Harmonic (Sinusoidal) Perturbations and Resonance

Shine light of frequency $\omega$ on an atom and it makes transitions — but only sharply, and only near the frequencies that match its level spacings. This article works out why. Take the general first-order transition amplitude and specialize the perturbation to a sinusoid $\hat V(t)=\hat V_0\cos(\omega t)$ . The single time integral splits into two rotating terms, one with denominator $\omega_{fi}-\omega$ (absorption, $E_f=E_i+\hbar\omega$ ) and one with $\omega_{fi}+\omega$ (stimulated emission, $E_f=E_i-\hbar\omega$ ). Near resonance one of them blows up and the other is negligible — that separation is the rotating-wave approximation, and keeping the near-resonant term alone is the whole practical content of driven transition theory. What this article owns is the resonance line shape and its scaling. The resonant probability is a $\mathrm{sinc}^2$ in the detuning $\delta=\omega_{fi}-\omega$ : its peak height grows as $t^2$ , its width shrinks as $1/t$ , and its area therefore grows linearly in $t$ — a constant rate. That last fact is the seed of Fermi's golden rule in the driven case, and we prove the enabling identity $\int\mathrm{sinc}^2$ -integral $=2\pi t$ symbolically. The width-shrinks-as- $1/t$ statement is the energy-time uncertainty relation wearing work clothes; the $\mathrm{sinc}^2/$ Lorentzian peak is the same line-shape family that resonant transmission wears — both recalled by name, not re-derived. We also mark the honest edges. The perturbative result fails exactly at resonance for long times, where the initial state depletes and the two-level problem must be solved exactly (that is the next article). The counter-rotating term, dropped by the rotating-wave approximation, is not zero — it produces the small AC-Stark and Bloch–Siegert shifts named here and left to advanced treatments. The two rotating terms, the rotating-wave approximation, and the resonance line-shape scaling are the deliverables.

Key takeaways

A sinusoidal perturbation $\hat V(t)=\hat V_0\cos(\omega t)$ gives a first-order amplitude with two rotating terms: $c_f^{(1)}\propto\frac{e^{i(\omega_{fi}-\omega)t}-1}{\omega_{fi}-\omega}+\frac{e^{i(\omega_{fi}+\omega)t}-1}{\omega_{fi}+\omega}$ .
The $\omega_{fi}-\omega$ term resonates when $\hbar\omega=E_f-E_i$ (absorption); the $\omega_{fi}+\omega$ term when $\hbar\omega=E_i-E_f$ (stimulated emission).
Rotating-wave approximation: near resonance keep only the co-rotating term; the counter-rotating one is smaller by $\sim\delta/\omega_{fi}$ and produces only small shifts (AC-Stark, Bloch–Siegert).
Owned result: the resonant probability $P=\frac{|V_{fi}|^2}{4\hbar^2}\frac{\sin^2(\delta t/2)}{(\delta/2)^2}$ has peak height $\propto t^2$ , width in $\delta$ $\propto 1/t$ , and area $\propto t$ — a constant rate. The enabling identity $\int\frac{\sin^2(\delta t/2)}{(\delta/2)^2}\mathrm{d}\delta=2\pi t$ is proved symbolically.
The width-shrinks-as- $1/t$ is the energy-time relation; the peak is the same $\mathrm{sinc}^2$ /Lorentzian family as a transmission resonance — both recalled by name.
Perturbation theory fails at exact resonance for long $t$ (the state depletes); the exact two-level solution is the next article.
Key idea: a periodic drive makes transitions only near $\hbar\omega=|E_f-E_i|$ ; the resonance sharpens as $1/t$ while its area grows as $t$ , delivering a constant transition rate.

The core insight

Feed a sinusoid into the first-order transition amplitude. The single integral produces two resonant denominators, $\omega_{fi}\mp\omega$ , one for absorption and one for stimulated emission. Near resonance one term dominates (the rotating-wave approximation), and the transition probability is a $\mathrm{sinc}^2$ in the detuning: a peak that grows as $t^2$ and narrows as $1/t$ , so its area — the rate — grows linearly in time. That is resonance, and it is the bridge from a single first-order amplitude to the golden-rule rate.

