---
title: Time-Dependent Perturbation Theory and the Golden Rule
module: Radiative Transitions and Spectral Lines
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  An atom in a weak oscillating field makes transitions between its stationary
  states. First-order time-dependent perturbation theory gives the transition
  amplitude as a Fourier component of the perturbation at the Bohr frequency, and
  the resulting probability is a sinc-squared resonance that sharpens as the field
  acts longer. For a two-level system the same coupling produces Rabi oscillations;
  for a transition into a continuum the long-time limit collapses the sinc-squared
  into a delta function and yields Fermi's golden rule, a constant transition rate
  set by the coupling strength and the density of final states.
topics: [Radiative Transitions and Spectral Lines]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 11 — Quantum Dynamics; §11.1 Two-Level Systems, §11.2 Emission and Absorption of Radiation"
  - book: Bransden & Joachain
    ref: "Ch. 4 — Interaction of One-Electron Atoms with EM Radiation; §4.5 Transition Rates; Ch. 11"
  - book: Foot
    ref: "Ch. 7 — The Interaction of Atoms with Radiation; §7.1–7.2"
draft: false
---

The stationary states of an atom are the eigenstates of a time-independent
Hamiltonian $H_0$, and left alone an atom stays in one of them forever. Spectral
lines exist because atoms do not stay alone: an electromagnetic field couples the
states and drives population from one to another. The field is weak compared with
the internal Coulomb binding — the electric field of the light in a spectroscopy
experiment is a tiny fraction of the $5\times 10^{11}\ \mathrm{V/m}$ that binds
the electron — so the coupling is a perturbation, and the machinery that computes
transition rates is time-dependent perturbation theory.[^gs-tdpt]

This lesson builds that machinery from the time-dependent Schrödinger equation and
carries it to the two results every later lesson uses: the resonance lineshape of
a driven transition, and Fermi's golden rule for the rate into a continuum. The
[dipole approximation](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients)
supplies the specific coupling; here the perturbation stays general.

## The equations for the expansion coefficients

Let $H_0$ have a complete orthonormal set of eigenstates,

$$
H_0\,\ket{n} = E_n\,\ket{n},
\qquad
\braket{m}{n} = \delta_{mn}.
$$

A general solution of the full time-dependent Schrödinger equation
$i\hbar\,\partial_t\ket{\Psi} = \big(H_0 + H'(t)\big)\ket{\Psi}$ is expanded in the
stationary states with their free phases factored out:

$$
\ket{\Psi(t)} = \sum_n c_n(t)\,e^{-iE_n t/\hbar}\,\ket{n}.
$$

Writing the phase explicitly is what makes the method work: if $H'=0$ the
coefficients $c_n$ are constant, so every effect of the perturbation is carried by
their slow time dependence. Substituting the expansion into the Schrödinger
equation, using $H_0\ket{n}=E_n\ket{n}$ to cancel the $E_n$ terms, projecting onto
$\bra{k}$, and defining the **Bohr angular frequency**

$$
\omega_{kn} \equiv \frac{E_k - E_n}{\hbar}
$$

gives an exact set of coupled first-order equations for the coefficients,

$$
\dot c_k(t) = \frac{1}{i\hbar}\sum_n H'_{kn}(t)\,e^{i\omega_{kn}t}\,c_n(t),
\qquad
H'_{kn}(t) \equiv \bra{k}H'(t)\ket{n}.
$$

Nothing has been approximated yet. The system is as hard as the Schrödinger
equation itself; the coupling matrix element $H'_{kn}$ mixes every pair of states,
and each mixing carries the oscillating phase $e^{i\omega_{kn}t}$ that records how
far the two levels are out of step.

