Time-Dependent Perturbation Theory and the Golden Rule

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 $5\times 10^{11}V/m$ that binds the electron — so the coupling is a perturbation, and the machinery that computes transition rates is time-dependent perturbation theory.¹

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 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,

A general solution of the full time-dependent Schrödinger equation $i\hslash {\partial }_t|\Psi ⟩=(H_0+H^{\prime }(t))|\Psi ⟩$ is expanded in the stationary states with their free phases factored out:

Writing the phase explicitly is what makes the method work: if $H^{\prime }=0$ 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 $H_0|n⟩=E_n|n⟩$ to cancel the $E_n$ terms, projecting onto $⟨k|$, and defining the Bohr angular frequency

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

Nothing has been approximated yet. The system is as hard as the Schrödinger equation itself; the coupling matrix element $H_{kn}^{\prime }$ 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 $|i⟩$ at $t=0$, so $c_i(0)=1$ and $c_{f\ne i}(0)=0$, 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 $f\ne i$ by direct integration:

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 $P_{i\to f}(t)={\left|c_f^{(1)}(t)\right|}^2$. The result is trustworthy only while $P_{i\to f}\ll 1$; once appreciable population has left $|i⟩$ the assumption $c_i\approx 1$ fails and higher orders matter.

The harmonic perturbation and resonance

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

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,

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 $\hslash \omega \approx E_f-E_i$.
• Stimulated emission. If $E_f<E_i$ then ${\omega }_{fi}<0$, and the first term resonates when $\omega \approx \left|{\omega }_{fi}\right|$: 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 ${\left|e^{i\Delta t}-1\right|}^2=4{\sin }^2(\Delta t/2)$,

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.

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.

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 $\{|i⟩,|f⟩\}$, 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

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

This is the Rabi formula.² 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, ${\sin }^2({\Omega }_{R}t/2)\approx ({\Omega }_{R}t/2{)}^2={\left|V_{fi}\right|}^2{t}^2/4{\hslash }^2$, recovers exactly the $t^2$ growth of the perturbative peak. First-order theory is the small-$t$ tangent to the Rabi oscillation.

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, $\left|V_{fi}\right|\ll \hslash \left|\Delta \right|$; 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 $\rho (E_f)\mathbf{d}{E}_f$ states in the interval $\mathbf{d}{E}_f$. For a harmonic perturbation the total probability in the upper band is

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

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 $\delta (\Delta )=\hslash \delta (E_f-E_i-\hslash \omega )$ and dividing by $t$ gives a probability that grows linearly in time, so the rate $\Gamma =P/t$ is constant:

The rate is a product of three factors: $2\pi /\hslash$, the squared coupling matrix element ${\left|V_{fi}/2\right|}^2$, 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 $\sim \hslash /t$, 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 analysis, and, applied to the vacuum field, the spontaneous-emission rate that fixes every excited-state lifetime.

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.

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.

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 $2\pi /\hslash$ 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), and read the finite width of the resonance as the observed line shape.