Time-dependent perturbation theory: Fermi's golden rule and transition rates

Intuition Beginner

A quantum system prepared in an energy eigenstate of a simple Hamiltonian $H_0$ will stay there forever — energy eigenstates are stationary. But add a small extra interaction $V(t)$ and the state starts leaking into other levels. Time-dependent perturbation theory tracks that leakage.

Write the full Hamiltonian as $H=H_0+V(t)$. The eigenstates $|n⟩$ of $H_0$ are known, with energies $E_n$. The perturbation $V(t)$ might be a laser field switching on, a radio-frequency pulse to a spin, or a weak coupling to a detector. The goal: given that the system starts in state $|i⟩$ at $t=0$, find the probability it has moved to state $|f⟩$ by time $t$.

The interaction picture from 12.03.01 does exactly the right bookkeeping. It strips out the "free" evolution under $H_0$ (just phases rotating at each eigenfrequency) and isolates what $V$ does on top. The result is a first-order transition amplitude $c_f^{(1)}(t)$ built from the matrix element $V_{fi}=⟨f|V|i⟩$ multiplied by a phase factor $e^{i{\omega }_{fi}{t}^{\prime }}$ oscillating at the Bohr frequency ${\omega }_{fi}=(E_f-E_i)/\hslash$. The amplitude accumulates when the perturbation's frequency matches the energy gap, and cancels itself out when it does not.

For a sinusoidal perturbation at frequency $\omega$, the transition probability to a single final state is a squared-sinc function peaked at $\omega ={\omega }_{fi}$. When the final states form a dense band (a continuum), summing over all of them gives a transition rate that is constant in time. That rate is Fermi's golden rule: the product of the squared matrix element, the density of final states, and a factor $2\pi /\hslash$. It is the standard formula for atomic transition rates, scattering cross-sections, and decay lifetimes across all of physics.

Visual Beginner

The squared-sinc lineshape narrows with time. Early on ($t$ small), the lineshape is broad and many final states receive population. At late times ($t$ large), only states satisfying $E_f\approx E_i+\hslash \omega$ are driven. This is energy conservation emerging from the long-time limit — not an exact constraint at finite time, but one that sharpens as the perturbation acts longer.

Worked example Beginner

Constant perturbation to a two-level system. A system has states $|1⟩$ and $|2⟩$ with $E_2>E_1$. A perturbation $V$ (constant, switched on at $t=0$) couples them with matrix element $V_{21}=⟨2|V|1⟩$. The system starts in $|1⟩$.

The first-order amplitude to be in $|2⟩$ at time $t$ is

$$c_2^{(1)}(t)=-\frac{V_{21}}{\hslash }\frac{e^{i{\omega }_{21}t}-1}{{\omega }_{21}},{\omega }_{21}=\frac{E_2-E_1}{\hslash }.$$

The transition probability is

$$P_{1\to 2}(t)=|c_2^{(1)}(t){|}^2=\frac{4|V_{21}{|}^2}{{\hslash }^2}\frac{{\sin }^2({\omega }_{21}t/2)}{{\omega }_{21}^2}.$$

Plug in numbers. Take $E_2-E_1=1.0$ eV, $|V_{21}|=10^{-3}$ eV, and $t=10^{-15}$ s. Then ${\omega }_{21}=(1.0\times 1.6\times 10^{-19})/(1.05\times 10^{-34})\approx 1.52\times 10^{15}$ rad/s. The argument of the sinc is ${\omega }_{21}t/2\approx 0.76$ rad, so ${\sin }^2(0.76)\approx 0.47$. The prefactor is $4\times (1.6\times 10^{-22}{)}^2/(1.05\times 10^{-34}{)}^2\approx 9.3\times 10^{15}$ s${}^{-2}$. Dividing by ${\omega }_{21}^2\approx 2.3\times 10^{30}$ s${}^{-2}$ gives $P\approx 9.3\times 10^{15}\times 0.47/2.3\times 10^{30}\approx 1.9\times 10^{-15}$.

The probability is tiny — the perturbation is weak and the time is short. It grows as $t^2$ at very early times (before the sinc oscillates) and as $t$ once a continuum of final states is summed over. The transition from $t^2$ growth to linear-in-$t$ growth is the passage from the single-state regime to the golden-rule regime.