## First-order amplitude

Suppose the atom starts in a definite state $\ket{i}$ at $t=0$, so
$c_i(0)=1$ and $c_{f\neq i}(0)=0$, and the perturbation is weak. Expand the
coefficients in powers of the perturbation, $c_k = c_k^{(0)} + c_k^{(1)} + \cdots$.
The zeroth order keeps the initial state fixed, $c_k^{(0)}=\delta_{ki}$.
Feeding that into the right-hand side gives the first-order coefficient for any
final state $f\neq i$ by direct integration:

$$
c_f^{(1)}(t)
= \frac{1}{i\hbar}\int_0^t H'_{fi}(t')\,e^{i\omega_{fi}t'}\,\d t'.
$$

The transition amplitude is the Fourier transform of the perturbation matrix
element, evaluated at the Bohr frequency $\omega_{fi}$ of the pair. The transition
probability is $P_{i\to f}(t) = \abs{c_f^{(1)}(t)}^2$. The result is trustworthy
only while $P_{i\to f}\ll 1$; once appreciable population has left $\ket{i}$ the
assumption $c_i\approx 1$ fails and higher orders matter.

> **Definition (Transition amplitude).** The first-order coefficient
> $c_f^{(1)}(t)$ is the probability amplitude to find the system in $\ket{f}$ at
> time $t$ given that it started in $\ket{i}$. It is the component of the
> perturbation matrix element $H'_{fi}(t)$ oscillating at the Bohr frequency
> $\omega_{fi}$, integrated over the time the perturbation has acted.

## The harmonic perturbation and resonance

Monochromatic light supplies a perturbation oscillating at a single angular
frequency $\omega$. Write it as

$$
H'(t) = V\cos\omega t = \tfrac12 V\big(e^{i\omega t} + e^{-i\omega t}\big),
$$

with $V$ a time-independent operator (the dipole coupling, in the next lesson).
The two exponentials are the whole story. Inserting into the first-order integral
and integrating each term,

$$
c_f^{(1)}(t)
= -\frac{V_{fi}}{2\hbar}
\left[
\frac{e^{i(\omega_{fi}+\omega)t}-1}{\omega_{fi}+\omega}
+
\frac{e^{i(\omega_{fi}-\omega)t}-1}{\omega_{fi}-\omega}
\right],
\qquad V_{fi} = \bra{f}V\ket{i}.
$$

Each fraction is small unless its denominator nearly vanishes. Two resonances
appear, and they are physically distinct:

- **Absorption.** If $E_f > E_i$ then $\omega_{fi}>0$, and the second term blows up
  when $\omega\approx\omega_{fi}$: the atom climbs by absorbing a photon of energy
  $\hbar\omega\approx E_f-E_i$.
- **Stimulated emission.** If $E_f < E_i$ then $\omega_{fi}<0$, and the first term
  resonates when $\omega\approx\abs{\omega_{fi}}$: the field drives the atom down,
  and the emitted photon adds to the driving field.

Near either resonance the non-resonant term is negligible — dropping it is the
**rotating-wave approximation**. Take absorption, keep the resonant term, and
define the detuning $\Delta \equiv \omega_{fi}-\omega$. Using
$\abs{e^{i\Delta t}-1}^2 = 4\sin^2(\Delta t/2)$,

$$
P_{i\to f}(t)
= \frac{\abs{V_{fi}}^2}{\hbar^2}\,
\frac{\sin^2(\Delta t/2)}{\Delta^2}.
$$

This is the central result of first-order theory. As a function of detuning it is
a sinc-squared curve: a tall central peak at $\Delta=0$ flanked by rapidly
shrinking side lobes.

