Constant Perturbation Switched On

The constant-perturbation result is the quiet foundation under three big structures. In scattering theory, a particle meeting a static potential is a constant perturbation between plane-wave states; the golden-rule rate with the free-particle density of states is exactly the first Born approximation, and the differential cross section becomes $|\tilde V(\mathbf q)|^2$ — the scattering chapter derives this by name. In condensed matter and quantum optics, a discrete level suddenly coupled to a continuum (an impurity to a band, an excited atom to the electromagnetic vacuum) decays at the constant golden-rule rate, and integrating that rate gives the exponential Weisskopf–Wigner decay and the natural linewidth. And conceptually, recognizing the constant case as the $\omega\to0$ limit of the harmonic drive unifies static scattering with resonant spectroscopy — the same peaked energy factor, sitting at $E_f=E_i$ instead of $E_f=E_i\pm\hbar\omega$. The abrupt-versus-slow spectrum of switch-on rates is bracketed by the sudden approximation (an instantaneous change, handled by overlaps) and the adiabatic theorem (an infinitely slow change, no transitions), both recalled here by name; the constant perturbation is the perturbative middle ground.

Think of the first-order amplitude as a sum of little phasors $e^{i\omega_{fi}t'}$, one for each instant $t'$ in $[0,t]$ that the perturbation acts. When the final state is degenerate with the initial one ($\omega_{fi}=0$) all the phasors point the same way and add coherently: the amplitude grows linearly in $t$, the probability as $t^2$. When $E_f\neq E_i$ the phasors wind around the clock and largely cancel, leaving only a small residual. Hence the sharp peak at energy conservation — it is a phase-coherence effect, not something put in by hand. Energy conservation emerges from the finite observation time. A static $V$ supplies no energy, so it cannot by itself change the energy; and indeed a measurement lasting only a time $t$ cannot resolve energy to better than $\Delta E\sim\hbar/t$. Transitions to states within $\hbar/t$ of $E_i$ are therefore allowed, and as $t\to\infty$ that window collapses to a delta function — exact energy conservation. The second figure makes this concrete: the same peaked function, drawn at increasing $t$, sharpens and grows toward $2\pi t\,\delta$. The $t^2$ peak height and the $1/t$ width are inseparable, and their product — the area — grows linearly in $t$. That linear growth is what lets us speak of a rate. But it only becomes a sensible, bounded rate after we integrate over a continuum of final states weighted by their density $\rho(E_f)$: a single discrete degenerate level would keep growing as $t^2$ (until first-order theory breaks and Rabi oscillations take over), whereas a continuum's many final states dephase, converting the would-be oscillation into steady, irreversible decay.

The single discrete level and the continuum behave completely differently, and it is worth stating the contrast sharply, because it is the whole conceptual content of the chapter. Discrete degenerate level: probability grows as $t^2$ at first order, then — once it is no longer small — turns into bounded, reversible Rabi oscillation (the exact two-level result owns this). Continuum: the peaked function collapses onto $E_f=E_i$, the density of states weights the accessible band, and the initial state decays irreversibly at a constant rate. Coherence survives for the single level and is lost to dephasing in the continuum. The last picture is the unification this article owns: the constant perturbation is the zero-frequency limit of a harmonic one. A sinusoidal drive at frequency $\omega$ produces two resonance peaks, absorption at $\omega_{fi}=+\omega$ and emission at $\omega_{fi}=-\omega$; slide $\omega$ down to zero and the two peaks march together and merge into a single, taller peak sitting exactly at energy conservation $\omega_{fi}=0$. The third figure draws that merger, and the derivation proves it term for term.

The starting point is the first-order interaction-picture amplitude, whose general form the first-order-transition article owns; here we specialize it to a time-independent $V$. Before each display, the object it defines.

where: $c_f^{(1)}$ — first-order amplitude of $|f\rangle$ in the interaction picture; $V_{fi}=\langle f|V|i\rangle$ — the (time-independent) matrix element; $\omega_{fi}=(E_f-E_i)/\hbar$ — the Bohr frequency between final and initial states. The physical read: the amplitude is the coherent sum over the on-time of the perturbation. Doing the elementary integral and squaring gives the transition probability — the article's central object.

where: $P_{i\to f}$ — probability of finding the system in $|f\rangle$ at time $t$; the factor $4\sin^2(\omega_{fi}t/2)/\omega_{fi}^2$ is the universal energy-mismatch function. The physical read: a fixed matrix element times a shape that depends only on the energy mismatch and the elapsed time. Its two defining scalings, at the peak and in width, make the "rate" language possible.

where: $F(0,t)=t^2$ — the peak height at exact resonance; $\Delta\omega\sim2\pi/t$ — the zero-to-zero width; the area identity $2\pi t$ — the fixed weight that, spread over a shrinking width, becomes a delta function. The physical read: height up, width down, area up linearly — a rate. Summed over a continuum this collapses to the golden rule, which the golden-rule article owns; here we own its constant-perturbation derivation.

where: $\Gamma$ — the time-independent transition rate; $\rho(E_f)$ — the density of final states, which the density-of-states article owns. The physical read: a constant rate, evaluated at energy conservation, into whatever density of states is available.

The behaviour is set by the energy mismatch $\omega_{fi}$, the observation time $t$, and whether the final states are discrete or continuous.