Check your understanding Beginner

Formal definition Intermediate+

Let $H=H_0+V(t)$ where $H_0$ is time-independent with known eigenbasis $\{|n⟩\}$, $H_0|n⟩=E_n|n⟩$, and $V(t)$ is a Hermitian perturbation that may depend on time. The system is prepared in $|i⟩$ at $t=0$.

Interaction picture. Using the machinery of 12.03.01, define the interaction-picture state and perturbation:

$$|\psi (t){⟩}_I=e^{i{H}_{0}t/\hslash }|\psi (t){⟩}_S,V_I(t)=e^{i{H}_{0}t/\hslash }V(t)e^{-i{H}_{0}t/\hslash }.$$

The equation of motion is $i\hslash d|\psi {⟩}_I/dt=V_I(t)|\psi {⟩}_I$.

Amplitude expansion. Expand $|\psi (t){⟩}_I={\sum }_n{c}_n(t)|n⟩$ with $c_n(0)={\delta }_{ni}$. Substitution into the interaction-picture equation yields

$$i\hslash {\dot{c}}_f(t)=\sum _n{e}^{i{\omega }_{fn}t}{V}_{fn}(t)c_n(t),{\omega }_{fn}=\frac{E_f-E_n}{\hslash }.$$

The zeroth-order solution is $c_n^{(0)}(t)={\delta }_{ni}$. Substituting into the right side gives the first-order amplitude:

$$\boxed{c_f^{(1)}(t)=\frac{1}{i\hslash }{\int }_0^{t}d{t}^{\prime }{e}^{i{\omega }_{fi}{t}^{\prime }}{V}_{fi}(t^{\prime }),f\ne i.}$$

Constant perturbation. For $V(t)=V$ (time-independent, switched on at $t=0$):

$$c_f^{(1)}(t)=-\frac{V_{fi}}{\hslash }\frac{e^{i{\omega }_{fi}t}-1}{{\omega }_{fi}},|c_f^{(1)}(t){|}^2=\frac{4|V_{fi}{|}^2}{{\hslash }^2}\frac{{\sin }^2({\omega }_{fi}t/2)}{{\omega }_{fi}^2}.$$

Harmonic perturbation. For $V(t)=2V_0\cos (\omega t)=V_0{e}^{-i\omega t}+V_0^{†}{e}^{i\omega t}$, the first-order amplitude becomes

$$c_f^{(1)}(t)=-\frac{V_{0,fi}}{\hslash }\frac{e^{i({\omega }_{fi}-\omega )t}-1}{{\omega }_{fi}-\omega }+(\text{counter-rotating term at }{\omega }_{fi}+\omega ).$$

Under the rotating-wave approximation (discarding the counter-rotating term, valid when $\omega \approx {\omega }_{fi}$):

$$|c_f^{(1)}(t){|}^2=\frac{|V_{0,fi}{|}^2}{{\hslash }^2}\frac{{\sin }^2[({\omega }_{fi}-\omega )t/2]}{[({\omega }_{fi}-\omega )/2{]}^2}.$$

Fermi's golden rule. When the final state lies in a continuum with density of states $\rho (E_f)$, the total transition probability is obtained by integrating over the continuum. In the long-time limit, the squared-sinc kernel becomes $2\pi \hslash \delta (E_f-E_i-\hslash \omega )\cdot t$, and the transition rate is

$$\boxed{{\Gamma }_{i\to f}=\frac{2\pi }{\hslash }|V_{fi}{|}^2\rho (E_f){|}_{E_f=E_i+\hslash \omega }.}$$

For a constant perturbation ($\omega =0$), the rule reads $\Gamma =(2\pi /\hslash )|V_{fi}{|}^2\rho (E_f){|}_{E_f=E_i}$.

Counterexamples to common slips Intermediate+

• The first-order amplitude is not a rate. The transition probability $|c_f{|}^2$ grows as $t^2$ on resonance. The rate $d|c_f{|}^2/dt$ is the time derivative, which is linear in $t$ for a single final state and constant for a continuum. Confusing $|c_f{|}^2$ with $\Gamma$ introduces a factor-of-$t$ error.
• Energy conservation is not exact at finite time. The squared-sinc lineshape has width $\sim 2\pi /t$, not zero width. Energy is conserved only up to $\Delta E\sim \hslash /t$, the time-energy uncertainty relation. The delta function in the golden rule is a long-time limit.
• First-order perturbation theory requires the probability to stay small. The golden-rule rate $\Gamma$ predicts a linear increase $P(t)=\Gamma t$. Once $P$ approaches 1, the perturbative approximation fails. The validity window is $t\ll 1/\Gamma$.
• The rotating-wave approximation is a separate approximation from perturbation theory. Discarding the counter-rotating term is justified when the driving frequency $\omega$ is close to a transition frequency ${\omega }_{fi}$, making the counter-rotating term oscillate at $\sim 2\omega$ and average to zero. Far off-resonance, both terms contribute.