$$
% caption: Transition probability versus detuning at fixed time. The central lobe
% peaks at exact resonance (Delta = 0) with height proportional to t^2 and full
% width between first zeros of 4 pi / t; the side lobes carry little weight.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->] (-5.4,0) -- (5.6,0) node[right, black] {detuning};
\draw[->] (0,-0.3) -- (0,3.5) node[above, black] {probability};
% sinc^2 curve, hand-plotted
\draw[acc, very thick] plot[smooth] coordinates{
 (-5,0) (-4.4,0.08) (-3.75,0.14) (-3,0.10) (-2.5,0)
 (-2,0.62) (-1.5,1.33) (-1,2.12) (-0.5,2.75) (0,3)
 (0.5,2.75) (1,2.12) (1.5,1.33) (2,0.62) (2.5,0)
 (3,0.10) (3.75,0.14) (4.4,0.08) (5,0)};
% zero markers
\draw[black] (-2.5,0.08) -- (-2.5,-0.08) node[below] {first zero};
\draw[black] (2.5,0.08) -- (2.5,-0.08) node[below] {first zero};
% peak label
\draw[black, dashed] (0,3) -- (2.9,3);
\node[black, anchor=west] at (2.95,3) {height grows as $t^2$};
\end{tikzpicture}
$$

Three features of $P_{i\to f}(\Delta,t)$ organize everything that follows. At exact
resonance $\sin^2(\Delta t/2)/\Delta^2 \to t^2/4$, so the peak probability grows as
$t^2$. The width in detuning between the first zeros is $4\pi/t$, so the resonance
sharpens as $1/t$: the longer the field acts, the more sharply the atom
discriminates the resonant frequency. The area under the peak grows as $t$, and
that linear-in-time growth is what becomes a constant rate.

> **Definition (Rotating-wave approximation).** Near a resonance one of the two
> counter-rotating terms $e^{\pm i\omega t}$ in a harmonic perturbation has a
> denominator of order $2\omega$ and is dropped; only the term whose frequency
> nearly matches the Bohr frequency is kept. It is accurate when the coupling
> $\abs{V_{fi}}/\hbar$ and the detuning are both small compared with $\omega$.

## Rabi oscillations in a two-level system

The $t^2$ growth of the peak cannot continue: probability is bounded by one.
First-order theory breaks down at resonance precisely when $P$ approaches unity.
For an isolated pair of levels the two-state problem can be solved exactly, and the
exact answer replaces the unbounded parabola with a bounded oscillation.

Restrict the Hilbert space to $\{\ket{i},\ket{f}\}$, apply the rotating-wave
approximation, and the coupled equations for $c_i,c_f$ become a linear system with
constant coefficients. Define the resonant **Rabi frequency**

$$
\Omega_R \equiv \frac{\abs{V_{fi}}}{\hbar}
$$

and the generalized Rabi frequency $\Omega \equiv \sqrt{\Omega_R^2 + \Delta^2}$.
Solving with the initial condition $c_i(0)=1$ gives the excited-state probability

$$
P_{i\to f}(t)
= \frac{\Omega_R^2}{\Omega_R^2 + \Delta^2}\,
\sin^2\!\left(\frac{\Omega t}{2}\right).
$$

This is the **Rabi formula**.[^gs-rabi] On resonance ($\Delta=0$) it reduces to
$P = \sin^2(\Omega_R t/2)$: the population cycles completely between the two levels
with period $2\pi/\Omega_R$, reaching unity at $t=\pi/\Omega_R$ (a $\pi$-pulse) and
returning to the ground state at $t=2\pi/\Omega_R$. Off resonance the oscillation
is faster (frequency $\Omega>\Omega_R$) but never reaches the top: the prefactor
$\Omega_R^2/(\Omega_R^2+\Delta^2)$ caps the excitation, a Lorentzian in detuning of
half-width $\Omega_R$. Expanding the resonant result for short times,
$\sin^2(\Omega_R t/2)\approx (\Omega_R t/2)^2 = \abs{V_{fi}}^2 t^2/4\hbar^2$,
recovers exactly the $t^2$ growth of the perturbative peak. First-order theory is
the small-$t$ tangent to the Rabi oscillation.