Where this lives today

The harmonic-perturbation resonance is the theoretical core of spectroscopy, quantum optics, and quantum control. Every absorption and emission line an atom shows is a resonance of exactly this kind; the $1/t$ width law is why long coherent interrogation gives narrow lines, the principle behind precision atomic spectroscopy and optical clocks. The rotating-wave approximation is the standard first step in the semiclassical theory of atom–light interaction, feeding into the Einstein A and B coefficients, laser physics, and the driven-qubit gates of quantum computing. The resonant line shape — $\mathrm{sinc}^2$ at finite time, Lorentzian for a decaying state — recurs everywhere a system is driven near a natural frequency, from NMR and ESR to cavity QED. The neglected counter-rotating term resurfaces at the precision frontier as the Bloch–Siegert and AC-Stark shifts, which modern clock and qubit experiments must model and subtract. Beyond first-order perturbation theory, the exact driven two-level dynamics — Rabi flopping — is the subject of the next article.

The physical picture

An atom in an energy eigenstate has no oscillating dipole to grab — it is a stationary state. A periodic field supplies the missing frequency: it can hand the atom a quantum $\hbar\omega$ (absorption) or coax one out (stimulated emission), but only if $\hbar\omega$ nearly matches a level spacing. The "nearly" is the whole story. Off resonance the amplitude oscillates and returns — nothing accumulates. On resonance the amplitude builds coherently, and the probability grows. Mathematically, the drive $\cos(\omega t)=\tfrac12(e^{i\omega t}+e^{-i\omega t})$ carries two frequencies. Combined with the interaction-picture phase $e^{i\omega_{fi}t}$ of the transition amplitude, one piece rotates at $\omega_{fi}-\omega$ and the other at $\omega_{fi}+\omega$ . When the drive frequency approaches the transition frequency, $\omega\to\omega_{fi}$ , the first piece stops rotating — its phase is stationary, so its time integral accumulates without cancelling — while the second rotates twice as fast as before and averages away. Keeping only the stationary piece is the rotating-wave approximation, and it turns a two-term mess into a clean single resonance.

Figure

The resonance sharpens: transition probability versus detuning at three times

The figure is the article's thesis. As $t$ increases the resonance peak climbs (height $\propto t^2$ ) and pinches in (width $\propto1/t$ ). The two scalings conspire so the area under the curve grows only linearly in $t$ : the total transition probability into a spread of nearby final states increases at a steady rate. That constant rate is what Fermi's golden rule packages, and the $2\pi t$ area is the exact number that puts the $2\pi$ in the rule.

The two rotating terms

We start from the first-order transition amplitude owned by First-Order Transition Amplitudes and Probabilities, and feed in a harmonic drive. The context: the amplitude is a single time integral of the perturbation dressed with the Bohr-frequency phase, and the whole effect of the drive being periodic is to split that integral into two.

c_f^{(1)}(t)=-\frac{i}{\hbar}\int_0^t V_{fi}(t')\,e^{i\omega_{fi}t'}\,\mathrm{d}t',\qquad V_{fi}(t)=V_{fi}\cos(\omega t).

where: $V_{fi}=\langle f|\hat V_0|i\rangle$ is the (time-independent) matrix element of the drive amplitude, $\omega$ is the drive frequency, and $\omega_{fi}=(E_f-E_i)/\hbar$ the transition frequency. The physical read: the phase $e^{i\omega_{fi}t'}$ oscillates at the transition frequency; multiplying by $\cos(\omega t')$ produces beats at the sum and difference frequencies $\omega_{fi}\pm\omega$ , and only a near-zero beat frequency lets the integral accumulate.

Derivation ASplitting the harmonic drive into absorption and emission

The goal is to evaluate the first-order integral for a cosine drive and read off the two resonances. We write the cosine as a sum of two exponentials, integrate each elementary exponential, and identify which final energies each term drives. This produces the two-denominator structure that every subsequent statement about resonance rests on.

Step 1. Expand $\cos(\omega t')=\tfrac12(e^{i\omega t'}+e^{-i\omega t'})$ , so the integrand becomes $\tfrac{V_{fi}}{2}\big(e^{i(\omega_{fi}+\omega)t'}+e^{i(\omega_{fi}-\omega)t'}\big)$ .

Step 2. Integrate each exponential: $\int_0^t e^{i\alpha t'}\mathrm{d}t'=(e^{i\alpha t}-1)/(i\alpha)$ . Why: elementary, and it exposes the denominators $\alpha=\omega_{fi}\pm\omega$ that control resonance.

Step 3. Assemble: $c_f^{(1)}(t)=-\dfrac{V_{fi}}{2\hbar}\left[\dfrac{e^{i(\omega_{fi}+\omega)t}-1}{\omega_{fi}+\omega}+\dfrac{e^{i(\omega_{fi}-\omega)t}-1}{\omega_{fi}-\omega}\right]$ .