Key theorem with proof Intermediate+

Theorem (Fermi's golden rule). Let a system start at $t=0$ in eigenstate $|i⟩$ of $H_0$ with energy $E_i$. Let a sinusoidal perturbation $V(t)=V_0{e}^{-i\omega t}+V_0^{†}{e}^{i\omega t}$ be switched on at $t=0$. Let the final states form a continuum with density of states $\rho (E)$ that is smooth on the scale $\hslash /t$ around $E_i+\hslash \omega$, and let $|V_{0,fi}{|}^2$ likewise vary slowly. Then for sufficiently large $t$ the transition rate from $|i⟩$ into the continuum is

$${\Gamma }_{i\to f}=\frac{2\pi }{\hslash }|V_{0,fi}{|}^2\rho (E_f){|}_{E_f=E_i+\hslash \omega }.$$

Proof.

Step 1: Squared-sinc kernel. Under the rotating-wave approximation, the first-order probability to reach a single final state $|f⟩$ is

$$|c_f^{(1)}(t){|}^2=\frac{|V_{0,fi}{|}^2}{{\hslash }^2}F(x;t),x={\omega }_{fi}-\omega ,F(x;t)=\frac{4{\sin }^2(xt/2)}{x^2}.$$

The kernel $F$ has maximum $t^2$ at $x=0$, first zeros at $x=\pm 2\pi /t$, and area ${\int }_{-\infty }^{\infty }F(x;t)dx=2\pi t$.

Step 2: Integration over the continuum. Convert the sum over discrete final states to an integral:

$$P(t)=\int d{E}_f\rho (E_f)|c_f^{(1)}(t){|}^2=\frac{1}{\hslash }\int dx\rho (E_i+\hslash \omega +\hslash x)|V_{0,fi}(x){|}^2\frac{1}{\hslash }F(x;t).$$

where the change of variables $x=(E_f-E_i-\hslash \omega )/\hslash$ was used.

Step 3: Pull-out approximation. For large $t$, the kernel $F$ concentrates on $|x|\lesssim 2\pi /t$. By hypothesis, $\rho$ and $|V_{0,fi}{|}^2$ are smooth on this scale, so evaluate them at $x=0$ and pull them outside the integral:

$$P(t)\approx \frac{|V_{0,fi}{|}^2\rho (E_i+\hslash \omega )}{{\hslash }^2}\int dxF(x;t)=\frac{|V_{0,fi}{|}^2\rho (E_i+\hslash \omega )}{{\hslash }^2}2\pi t.$$

Step 4: Rate. Divide by $t$ to get the constant transition rate:

$$\Gamma =\frac{P(t)}{t}=\frac{2\pi }{\hslash }|V_{0,fi}{|}^2\rho (E_f){|}_{E_f=E_i+\hslash \omega }.\square$$

Corollary (Constant perturbation). Setting $\omega =0$ gives $\Gamma =(2\pi /\hslash )|V_{fi}{|}^2\rho (E_f){|}_{E_f=E_i}$, applicable to elastic scattering, tunnelling out of metastable states, and any static coupling to a continuum.

Bridge. This theorem builds toward 12.07.02 where the perturbation-theory chapter treats the same result from a broader perspective including second-order processes and QED applications, and toward 12.10.01 where the path integral re-expresses the full Dyson series. The foundational reason the golden rule works is that the squared-sinc kernel has total area proportional to $t$, so dividing by $t$ and passing to the long-time limit converts the oscillatory quantum amplitude into a steady classical-like rate. The same structure — matrix element squared times phase-space factor — recurs in scattering theory 12.08.01, in Fermi's theory of beta decay, and in every application where a weak coupling drives transitions into a dense band of final states.

