Time-dependent perturbation theory

Same particle-in-a-box eigenbasis as before (ħ = m = L = 1, Eₙ = n²π²/2), now driven by an explicit perturbation H'(x,t) = A·x·f(t). The spatial part is the dipole-like coupling x (so the usual opposite-parity selection rule applies); switch f(t) between off, constant, a polynomial ramp, or an oscillating drive, and compare the exact numerical population transfer against the first-order perturbative estimate.

System

Initial state n₀

System starts purely in φ_n₀ at t = 0.

Perturbation f(t) — H'(x,t)=A·x·f(t)

V = 0 — unperturbed reference, f(t)=0 V = const — suddenly switched on, f(t)=1 V = f(t), polynomial ramp, f(t)=tᵖ V = f(t), oscillating drive, f(t)=sin/cos(ωt)

amplitude A

frequency ω

cos(ωt) sin(ωt)

speed t = 0.00 total norm Σ|c|² = 1.000

drive envelope f(t)

energy levels & current population

Population transfer P_m(t) = |c_m(t)|²

Solid = exact (numerically propagated). Dashed = first-order TDPT estimate, P_m⁽¹⁾(t)=|c_m⁽¹⁾(t)|², shown only for levels the selection rule actually allows (opposite parity to n₀). Where solid and dashed separate, first-order theory has broken down.

|Ψ(x,t)|² under the drive

Reading the four cases

V = 0. Reference case — c_n₀ just spins in phase, every P_m(t) stays flat. No transitions, by construction.
V = const, switched on suddenly. A textbook "sudden perturbation": population leaks out of n₀ into opposite-parity neighbours immediately, growing roughly as t² at very short times (first-order theory), then — for large A — swinging into full Rabi-type oscillations that first-order theory cannot capture.
V = f(t) = tᵖ, polynomial ramp. A crude model of adiabatic switching. Larger p delays the drive longer near t=0, suppressing early transitions; compare p=1 vs p=3 at the same amplitude to see the sudden-vs-adiabatic contrast — slower switch-on transfers less population for the same elapsed time.
V = f(t) = sin/cos(ωt), oscillating drive. Set ω to the level spacing |E_m−E_n₀| of an allowed (opposite-parity) neighbour and watch resonant, large-amplitude transfer even at modest A; move ω away from resonance and transitions stay small and oscillatory — exactly the resonance condition ω ≈ ω_mn that first-order TDPT predicts.

In every case the same first-order formula is being evaluated live: c_m⁽¹⁾(t) = −i∫₀ᵗ H'_{m n₀}(t′) e^{iω_{mn₀}t′} dt′, with H'_{mn₀}(t)=A·x_{mn₀}·f(t) and x_{mn₀} the closed-form PIB dipole matrix element (nonzero only for m+n₀ odd). The exact curve instead numerically integrates the full Schrödinger equation in this basis — no rotating-wave or weak-field approximation — so any gap between the two is genuine breakdown of perturbation theory, not numerical error.