$$
% caption: Rabi flopping of the excited-state population. On resonance (solid) the
% population cycles fully between the two levels; off resonance (dashed) it
% oscillates faster but saturates below one, capped by the ratio of Rabi frequency
% to detuning.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->] (-0.3,0) -- (9.6,0) node[right, black] {time};
\draw[->] (0,-0.2) -- (0,3.3) node[above, black] {excited population};
% gridline at 1
\draw[black, dashed] (0,3) -- (9.2,3);
\node[anchor=east] at (-0.1,3) {$1$};
% on-resonance sin^2, full swing, period 4 units
\draw[acc, very thick] plot[smooth] coordinates{
 (0,0) (0.5,0.44) (1,1.5) (1.5,2.56) (2,3) (2.5,2.56)
 (3,1.5) (3.5,0.44) (4,0) (4.5,0.44) (5,1.5) (5.5,2.56)
 (6,3) (6.5,2.56) (7,1.5) (7.5,0.44) (8,0) (8.5,0.44) (9,1.5)};
\node[acc, anchor=south west] at (2,3.02) {on resonance};
% off-resonance: faster, capped at 1.35
\draw[black, very thick, dashed] plot[smooth] coordinates{
 (0,0) (0.35,0.35) (0.7,1.0) (1.05,1.35) (1.4,1.0) (1.75,0.35)
 (2.1,0) (2.45,0.35) (2.8,1.0) (3.15,1.35) (3.5,1.0) (3.85,0.35)
 (4.2,0) (4.55,0.35) (4.9,1.0) (5.25,1.35) (5.6,1.0) (5.95,0.35)
 (6.3,0) (6.65,0.35) (7,1.0) (7.35,1.35) (7.7,1.0) (8.05,0.35) (8.4,0)};
\node[black, anchor=south west] at (6.5,1.4) {off resonance};
\end{tikzpicture}
$$

> **Worked example (A resonant $\pi$-pulse).** A two-level atom sits exactly on
> resonance in a field whose dipole coupling gives a Rabi frequency
> $\Omega_R = 2\pi\times 1\ \mathrm{MHz}$, with $\Omega_R = \abs{\vec d_{fi}\cdot
> \vec E_0}/\hbar$. Complete inversion occurs when the pulse area reaches $\pi$:
> $$
> \Omega_R\,t_\pi = \pi
> \quad\Longrightarrow\quad
> t_\pi = \frac{\pi}{\Omega_R} = \frac{1}{2\times 10^{6}\ \mathrm{s^{-1}}}
> = 0.5\ \mu\mathrm{s}.
> $$
> After $t_\pi$ the excited state holds the whole population ($P=1$); after
> $2t_\pi$ the atom is back in the ground state. A pulse of half that length, a
> $\pi/2$-pulse, leaves an equal superposition, the operation underneath Ramsey
> interferometry and the
> [optical clock](/atomic-physics/modern-atomic-physics/optical-clocks-precision).

### Validity of the perturbative result

First-order theory holds only while the transferred probability stays small,
$P_{i\to f}\ll 1$, so that $c_i\approx 1$ remains a good approximation. Off
resonance the amplitude never grows dangerous provided the coupling is weak
compared with the detuning, $\abs{V_{fi}}\ll\hbar\abs{\Delta}$; the prefactor
$\Omega_R^2/(\Omega_R^2+\Delta^2)$ of the Rabi formula is then small and the
excitation is bounded for all time. On resonance there is no such protection: the
population reaches unity at $t=\pi/\Omega_R$, and the perturbative parabola tracks
the true oscillation only for $t\ll 1/\Omega_R$. The two failures are the same
statement seen from the single-state and continuum sides; the golden rule below is
what survives when a continuum drains the amplitude before it can return.

## Fermi's golden rule

Between the perturbative parabola and the bounded Rabi cycle sits the case that
governs real spectral lines: a transition not into a single discrete state but
into a **continuum** of final states — the continuum of field modes into which a
photon can be emitted, or a band of closely spaced levels. When many final states
lie within the resonance width, the $t^2$ growth of any one state is replaced by a
steady rate, because the sharpening peak sweeps through more and more states as it
narrows.