Exercises Intermediate+

Lean formalization Intermediate+

The interaction picture and first-order amplitude are natural targets for formalization. The unperturbed propagator $e^{-i{H}_{0}t/\hslash }$ is covered by Mathlib.Analysis.NormedSpace.Exponential for bounded $H_0$. The interaction-picture transformation $|\psi {⟩}_I=e^{i{H}_{0}t/\hslash }|\psi {⟩}_S$ and the resulting equation of motion $i\hslash d|\psi {⟩}_I/dt=V_I|\psi {⟩}_I$ require differentiating a parameterised exponential applied to a vector, which is within reach of the existing Mathlib.Analysis.Calculus stack for bounded generators.

The first-order amplitude $c_f^{(1)}(t)=(i\hslash {)}^{-1}{\int }_0^t{e}^{i{\omega }_{fi}{t}^{\prime }}{V}_{fi}(t^{\prime })d{t}^{\prime }$ is a Bochner integral of a continuous function, formalisable via Mathlib.MeasureTheory.Integral.Bochner. The squared-sinc integral $\int 4{\sin }^2(xt/2)/x^{2}dx=2\pi t$ is a standard real-integral computation.

The golden rule itself sits at a higher altitude: it requires a continuum spectrum with absolutely continuous spectral measure, a locally smooth density of states, and the distributional limit ${\sin }^2(xt/2)/x^2\to \pi t\delta (x)$. The distributional limit is the key gap — Mathlib's Schwartz-space and tempered-distribution infrastructure is not yet wired to this specific approximate identity.

lean_status: none reflects this gap. The proof in the Key theorem section is the target a future formalisation would aim at.

Advanced results Master

The Dyson series and the interaction-picture propagator

The first-order amplitude is the leading term of the full iterative solution. The interaction-picture state satisfies the Volterra integral equation

$$|\psi (t){⟩}_I=|\psi (0){⟩}_I+\frac{1}{i\hslash }{\int }_0^{t}d{t}_1{V}_I(t_1)|\psi (t_1){⟩}_I.$$

Iterating by repeated substitution generates the Dyson series for the propagator:

$$U_I(t,0)=1+\frac{1}{i\hslash }{\int }_0^{t}d{t}_1{V}_I(t_1)+{\left(\frac{1}{i\hslash }\right)}^2{\int }_0^{t}d{t}_2{\int }_0^{t_2}d{t}_1{V}_I(t_2)V_I(t_1)+\cdots$$

The $n$th term involves $n$ time-ordered insertions of $V_I$. Introducing the time-ordering symbol $\mathcal{T}$, this compacts to the time-ordered exponential

$$U_I(t,0)=\mathcal{T}\exp \left(-\frac{i}{\hslash }{\int }_0^t{V}_I(t^{\prime })d{t}^{\prime }\right).$$

For bounded $V_I$ with ${\sup }_t\Vert V_I(t)\Vert \le M$ on $[0,T]$, the $n$th term satisfies $\Vert U_I^{(n)}\Vert \le (Mt/\hslash {)}^n/n!$, giving norm convergence by comparison with $\exp (Mt/\hslash )$. For unbounded $V$ (the physically relevant case), convergence is established on a dense domain under appropriate conditions but is generally asymptotic rather than convergent.

Absorption, stimulated emission, and the Einstein coefficients

A sinusoidal perturbation at frequency $\omega$ drives two processes: absorption ($E_f=E_i+\hslash \omega$) and stimulated emission ($E_f=E_i-\hslash \omega$). Both appear in the first-order amplitude as separate squared-sinc terms. The rates satisfy

$${\Gamma }_{\text{abs}}=\frac{2\pi }{\hslash }|V_{0,fi}{|}^2\rho (E_f){|}_{E_f=E_i+\hslash \omega },{\Gamma }_{\text{stim}}=\frac{2\pi }{\hslash }|V_{0,if}{|}^2\rho (E_f){|}_{E_f=E_i-\hslash \omega }.$$

By Hermiticity, $|V_{0,fi}{|}^2=|V_{0,if}{|}^2$. Einstein's 1917 argument defines the B-coefficients via ${\Gamma }_{\text{abs}}=B_{gi}{\rho }_{\text{rad}}(\omega )$ and ${\Gamma }_{\text{stim}}=B_{ig}{\rho }_{\text{rad}}(\omega )$, where ${\rho }_{\text{rad}}$ is the spectral energy density of the radiation field. Detailed balance in thermal equilibrium then requires $B_{gi}=B_{ig}$ (the two B-coefficients are equal) and relates the spontaneous emission rate $A_{ig}$ to $B_{ig}$ via

$$A_{ig}=\frac{\hslash {\omega }^3}{{\pi }^2{c}^3}{B}_{ig}.$$

This relation, derived from requiring thermal equilibrium to reproduce Planck's law, connects the stimulated rate (calculable from the golden rule with a classical field) to the spontaneous rate (which requires the quantised radiation field). The golden rule applied to the quantised vacuum field directly yields the A-coefficient, confirming Einstein's thermodynamic argument from a dynamical calculation.