Step 4. Identify the physics. The $\omega_{fi}-\omega$ denominator vanishes at $\omega=\omega_{fi}$ , i.e. $E_f=E_i+\hbar\omega$ — the atom absorbs a quantum. The $\omega_{fi}+\omega$ denominator vanishes when $\omega=-\omega_{fi}$ , i.e. for a downward transition $E_f=E_i-\hbar\omega$ — stimulated emission. Each physical process gets one resonant term; the other is off-resonant and small. Why the factor $\tfrac12$ : it comes directly from $\cos=\tfrac12(e^{+}+e^{-})$ , and it is the source of the factor-of-4 in the resonant probability below — a convention worth pinning, since some texts write the drive as $2V_{fi}\cos\omega t$ to cancel it. Verified: verify_aq-qm-harmonic-perturbation-resonance.py confirms the two-term closed form against the direct symbolic integral on a numeric grid of $(V,\hbar,\omega_{fi},\omega,t)$ to $10^{-10}$ .

The second figure sweeps the drive frequency $\omega$ across a fixed transition $\omega_{fi}$ and plots the transition probability. It is flat and small almost everywhere, then spikes sharply when $\omega$ hits $\omega_{fi}$ — the absorption resonance. The counter-rotating term would produce a mirror spike at $\omega=-\omega_{fi}$ (the emission resonance for a downward transition), off the physical positive- $\omega$ axis; its only effect on the physical side is the tiny non-resonant background and the Bloch–Siegert shift.

Figure

Sweeping the drive frequency: a single sharp absorption spike at the transition frequency

The rotating-wave approximation and the resonance line shape

Near resonance the two terms are wildly unequal: the co-rotating term has a tiny denominator $\delta=\omega_{fi}-\omega\to0$ , while the counter-rotating term has denominator $\omega_{fi}+\omega\approx2\omega_{fi}$ . Dropping the small term is the rotating-wave approximation. The context for the next equation: with only the resonant term kept, the probability is the squared modulus of a single elementary integral.

P_{i\to f}(t)=|c_f^{(1)}|^2\approx\frac{|V_{fi}|^2}{4\hbar^2}\,\frac{\sin^2(\delta t/2)}{(\delta/2)^2},\qquad \delta=\omega_{fi}-\omega.

where: $\delta$ is the detuning, and the $\tfrac14$ is the square of the $\tfrac12$ from the cosine. The physical read: this is a $\mathrm{sinc}^2$ centred on $\delta=0$ ; on resonance it reduces to $(|V_{fi}|^2/4\hbar^2)\,t^2$ , growing quadratically, while its width in $\delta$ collapses as $1/t$ .

Go deeper · Derivation B · 1 minThe resonance line shape and why its area is a rate

Derivation BThe resonance line shape and why its area is a rate

This is the owned structural result: the scaling of the resonant $\mathrm{sinc}^2$ peak — height, width, and area — and the exact $2\pi t$ area integral that seeds the golden rule. We establish the three scalings and prove the area identity symbolically.

Step 1. Peak height. As $\delta\to0$ , $\sin^2(\delta t/2)/(\delta/2)^2\to t^2$ , so $P(\delta=0)=(|V_{fi}|^2/4\hbar^2)\,t^2$ — the probability at exact resonance grows as $t^2$ . Why: $\sin x\approx x$ , so the ratio $\to (\delta t/2)^2/(\delta/2)^2=t^2$ .

Step 2. Width. The first zeros of $\sin^2(\delta t/2)$ sit at $\delta t/2=\pm\pi$ , i.e. $\delta=\pm2\pi/t$ . The central peak therefore has width $\propto1/t$ : it sharpens as time goes on. Why: this is the energy-time relation in disguise — resolving the transition energy to $\hbar\delta$ requires observing for $t\gtrsim1/\delta$ , exactly the $\Delta E\,\Delta t\gtrsim\hbar$ bound owned by The Energy-Time Uncertainty Relation.

Step 3. Area. Integrate over all detunings: $\int_{-\infty}^{\infty}\frac{\sin^2(\delta t/2)}{(\delta/2)^2}\,\mathrm{d}\delta=2\pi t$ , proved symbolically. Why it matters: height $\times$ width $\sim t^2\times(1/t)=t$ , so the area — the total probability into a band of final states — grows linearly in $t$ . A probability growing linearly in time is a constant rate; this is the golden-rule seed, cashed by Fermi's Golden Rule.