Sum the transition probability over final states with energies distributed
according to a density $\rho(E_f)$, meaning $\rho(E_f)\,\d E_f$ states in the
interval $\d E_f$. For a harmonic perturbation the total probability in the upper
band is

$$
P(t) = \int \frac{\abs{V_{fi}}^2}{\hbar^2}\,
\frac{\sin^2(\Delta t/2)}{\Delta^2}\,\rho(E_f)\,\d E_f,
\qquad \Delta = \omega_{fi} - \omega.
$$

At large $t$ the sinc-squared factor is sharply peaked at $\Delta=0$, so
$\abs{V_{fi}}^2$ and $\rho$ can be pulled out at the resonant energy
$E_f = E_i + \hbar\omega$. The remaining integral is a standard representation of
the delta function,

$$
\frac{\sin^2(\Delta t/2)}{\Delta^2} \;\xrightarrow{\ t\to\infty\ }\;
\frac{\pi t}{2}\,\delta(\Delta),
$$

which follows because the function has area $\pi t/2$, height $t^2/4$, and width
$\sim 1/t$, so it acts as $(\pi t/2)\,\delta(\Delta)$ under integration. Converting
$\delta(\Delta) = \hbar\,\delta(E_f - E_i - \hbar\omega)$ and dividing by $t$ gives
a probability that grows linearly in time, so the **rate** $\Gamma = P/t$ is
constant:

> **Theorem (Fermi's golden rule).** For a perturbation with amplitude operator
> $V$ oscillating at frequency $\omega$, the transition rate from $\ket{i}$ into a
> continuum of final states of density $\rho(E_f)$ is
> $$
> \Gamma_{i\to f}
> = \frac{2\pi}{\hbar}\,
> \abs{\bra{f}\tfrac12 V\ket{i}}^2\,\rho(E_f)
> \Big|_{E_f = E_i + \hbar\omega}.
> $$
> For a perturbation switched on suddenly and held constant, the same derivation
> with $\omega=0$ gives $\Gamma_{i\to f} = (2\pi/\hbar)\,\abs{\bra{f}H'\ket{i}}^2
> \rho(E_f)$ at $E_f = E_i$.

The rate is a product of three factors: $2\pi/\hbar$, the squared coupling matrix
element $\abs{V_{fi}/2}^2$, and the density of final states at the energy fixed by
conservation. Energy is conserved exactly only in the long-time limit; at finite
$t$ the resonance has width $\sim\hbar/t$, the time-energy uncertainty relation in
concrete form. The golden rule is the workspace for the entire module: it gives the
absorption and stimulated-emission rates in the
[Einstein-coefficient](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients)
analysis, and, applied to the vacuum field, the spontaneous-emission rate that
fixes every excited-state
[lifetime](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes).

$$
% caption: The golden rule for a transition into a continuum. A discrete initial
% level couples to a band of final states of density rho(E); only states within
% the resonance width (shaded) contribute, and the rate is set by the coupling and
% by rho at the energy that conserves energy.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% initial discrete level (left)
\draw[acc, very thick] (-0.2,0) -- (1.8,0);
\node[anchor=east] at (-0.3,0) {$E_i$};
\node[acc, anchor=south] at (0.8,0.05) {initial state};
% continuum band (right)
\fill[acc!8] (5,1.4) rectangle (8.6,4.2);
\draw[black] (5,1.4) rectangle (8.6,4.2);
% many close levels
\foreach \yy in {1.5,1.7,1.9,2.1,2.3,2.5,2.7,2.9,3.1,3.3,3.5,3.7,3.9,4.1}
  \draw[black] (5,\yy) -- (8.6,\yy);
% resonance window shaded
\fill[acc!25] (5,2.55) rectangle (8.6,3.05);
\node[acc, anchor=west] at (8.7,2.8) {resonance width};
\node[anchor=west] at (8.7,4.2) {level density};
% coupling arrows into the window
\draw[acc, very thick, ->] (1.8,0) .. controls (3.4,0.4) and (4.0,2.2) .. (4.95,2.7);
\draw[acc, very thick, ->] (1.8,0) .. controls (3.4,0.2) and (4.0,2.4) .. (4.95,2.9);
\node[acc, anchor=south, rotate=32] at (3.3,1.1) {coupling $V$};
\end{tikzpicture}
$$