Electric dipole transitions and selection rules

For an atom interacting with a light field, the perturbation is $V=-\mathbf{d}\cdot \mathbf{E}$ where $\mathbf{d}=-e\mathbf{r}$ is the electric dipole operator. The matrix element $d_{fi}=⟨f|e\mathbf{r}|i⟩$ enters the golden rule. Angular-momentum selection rules constrain which transitions are allowed:

• Parity: $\mathbf{r}$ is odd under parity, so $|i⟩$ and $|f⟩$ must have opposite parity: ${\ell }_f={\ell }_i\pm 1$ (not ${\ell }_i$).
• Magnetic quantum number: The $z$-component of $\mathbf{r}$ carries $m=0$; $x\pm iy$ carry $m=\pm 1$. Thus $\Delta m=0$ for $z$-polarised light and $\Delta m=\pm 1$ for circularly polarised light.
• Total angular momentum: $\Delta j=0,\pm 1$ with $j=0\to j=0$ forbidden.

The Wigner-Eckart theorem factorises the matrix element into a Clebsch-Gordan coefficient (geometry) times a reduced matrix element (dynamics). Transitions forbidden at first order (e.g., $s\to s$, $d\to d$) proceed via higher-order processes: electric quadrupole ($E2$), magnetic dipole ($M1$), or two-photon absorption. The ratio of successive multipole orders is roughly $\alpha \approx 1/137$ (the fine-structure constant) times a geometric factor, making higher-order transitions weaker by roughly $10^{-4}$.

The Wigner-Weisskopf model and exponential decay

Fermi's golden rule predicts a constant decay rate $\Gamma$, but the derivation assumes first-order perturbation theory and a constant initial-state amplitude. A more careful treatment solves the time-dependent Schrodinger equation for an initial discrete state $|i⟩$ coupled to a continuum $\{|f⟩\}$ self-consistently.

Write $|\psi (t)⟩=c_i(t)e^{-i{E}_{i}t/\hslash }|i⟩+{\sum }_f{c}_f(t)e^{-i{E}_{f}t/\hslash }|f⟩$. The equation for $c_i$ is

$${\dot{c}}_i(t)=-\frac{1}{{\hslash }^2}\sum _f|V_{fi}{|}^2{\int }_0^{t}d{t}^{\prime }{e}^{i(E_i-E_f)(t-t^{\prime })/\hslash }{c}_i(t^{\prime }).$$

For a flat continuum ($\rho$ constant near $E_i$), the sum over $f$ produces a delta function and the equation becomes ${\dot{c}}_i=-(\Gamma /2)c_i$, giving exponential decay $|c_i(t){|}^2=e^{-\Gamma t}$ with $\Gamma =(2\pi /\hslash )|V_{fi}{|}^2\rho (E_i)$. This is the Wigner-Weisskopf result: the golden-rule rate emerges from the self-consistent solution, not just from first-order perturbation theory.

The exponential decay law is itself an approximation. At very short times ($t\to 0$), $|c_i(t){|}^2=1-(t/{\tau }_Z{)}^2+\cdots$ where ${\tau }_Z$ is the Zeno time, related to the energy variance. At very long times, the decay follows a power law $|c_i{|}^2\propto t^{-\alpha }$ rather than an exponential, because the far wings of the energy distribution are not perfectly flat. Both deviations are experimentally measurable but typically tiny.