Step 4. The line-shape family. The function $\sin^2(\delta t/2)/(\delta/2)^2$ is the finite-time diffraction kernel; as $t\to\infty$ it tends to $2\pi t\,\delta(\omega_{fi}-\omega)$ (a sharp energy-conserving spike), and for a decaying state it is dressed into the Lorentzian of width $\Gamma=\hbar/\tau$ — the same Breit–Wigner line shape owned by Resonant Transmission and Tunnelling Resonances, where linewidth and lifetime are Fourier partners. Verified: verify_...py confirms the peak $\propto t^2$ (at $t=4,8,16$ ), the first zero at $\delta=2\pi/t$ , and the area $=2\pi t$ both numerically (quad over $[-\infty,\infty]$ ) and symbolically ( $\int=2\pi t$ exactly).

Limiting cases

The behaviour is set by the detuning, the observation time, and the coupling strength.

Case	Result	Meaning
Exact resonance, $\delta=0$	$P=\frac{\|V_{fi}\|^2}{4\hbar^2}t^2$	Probability grows quadratically — but only while $P\ll1$ ; beyond that the exact two-level solution takes over (next article).
Large detuning, $\|\delta\|\gg1/t$	$P\lesssim\frac{\|V_{fi}\|^2}{\hbar^2\delta^2}$	Small, non-accumulating oscillations; no net rate. Off-resonant driving only shifts levels (AC-Stark).
Long time, band of final states	area $=2\pi t$ ⇒ constant rate	The $\mathrm{sinc}^2$ tends to $2\pi t\,\delta(\omega_{fi}-\omega)$ ; summing over a continuum gives Fermi's golden rule.
Deep rotating-wave regime, $\|\delta\|\ll\omega_{fi}$	single resonant term	Counter-rotating term smaller by $\sim\delta/\omega_{fi}$ ; the RWA is excellent (verify: rel. error $\approx0.3\%$ at $\omega_{fi}=50$ , $\delta=0.05$ ).
Static limit, $\omega\to0$	resonance moves to $\omega_{fi}=0$	The harmonic case reduces to the constant perturbation (energy-conserving transitions) — the subject of Constant Perturbation Switched On.

Common trapThe transition probability cannot grow past one

The resonant probability $P=(|V_{fi}|^2/4\hbar^2)t^2$ grows without bound in first-order perturbation theory, which is nonsense — a probability cannot exceed one. The formula is only valid while $P\ll1$ , i.e. for times short compared to the Rabi period. At and beyond $P\sim1$ the initial state depletes, first-order theory breaks down, and you must solve the two-level problem exactly (the Rabi solution, next article), which correctly bounds the population by oscillating it between $0$ and $1$ . A related trap: the rotating-wave approximation is not exact. The dropped counter-rotating term produces the small Bloch–Siegert shift of the resonance and the AC-Stark shift of the levels; these matter for precision spectroscopy even though they are invisible in the leading resonance line shape.

Worked examples

Example 1 — an atom driven near an optical transition. Take a transition at wavelength $750\,\mathrm{nm}$ , so $\omega_{fi}=2\pi c/\lambda$ , driven by a laser pulse of duration $t=1\,\mathrm{ns}$ . Compute the transition frequency, the resonance width in the drive frequency, and the first-zero detuning.

Step 1. Transition frequency: $\omega_{fi}=2\pi c/\lambda=2\pi(2.998\times10^8)/(750\times10^{-9})=\mathbf{2.51\times10^{15}\ \mathrm{rad/s}}$ (an optical frequency, as expected).

Step 2. The resonance line shape is $\mathrm{sinc}^2(\delta t/2)$ with $\delta=\omega_{fi}-\omega$ . Its central peak is bounded by the first zeros at $\delta=\pm2\pi/t$ .

Step 3. Compute the width for a $1\,\mathrm{ns}$ pulse: $\delta_{\text{zero}}=2\pi/t=2\pi/(10^{-9})=\mathbf{6.28\times10^{9}\ \mathrm{rad/s}}$ , about $1\,\mathrm{GHz}$ .

Step 4. Read the ratio: the resonance is sharp to a part in $\omega_{fi}/\delta_{\text{zero}}\approx2.5\times10^{15}/6.3\times10^{9}\approx\mathbf{4\times10^{5}}$ . A nanosecond pulse resolves the optical transition to about one part in $10^5$ — the longer you drive, the sharper the resonance, exactly the $1/t$ width law. This is why long, coherent pulses give narrow spectroscopic lines.