## The regimes side by side

The three results are one calculation read at different time scales and different
final-state structures. Which applies depends on how the coupling strength, the
detuning, and the spacing of final states compare.

| Regime | Final states | Time behavior of $P$ | Governing quantity |
| --- | --- | --- | --- |
| First-order resonance | single level | grows as $t^2$ at $\Delta=0$ | $\abs{V_{fi}}^2 t^2/4\hbar^2$ |
| Rabi oscillation | single level | bounded oscillation | Rabi frequency $\Omega_R$ |
| Golden rule | dense continuum | linear in $t$ (constant rate) | $\abs{V_{fi}}^2\rho(E_f)$ |

The perturbative parabola is the short-time limit of the Rabi cycle; the golden
rule is the long-time limit once a continuum absorbs the population before it can
return. A closed two-level atom in a coherent laser field flops; an atom that can
emit into the continuum of vacuum modes decays at the golden-rule rate. The next
lesson supplies the coupling operator $V$ — the electric dipole interaction — and
turns these abstract matrix elements into the oscillator strengths and Einstein
coefficients that are measured.

$$
% caption: The two limits of the driven transition. Short-time perturbation theory
% (parabola) is the tangent to the exact resonant Rabi oscillation (solid); a
% continuum of final states damps the return and linearizes the growth into the
% constant golden-rule rate (straight line).
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.3,0) -- (8.0,0) node[right, black] {time};
\draw[->] (0,-0.2) -- (0,3.4) node[above, black] {probability};
\draw[black, dashed] (0,3) -- (7.6,3);
\node[anchor=east] at (-0.1,3) {$1$};
% Rabi oscillation on resonance
\draw[acc, very thick] plot[smooth] coordinates{
 (0,0) (0.6,0.5) (1.2,1.6) (1.8,2.7) (2.4,3) (3.0,2.7)
 (3.6,1.6) (4.2,0.5) (4.8,0) (5.4,0.5) (6.0,1.6) (6.6,2.7) (7.0,3)};
\node[acc, anchor=south east] at (2.4,3.02) {Rabi};
% perturbative parabola (tangent at origin)
\draw[black, very thick, dashed] plot[smooth] coordinates{
 (0,0) (0.5,0.16) (1.0,0.64) (1.5,1.44) (2.0,2.56) (2.35,3.4)};
\node[black, anchor=south west] at (1.9,2.6) {perturbation};
% golden-rule linear growth
\draw[black, very thick, densely dotted] (0,0) -- (5.0,3.0);
\node[black, anchor=north west] at (4.0,2.4) {golden rule};
\end{tikzpicture}
$$

The golden rule earns its central place because it converts the quantum-mechanical
amplitude into a measurable rate with a transparent structure. Everything specific
to atoms and light enters through the matrix element $V_{fi}$ and the mode density
$\rho$; the $2\pi/\hbar$ and the energy-conserving delta function are universal.
The remaining lessons evaluate $V_{fi}$ for the dipole coupling, work out which
matrix elements vanish (the [selection
rules](/atomic-physics/radiative-transitions-and-line-shapes/selection-rules-forbidden-transitions)),
and read the finite width of the resonance as the observed
[line shape](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes).

[^gs-tdpt]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_, 3rd ed., §11.1–11.2. The coupled equations for the expansion coefficients and the first-order amplitude are derived there; the bound-electron field strength $\sim 5\times 10^{11}\ \mathrm{V/m}$ is the Coulomb field at the Bohr radius, $e/4\pi\varepsilon_0 a_0^2$.
[^gs-rabi]: **Griffiths & Schroeter**, §11.1.2 — the exact two-level solution in the rotating-wave approximation and the Rabi formula. See also **Foot**, _Atomic Physics_, §7.3 for the Rabi problem in the density-matrix language, and **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §4.5 for the golden-rule derivation from the sinc-squared limit.