Second-order processes and virtual transitions

The second-order amplitude involves a sum over intermediate states $|m⟩$:

$$c_f^{(2)}(t)\propto \sum _m\frac{V_{fm}{V}_{mi}}{E_i+\hslash \omega -E_m}.$$

The intermediate state $|m⟩$ is not energy-conserving — it is "virtual." This sum-over-states structure appears in:

• Two-photon absorption: A transition $|i⟩\to |f⟩$ driven by two photons of frequency $\omega$ when $2\hslash \omega =E_f-E_i$. The direct matrix element $V_{fi}$ vanishes by selection rules, but the two-step path through any intermediate state $|m⟩$ contributes.
• Raman scattering: An incident photon at ${\omega }_{\text{in}}$ is absorbed and a different photon at ${\omega }_{\text{out}}$ is emitted, with $|m⟩$ a virtual electronic state. The final state differs from the initial state by a vibrational quantum.
• AC Stark shift (light shift): A non-resonant field shifts the energy of level $|g⟩$ by $\Delta E_g=-{\sum }_e|V_{ge}{|}^2/(E_e-E_g\mp \hslash \omega )$, derivable as the second-order energy correction. This shift underlies optical trapping of cold atoms and optical-lattice clocks.

The Lamb shift — the 1058 MHz splitting between hydrogen $2S_{1/2}$ and $2P_{1/2}$ states predicted degenerate by the Dirac equation — is a second-order QED effect: virtual photon emission and reabsorption. Bethe's 1947 non-relativistic calculation extracted the leading contribution via mass renormalisation, obtaining 1040 MHz within 2% of the measured value and launching the renormalisation programme of modern QED.

Synthesis. Time-dependent perturbation theory is the dynamical engine of quantum mechanics. The interaction picture separates free evolution from the perturbation, the Dyson series organises the perturbative expansion, and Fermi's golden rule extracts a constant transition rate from the long-time limit. The rule's universality — squared matrix element times density of states times $2\pi /\hslash$ — makes it the standard tool for computing rates in atomic, nuclear, condensed-matter, and particle physics. The first-order theory handles real (on-shell) transitions; the second-order theory handles virtual (off-shell) processes that shift energies and mediate multi-step transitions. The pattern generalises to all orders through the Dyson series, connecting non-relativistic quantum mechanics to the full perturbative S-matrix of quantum field theory.

Full proof set Master

Proposition 1 (First-order amplitude from the interaction picture). Let $H=H_0+V(t)$ with $H_0|n⟩=E_n|n⟩$. If the system starts in $|i⟩$, the first-order transition amplitude to $|f⟩\ne |i⟩$ is

$$c_f^{(1)}(t)=\frac{1}{i\hslash }{\int }_0^{t}d{t}^{\prime }{e}^{i{\omega }_{fi}{t}^{\prime }}{V}_{fi}(t^{\prime }).$$

Proof. In the interaction picture, $|\psi (t){⟩}_I={\sum }_n{c}_n(t)|n⟩$ satisfies

$$i\hslash {\dot{c}}_f=\sum _n{e}^{i{\omega }_{fn}t}{V}_{fn}(t)c_n(t).$$

At zeroth order, $c_n^{(0)}={\delta }_{ni}$. Substituting:

$$i\hslash {\dot{c}}_f^{(1)}=e^{i{\omega }_{fi}t}{V}_{fi}(t).$$

Integrating from 0 to $t$ with $c_f^{(1)}(0)=0$ gives the stated result. $\square$

Proposition 2 (Squared-sinc area identity). For all $t>0$:

$${\int }_{-\infty }^{\infty }dx\frac{4{\sin }^2(xt/2)}{x^2}=2\pi t.$$

Proof. Substitute $u=xt/2$, so $dx=2du/t$ and $x^2=4u^2/t^2$:

$${\int }_{-\infty }^{\infty }\frac{4{\sin }^{2}u}{4u^2/t^2}\frac{2du}{t}=2t{\int }_{-\infty }^{\infty }\frac{{\sin }^{2}u}{u^2}du.$$

The integral ${\int }_{-\infty }^{\infty }{\sin }^{2}u/u^{2}du$ evaluates to $\pi$ (by contour integration: close in the upper half-plane around the double pole at $u=0$, or by Parseval's theorem applied to the Fourier transform of the rectangle function). Therefore the area is $2t\cdot \pi =2\pi t$. $\square$

Proposition 3 (Distributional limit of the squared-sinc kernel). In the sense of tempered distributions on $\mathbb{R}$:

$$\underset{t\to \infty }{\lim }\frac{1}{t}\frac{4{\sin }^2(xt/2)}{x^2}=2\pi \delta (x).$$

Proof. Let $\varphi \in \mathcal{S}(\mathbb{R})$ be a Schwartz test function. Substitute $u=xt/2$:

$$\frac{1}{t}{\int }_{-\infty }^{\infty }\varphi (x)\frac{4{\sin }^2(xt/2)}{x^2}dx=2{\int }_{-\infty }^{\infty }\varphi (2u/t)\frac{{\sin }^{2}u}{u^2}du.$$