Example 2 — height, width, and area at three times. Confirm the three scalings on the dimensionless $\mathrm{sinc}^2$ line shape at $t=4,8,16$ (natural units, $|V_{fi}|/\hbar=1$ ).

Step 1. Peak height at $\delta=0$ : $P_{\text{peak}}=t^2/4$ , giving $\mathbf{4,\ 16,\ 64}$ for $t=4,8,16$ — quadrupling as $t$ doubles, the $t^2$ law.

Step 2. First-zero width: $\delta_{\text{zero}}=2\pi/t=\mathbf{1.571,\ 0.785,\ 0.393}$ — halving as $t$ doubles, the $1/t$ law.

Step 3. Area: $\tfrac14\times2\pi t=(\pi/2)t$ , giving $\mathbf{6.28,\ 12.57,\ 25.13}$ — doubling as $t$ doubles, the linear-in- $t$ law.

Step 4. Check the conspiracy: height $\times$ width $\approx t^2/4\times2\pi/t=(\pi/2)t$ , matching the area to the geometric factor. The peak grows as $t^2$ and narrows as $1/t$ , so the area — the rate — grows as $t$ , which is precisely the constant-rate statement that becomes Fermi's golden rule.

Physicist

Origins · 1 minRotating Wave Approximation and how this was discovered

Rotating Wave Approximation

1937–1954 · molecular-beam and magnetic-resonance laboratories

The near-resonant simplification that keeps only the co-rotating term grew out of the magnetic-resonance experiments of the late 1930s, where Isidor Isaac Rabi's molecular-beam method first drove transitions with an oscillating field tuned to a level spacing — the direct physical realization of a harmonic perturbation on a two-level system. The rotating-wave approximation itself became a standard tool in the 1940s–50s as nuclear and atomic magnetic resonance matured; its neglected counter-rotating term was pinned down by Bloch and Siegert in 1940, whose small frequency shift bears their names. The approximation is now the default first move in quantum optics and NMR: transform to the frame co-rotating with the drive, drop the fast-oscillating remainder, and read off the resonance.

History

1937

— Rabi's molecular-beam magnetic-resonance method drives transitions with a tuned oscillating field

1940

— Bloch and Siegert compute the shift from the neglected counter-rotating term

1944

— Rabi awarded the Nobel Prize for the resonance method

1946

— magnetic resonance in bulk matter (Bloch; Purcell) makes the driven two-level picture ubiquitous

1950

s — the rotating-wave approximation is standardized across nuclear and atomic magnetic resonance

Going further

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 harmonic-perturbation resonance: the two rotating terms, the rotating-wave approximation, and the $\mathrm{sinc}^2$ line-shape scaling (height $t^2$ , width $1/t$ , area $2\pi t$ ). The first-order amplitude it starts from is First-Order Transition Amplitudes and Probabilities; the energy-time width law it invokes is The Energy-Time Uncertainty Relation; the Lorentzian line-shape family it joins is Resonant Transmission and Tunnelling Resonances. The constant transition rate into a continuum is developed in Fermi's Golden Rule, and the exact non-perturbative driven two-level dynamics — Rabi flopping — is Transition Probability for a Periodic Two-Level System. The atom-field dipole coupling and its selection rules are taken up in the semiclassical-radiation and electric-dipole-selection-rules articles.

Provenance. Equations follow Griffiths (Intro to QM, 3rd ed., Ch. 11), Sakurai & Napolitano (Modern QM, 2nd ed., §5.6), and Cohen-Tannoudji Vol. II Ch. XIII. A verifier should: (1) reproduce the first-order amplitude integral and confirm the two resonant denominators $\omega_{fi}\mp\omega$ ; (2) check $|e^{i\theta}-1|^2=4\sin^2(\theta/2)$ gives the sinc-squared line shape; (3) verify the delta-function limit using $\int\sin^2(\alpha t/2)/\alpha^2\,\mathrm d\alpha=\pi t/2$ to recover $\Gamma=2\pi|V_{fi}|^2\rho/\hbar$ ; (4) confirm the exact two-level RWA solution and its short-time $t^2$ limit. All numbers (factors of 2, $2\pi$ ) are standard textbook values; amplitude conventions ( $2\cos\omega t$ vs $\cos\omega t$ ) are stated explicitly to fix factor-of-2 ambiguities.

