbation result (Griffiths Eq. 11.35); (3) check the harmonic-perturbation two-denominator form (Griffiths Eqs. 11.41–11.45); (4) verify the second-order unitarity cancellation using Hermiticity of . Golden-rule and Einstein-coefficient values are stated in their dedicated entries.
Why expand in the interaction picture rather than the Schrödinger picture?
Because the interaction picture removes the trivial, fast phase oscillations e^{−iE_n t/ħ} that every eigenstate carries under H₀, leaving coefficients that change only because of the perturbation. In the Schrödinger picture the coefficients wobble at the full Bohr frequencies even when nothing physical is happening; in the interaction picture a constant coefficient means "no transition." That makes the perturbative bookkeeping clean: ċ_m is directly proportional to V, so the order in V equals the number of times the perturbation has acted.
What is the bookkeeping parameter λ, and does it have physical meaning?
λ is a formal marker that tracks powers of the perturbation; you set it to 1 at the end. Its only job is to let you sort terms by how many times V acts, so that "first order" means "linear in V." It has no physical value of its own — the physical small parameter is the actual dimensionless ratio (like V/ΔE or Vt/ħ) that must be small for the series to be useful. λ organises the algebra; the physical ratio decides convergence.
Why are there no energy denominators here, unlike static perturbation theory?
Because we are computing a transition over a finite time, not a correction to a stationary state. Static theory sums over intermediate states with weights 1/(E_i − E_m) because it dresses a fixed level; those denominators come from integrating e^{i(E_i−E_m)t/ħ} over all time. In the time-dependent problem the integral runs only from 0 to t, producing (e^{iω t}−1)/ω rather than a bare 1/ω — a finite-time factor, not a denominator. Only when you take t → ∞ against a continuum does something denominator-like re-emerge, and even then it is a principal-value/delta structure, not a static gap.
Does second order matter, or can I always stop at first?
Stop at first order when only one intermediate path matters and the transferred population is small. Second order becomes essential when the direct matrix element V_fi vanishes by a selection rule but a two-step path i→m→f is allowed — then the leading transition is second order, and the intermediate ("virtual") state m carries the process. Second-order processes are generically weaker by a factor ~(V/ΔE)², which is why forbidden lines and two-photon transitions are faint compared with allowed ones.
Is the "transferred population ≈ 26%" boundary universal?
The 10%-error-at-26%-transfer figure is exact for the on-resonance two-level case, which is the worst case (fastest divergence between first order and exact). Off resonance the transfer is capped well below that and first order stays excellent indefinitely. So 26% is a conservative rule of thumb: if your first-order calculation predicts more than roughly a quarter of the population moving, distrust it and reach for the exact or higher-order treatment. Below that, first order is typically good to ten percent or better.
How does this connect to Fermi's golden rule?
The golden rule is what you get by taking the first-order probability computed in the next article, folding it against a continuum of final states, and using the long-time delta-function limit. The result is a probability that grows linearly in time — a constant rate. So the golden rule is not a separate theory: it is first-order time-dependent perturbation theory applied to a discrete-to-continuum transition. The framework here supplies the amplitudes; the golden-rule article supplies the continuum average.
Why does the exact resonant probability stay bounded while first order runs away?
Unitarity. The exact two-level evolution conserves total probability, so amplitude that flows from i to f must eventually flow back — hence the bounded oscillation sin²(Vt/ħ). First order, by truncating the series, loses this feedback: it lets amplitude leave i without ever depleting the source, so its prediction φ² grows without bound. The runaway is the mathematical fingerprint of a truncation that has violated unitarity; the cure is either more orders or the exact solution.