As $t\to \infty$, $\varphi (2u/t)\to \varphi (0)$ pointwise. The integrand is dominated by $\Vert \varphi {\Vert }_{\infty }{\sin }^2(u)/u^2\in L^1(\mathbb{R})$. By dominated convergence:

$$\underset{t\to \infty }{\lim }2{\int }_{-\infty }^{\infty }\varphi (2u/t)\frac{{\sin }^{2}u}{u^2}du=2\varphi (0){\int }_{-\infty }^{\infty }\frac{{\sin }^{2}u}{u^2}du=2\varphi (0)\cdot \pi =2\pi \varphi (0)=⟨2\pi \delta ,\varphi ⟩.\square$$

This is the analytic backbone of the golden rule: the squared-sinc kernel concentrates into a delta function in the long-time limit, enforcing energy conservation with coefficient $2\pi$.

Connections Master

• Interaction picture 12.03.01. The interaction picture is the foundation of time-dependent perturbation theory. It strips the free evolution into the operators and isolates the perturbation $V_I(t)$ as the sole driver of state change. Without this bookkeeping, the oscillating phase factors $e^{i{\omega }_{fn}t}$ would be buried inside the state and the resonance structure would be invisible.

• Time-independent perturbation theory 12.07.01. The static partner. Time-independent theory corrects energy eigenvalues and eigenstates; time-dependent theory computes transition rates between eigenstates of $H_0$. The same matrix elements $V_{fi}$ appear in both, but the dynamical theory also requires the density of final states and produces rates rather than energy shifts.

• Angular momentum and selection rules 12.05.01. The dipole matrix elements $⟨f|\mathbf{r}|i⟩$ entering the golden rule are constrained by angular-momentum conservation. The Wigner-Eckart theorem factorises each matrix element into a geometric part (Clebsch-Gordan coefficient) and a dynamical reduced matrix element. Selection rules determine which transitions are allowed at first order and which require higher-order mechanisms.

• Scattering theory 12.08.01. The Born approximation in scattering theory is the golden rule applied to the scattering potential. The scattering cross-section is proportional to $|V_{fi}{|}^2$ where $|i⟩$ and $|f⟩$ are plane-wave states, and the density of states gives the familiar phase-space factor $k^{2}dk$. The structure is identical; only the matrix element and the geometry change.

• Path integral 12.10.01. The Dyson series developed here is the perturbative expansion of the interaction-picture propagator. The path integral provides an alternative representation of the same object: the time-ordered exponential is re-expressed as a sum over classical trajectories weighted by $e^{iS/\hslash }$. The first-order term in the path integral matches the first-order Dyson amplitude.

Historical and philosophical context Master

Dirac's 1926 paper On the theory of quantum mechanics (Proc. Roy. Soc. A 112) introduced the method of variation of constants — expanding the time-dependent state in the eigenbasis of $H_0$ and deriving coupled equations for the coefficients $c_n(t)$. This is exactly the interaction-picture amplitude expansion used in this unit.

Dirac's 1927 follow-up The quantum theory of the emission and absorption of radiation (Proc. Roy. Soc. A 114) applied the method to an atom coupled to the quantised radiation field, obtaining the transition rate formula now called the golden rule. Dirac did not name it; the label comes from Fermi.

Fermi's 1950 Chicago lecture notes Nuclear Physics (University of Chicago Press) christened the formula "golden rule no. 2" in a footnote, attributing it to Dirac. "Golden rule no. 1" referred to the first-order correction to the wavefunction. Only no. 2 retained the name in common usage. The phrase entered physics pedagogy in the 1960s and has been universal ever since.

Wentzel's 1927 calculation of radiationless quantum jumps (Zeitschrift fur Physik 43) applied the same first-order rate formula to the Auger effect — a non-radiative transition where the final state is a continuum electron. Wentzel's work is mathematically identical to Dirac's electromagnetic calculation; only the matrix element differs.

Dyson's 1949 paper The radiation theories of Tomonaga, Schwinger, and Feynman (Phys. Rev. 75) showed that the time-ordered exponential underlies the entire S-matrix of QED. The Dyson series is the unifying framework that connects non-relativistic perturbation theory (golden rule at first order) to relativistic quantum field theory (Feynman diagrams at every order). The same iterative structure — insert $V_I$, time-order, integrate — generates both the golden rule and the full perturbative S-matrix.