Builds on: Time-Dependent Perturbation Theory · First-Order Transition Amplitudes and Probabilities · The Interaction Picture and Time-Dependent Hamiltonians · Constant Perturbation Switched On Leads to: Fermi's Golden Rule · Transition Probability for a Periodic Two-Level System · Absorption and Stimulated Emission · Semiclassical Theory of Atom–Radiation Interaction · Spontaneous Emission and the Einstein A Coefficient

Why keep only the near-resonant term — isn't that throwing away physics?

It is throwing away small physics in a controlled way. Near resonance the co-rotating denominator $\delta=\omega_{fi}-\omega$ is tiny while the counter-rotating one is $\approx2\omega_{fi}$ , so the dropped term is smaller by a factor $\sim\delta/\omega_{fi}$ — often $10^{-6}$ for an optical transition driven within its natural linewidth. What you throw away is not zero: it is the Bloch–Siegert shift of the resonance and the AC-Stark shift of the levels. For leading-order line shapes the RWA is excellent; for precision metrology you put the counter-rotating term back.

What is the difference between the two resonant denominators physically?

The $\omega_{fi}-\omega$ term resonates when the drive supplies exactly the energy gap, $\hbar\omega=E_f-E_i$ : the atom absorbs a quantum and goes up. The $\omega_{fi}+\omega$ term resonates for a downward transition, $\hbar\omega=E_i-E_f$ : the drive stimulates the atom to emit a quantum and go down. Both processes are driven by the same field; which one is resonant depends on whether the final state is above or below the initial one. Stimulated emission is the process behind lasers.

Why does the resonance get narrower the longer I drive?

Because resolving the transition energy requires time. The finite-time line shape $\sin^2(\delta t/2)/(\delta/2)^2$ has width $\Delta\delta\sim2\pi/t$ in the detuning, so $\Delta E\cdot t\sim h$ . This is the energy-time uncertainty relation in action: a short drive is spectrally broad and excites a wide band of near-resonant transitions; a long drive is spectrally sharp and picks out the resonance precisely. It is the same reason a short laser pulse is broadband and a long one is monochromatic.

If the peak grows as t², won't the probability exceed one?

Yes, and that is the signal that first-order perturbation theory has broken down. The $t^2$ growth is only valid while $P\ll1$ , i.e. for times short compared to the Rabi period $\pi\hbar/|V_{fi}|$ . Once a significant fraction of the population has transferred, the initial state is depleted and the linear-response assumption fails. The exact two-level solution replaces the unbounded $t^2$ with a bounded $\sin^2(\Omega_R t/2)$ that oscillates between $0$ and $1$ — Rabi flopping — which the next article derives.

Where exactly does the 2π in Fermi's golden rule come from?

From the area under the resonance line shape. Summing the first-order probability over a continuum of final states replaces the sum by $\int\rho(E_f)\,\mathrm{d}E_f$ , and the $\mathrm{sinc}^2$ factor integrates to $\int\frac{\sin^2(\delta t/2)}{(\delta/2)^2}\mathrm{d}\delta=2\pi t$ . The $t$ makes the probability grow linearly (a rate), and the $2\pi$ is the exact numerical factor that lands in $\Gamma=(2\pi/\hbar)|V_{fi}|^2\rho$ . So the $2\pi$ is not an arbitrary convention — it is the integral of a squared sinc.

Does the harmonic case include the constant perturbation as a limit?

Yes. Set $\omega\to0$ and the cosine becomes a constant. The two resonant denominators $\omega_{fi}\pm\omega$ merge into $\omega_{fi}$ , and the resonance sits at $\omega_{fi}=0$ , i.e. $E_f=E_i$ — energy-conserving transitions to degenerate or continuum states. That is exactly the constant-perturbation case (its own article), which is therefore the zero-frequency limit of this one. The harmonic case shifts the energy-conserving peak to $E_f=E_i\pm\hbar\omega$ ; the constant case leaves it at $E_f=E_i$ .

Is the resonance line shape Lorentzian or sinc-squared?

Both, in different regimes. For a coherent drive over a fixed finite time $t$ with no decay, the exact first-order line shape is $\mathrm{sinc}^2(\delta t/2)$ — a central peak with oscillating side lobes. If the final (or initial) state decays with lifetime $\tau$ , the sharp-in-time cutoff is replaced by an exponential, and the Fourier transform turns the $\mathrm{sinc}^2$ into a Lorentzian of width $\Gamma=\hbar/\tau$ — the Breit–Wigner shape. The $\mathrm{sinc}^2$ is the idealized coherent case; the Lorentzian is what you measure when lifetimes matter.