Example 1 — a two-level box, degenerate and non-degenerate. Take a coupling of strength $|V_{fi}|/\hbar=0.10$ (in inverse time units) switched on at $t=0$, and evolve for $t=2.0$. Compute the transition probability for a degenerate final state and for a non-degenerate one, and locate the first zero of the non-degenerate line.

Step 1. Degenerate case, $\omega_{fi}=0$: $P=(|V_{fi}|/\hbar)^2 t^2=(0.10)^2(2.0)^2=\mathbf{0.040}$. Small, so first-order theory is valid here.

Step 2. Non-degenerate case, $\omega_{fi}=1.0$: $P=(|V_{fi}|/\hbar)^2\,4\sin^2(\omega_{fi}t/2)/\omega_{fi}^2=0.01\times4\sin^2(1.0)/1.0$. With $\sin(1.0)=0.84147$, $\sin^2(1.0)=0.70807$, so $P=0.04\times0.70807=\mathbf{0.02832}$.

Step 3. So the degenerate transition ($0.040$) is more probable than the detuned one ($0.02832$) at this time — energy conservation is already favouring $E_f=E_i$, even though the mismatch is only $\omega_{fi}=1$.

Step 4. The first zero of the energy-mismatch factor is at $\omega_{fi}t=2\pi$, i.e. $\omega_{fi}=2\pi/t=2\pi/2.0=\mathbf{\pi\approx3.14}$. Final states with $|\omega_{fi}|<\pi$ lie inside the central peak; the window half-width $\sim\pi/t$ is the $\Delta E\sim\hbar/t$ time-energy statement in disguise.

Example 2 — a discrete state decaying into a band. A discrete level is weakly coupled to a continuum with matrix element $|V_{fi}|=1\ \mu\mathrm{eV}=10^{-6}\ \mathrm{eV}$ and density of final states $\rho=10^{4}\ \mathrm{states/eV}$. Compute the golden-rule decay rate and lifetime, and check the rate regime is real.

Step 1. The constant-perturbation golden rule gives $\Gamma=(2\pi/\hbar)|V_{fi}|^2\rho(E_i)$. Use $\hbar=6.582\times10^{-16}\ \mathrm{eV\,s}$.

Step 2. Compute: $\Gamma=(2\pi/6.582\times10^{-16})\times(10^{-6})^2\times10^{4}=(9.546\times10^{15})\times10^{-12}\times10^{4}$, giving $\Gamma=\mathbf{9.55\times10^{7}\ s^{-1}}$.

Step 3. The lifetime is $\tau=1/\Gamma=\mathbf{1.05\times10^{-8}\ s}\approx10.5\ \mathrm{ns}$ — a physically sensible radiative-scale lifetime.

Step 4. Check the golden-rule window $1/\Delta\omega_{\text{band}}\ll t\ll1/\Gamma=10.5$ ns: for a band of width even $\sim1$ meV, $1/\Delta\omega_{\text{band}}\sim\hbar/\Delta E\sim10^{-12}$ s, so there is a wide window $10^{-12}\ \mathrm{s}\ll t\ll10^{-8}\ \mathrm{s}$ over which the rate is constant. The constant rate is real; only beyond $\tau$ does depletion of the initial state matter (exponential decay $e^{-\Gamma t}$, the golden-rule article's result).

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 18 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. XIII · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 14. This article owns the constant-perturbation transition probability, its $t^2$-versus-continuum fates, and the proof that the static case is the $\omega\to0$ limit of the harmonic drive. The general first-order amplitude it specializes belongs to First-Order Transition Amplitudes and Probabilities; the harmonic case with its two resonances belongs to Harmonic (Sinusoidal) Perturbations and Resonance; the continuum rate it points to is Fermi's Golden Rule, and the density factor is Density of Final States. The exact two-level treatment that replaces the runaway $t^2$ is Transition Probability for a Periodic Two-Level System. The static-potential scattering application — the first Born approximation — is derived by name in the scattering chapter; the sudden and adiabatic extremes of switch-on are The Sudden Approximation and The Adiabatic Theorem.

Provenance. Derivations follow standard time-dependent perturbation theory. A verifier should reproduce: (1) the amplitude integral $\int_0^t e^{i\omega t'}\mathrm{d}t'=(e^{i\omega t}-1)/(i\omega)$ and the resulting $P=4|V_{fi}|^2\sin^2(\omega_{fi}t/2)/(\hbar^2\omega_{fi}^2)$ (Griffiths Ch.11; Sakurai §5.6); (2) the area identity $\int\sin^2(u)/u^2\,\mathrm{d}u=\pi$ giving the $2\pi t\,\delta$ limit; (3) the Golden Rule $\Gamma=(2\pi/\hbar)|V_{fi}|^2\rho(E_i)$ (Sakurai Eq.5.6.34; Cohen-Tannoudji Vol.II Ch.XIII; Shankar Ch.18). No empirical numbers are quoted; all relations are exact analytic results.

Builds on: Time-Dependent Perturbation Theory: General Formalism · First-Order Transition Amplitudes and Probabilities · The Interaction Picture and Time-Dependent Hamiltonians Leads to: Fermi's Golden Rule · Density of Final States · Harmonic (Sinusoidal) Perturbations and Resonance · Transition Probability for a Periodic Two-Level System