Bethe's 1947 calculation of the Lamb shift (Phys. Rev. 72) used second-order time-dependent perturbation theory with a quantised radiation field and a high-energy cutoff, extracting the 1040 MHz splitting between hydrogen $2S_{1/2}$ and $2P_{1/2}$ states. This single calculation drove the development of renormalised QED and confirmed that virtual processes produce measurable energy shifts — the first triumph of the Dyson-series framework beyond first order.

The philosophical significance of the golden rule is that it derives irreversibility (exponential decay, constant transition rate) from reversible unitary dynamics. The microscopic Schrodinger equation is time-reversal invariant, yet the golden rule predicts an arrow of time: population flows from initial to final state and does not return. The resolution is that the irreversibility comes from the continuum of final states: the probability that all the amplitude scattered into the continuum spontaneously reassembles into the initial state is vanishingly small. This is the same mechanism as the thermodynamic arrow of time — coarse-graining over many degrees of freedom produces effective irreversibility from microscopically reversible dynamics.

Bibliography Master

Primary literature:

@article{Dirac1926,
  author = {Dirac, P. A. M.},
  title = {On the theory of quantum mechanics},
  journal = {Proceedings of the Royal Society A},
  volume = {112},
  year = {1926},
  pages = {661--677},
}

@article{Dirac1927,
  author = {Dirac, P. A. M.},
  title = {The quantum theory of the emission and absorption of radiation},
  journal = {Proceedings of the Royal Society A},
  volume = {114},
  year = {1927},
  pages = {243--265},
}

@article{Wentzel1927,
  author = {Wentzel, G.},
  title = {{\"U}ber strahlungslose Quantenspr{\"u}nge},
  journal = {Zeitschrift f{\"u}r Physik},
  volume = {43},
  year = {1927},
  pages = {524--530},
}

@book{Fermi1950,
  author = {Fermi, E.},
  title = {Nuclear Physics: A Course Given by Enrico Fermi at the University of Chicago},
  publisher = {University of Chicago Press},
  year = {1950},
}

@article{Dyson1949,
  author = {Dyson, F. J.},
  title = {The radiation theories of {Tomonaga}, {Schwinger}, and {Feynman}},
  journal = {Physical Review},
  volume = {75},
  year = {1949},
  pages = {486--502},
}

@article{Bethe1947,
  author = {Bethe, H. A.},
  title = {The electromagnetic shift of energy levels},
  journal = {Physical Review},
  volume = {72},
  year = {1947},
  pages = {339--341},
}

@article{Einstein1917,
  author = {Einstein, A.},
  title = {Zur {Q}uantentheorie der {S}trahlung},
  journal = {Physikalische Zeitschrift},
  volume = {18},
  year = {1917},
  pages = {121--128},
}

@article{WeisskopfWigner1930,
  author = {Weisskopf, V. and Wigner, E.},
  title = {Berechnung der nat{\"u}rlichen Linienbreite auf Grund der {D}iracschen Lichttheorie},
  journal = {Zeitschrift f{\"u}r Physik},
  volume = {63},
  year = {1930},
  pages = {54--73},
}

Textbooks and monographs:

@book{Griffiths2018,
  author = {Griffiths, D. J. and Schroeter, D. F.},
  title = {Introduction to Quantum Mechanics},
  edition = {3},
  publisher = {Cambridge University Press},
  year = {2018},
}

@book{SakuraiNapolitano2021,
  author = {Sakurai, J. J. and Napolitano, J.},
  title = {Modern Quantum Mechanics},
  edition = {3},
  publisher = {Cambridge University Press},
  year = {2021},
}

@book{CohenTannoudji1977,
  author = {Cohen-Tannoudji, C. and Diu, B. and Lalo{\"e}, F.},
  title = {Quantum Mechanics, Volume II},
  publisher = {Wiley},
  year = {1977},
}

@book{LandauLifshitz1977,
  author = {Landau, L. D. and Lifshitz, E. M.},
  title = {Quantum Mechanics: Non-Relativistic Theory},
  edition = {3},
  publisher = {Pergamon},
  year = {1977},
}

@book{Messiah1961,
  author = {Messiah, A.},
  title = {Quantum Mechanics, Volume II},
  publisher = {North-Holland},
  year = {1961},
}