Fermi's Golden Rule

Derivation

Fermi's Golden Rule

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Statement

For a system prepared in an unperturbed eigenstate \( |i\rangle \) and subjected to a weak, effectively constant perturbation \( \hat{V} \) that connects \( |i\rangle \) to a dense set of final states \( |f\rangle \) near the same energy, the probability per unit time for a transition into that continuum settles to the constant rate \( \Gamma_{i\to f} = \dfrac{2\pi}{\hbar}\,|M_{fi}|^{2}\,\rho(E_f) \), where \( M_{fi} = \langle f|\hat{V}|i\rangle \) and \( \rho(E_f) \) is the density of final states per unit energy evaluated at \( E_f = E_i \).

Why it matters

Fermi's Golden Rule is the workhorse formula for almost every rate in quantum physics: spontaneous emission and absorption of photons, radioactive alpha and beta decay, scattering cross-sections, Auger processes, and electron transport in solids. It converts a matrix element (structure) and a density of states (available phase space) into an observable lifetime, so it is the bridge between a Hamiltonian written on paper and a number measured in a lab.

Its deeper importance is conceptual: it shows that a genuinely irreversible, exponential-looking decay rate emerges from a strictly reversible, unitary Schrödinger equation, provided the final states form a continuum. The rule is where "reversible microscopic dynamics" meets "irreversible macroscopic decay."

Assumptions

The perturbation is weak (first-order theory suffices).If \( \hat{V} \) is not small, the initial-state amplitude depletes appreciably during the process, higher orders and back-transitions matter, and the simple product \( |M|^2\rho \) no longer gives the rate.

The perturbation is switched on at \( t=0 \) and is time-independent (or harmonic) thereafter.If \( \hat{V}(t) \) varies on the timescale of the transition, the energy-conserving delta function broadens or shifts and the rate acquires explicit time dependence.

The final states form a dense continuum with a smooth density \( \rho(E) \).If final states are discrete and well separated, the sum does not become an integral; one gets Rabi oscillations rather than a constant decay rate.

The observation time \( t \) lies in the Golden Rule window \( \hbar/\Delta E_{\text{band}} \ll t \ll \hbar/|M|\sqrt{\rho} \).Too short and the probability grows quadratically (Zeno regime, energy conservation not yet resolved); too long and depletion or recurrences invalidate first order.

\( M_{fi} \) and \( \rho \) vary slowly across the energy width \( \sim \hbar/t \) of the resonance.If they vary sharply, they cannot be pulled outside the energy integral and the clean factorised form fails.

Derivation

\[ i\hbar\,\dot{c}_f(t) = \sum_{n} c_n(t)\, V_{fn}\, e^{\,i\omega_{fn} t}, \qquad \omega_{fn} = \frac{E_f - E_n}{\hbar} \]

Exact equations of motion for the expansion coefficients \( c_f \) in the interaction picture, from time-dependent perturbation theory (prior result). No approximation yet. A

\[ c_i(t) \approx 1, \qquad c_{f\neq i}(t)\approx 0 \quad (t=0^+) \;\Longrightarrow\; i\hbar\,\dot{c}_f(t) \approx V_{fi}\, e^{\,i\omega_{fi} t} \]

First-order approximation: keep only the initial state on the right-hand side, since all other amplitudes are still small. This is where "weak \( \hat V \)" enters. A

\[ c_f(t) = \frac{1}{i\hbar}\int_0^{t} V_{fi}\, e^{\,i\omega_{fi} t'}\,\mathrm{d}t' = \frac{V_{fi}}{i\hbar}\,\frac{e^{\,i\omega_{fi} t}-1}{i\omega_{fi}} \]

Direct integration from \( 0 \) to \( t \), taking \( V_{fi} \) constant in time (constant perturbation switched on at \( t=0 \)). A

\[ P_{i\to f}(t) = |c_f(t)|^2 = \frac{|V_{fi}|^2}{\hbar^2}\,\left|\frac{e^{\,i\omega_{fi} t}-1}{\omega_{fi}}\right|^2 = \frac{|V_{fi}|^2}{\hbar^2}\,\frac{4\sin^2\!\left(\omega_{fi} t/2\right)}{\omega_{fi}^{2}} \]

Modulus squared, using \( |e^{i\theta}-1|^2 = 2-2\cos\theta = 4\sin^2(\theta/2) \). This is the transition probability to a single final state. A

\[ P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2}\; t^2\, \mathrm{sinc}^2\!\left(\frac{\omega_{fi} t}{2}\right), \qquad \mathrm{sinc}(x)\equiv \frac{\sin x}{x} \]

Rewrite as an amplitude \( \times \) a function sharply peaked at \( \omega_{fi}=0 \). The peak height scales as \( t^2 \); its width in \( \omega_{fi} \) scales as \( 1/t \). B

\[ \int_{-\infty}^{\infty} \frac{4\sin^2(\omega t/2)}{\omega^{2}}\,\mathrm{d}\omega = 2\pi t \quad\Longrightarrow\quad \frac{4\sin^2(\omega_{fi} t/2)}{\omega_{fi}^{2}} \;\xrightarrow[t\ \text{large}]{}\; 2\pi\, t\,\delta(\omega_{fi}) \]

The peaked function is a nascent Dirac delta: its integral is \( 2\pi t \) and its width shrinks as \( 1/t \), so for large \( t \) it acts as \( 2\pi t\,\delta(\omega_{fi}) \) inside a smooth integral. This is the mathematical heart of energy conservation. C

\[ P_{i\to\{f\}}(t) = \sum_{f} P_{i\to f}(t) \;\longrightarrow\; \int \mathrm{d}E_f\,\rho(E_f)\, P_{i\to f}(t) \]

Because the final states form a dense continuum, replace the sum over discrete \( f \) by an energy integral weighted by the density of states \( \rho(E_f) \) (prior result: density of states of a 3D box). B

\[ P_{i\to\{f\}}(t) = \int \mathrm{d}E_f\,\rho(E_f)\,\frac{|V_{fi}|^2}{\hbar^2}\,\frac{4\sin^2\!\big((E_f-E_i)t/2\hbar\big)}{\big((E_f-E_i)/\hbar\big)^{2}} \]

Substitute \( \omega_{fi} = (E_f-E_i)/\hbar \). The sinc-squared factor confines the integrand to \( |E_f-E_i|\lesssim \hbar/t \), a window that narrows as \( t \) grows. C

\[ P_{i\to\{f\}}(t) \approx |V_{fi}|^2\,\rho(E_i)\,\frac{1}{\hbar^2}\int_{-\infty}^{\infty} \frac{4\sin^2(\omega_{fi} t/2)}{\omega_{fi}^{2}}\,\hbar\,\mathrm{d}\omega_{fi} = \frac{2\pi}{\hbar}\,|V_{fi}|^2\,\rho(E_i)\, t \]

Pull the slowly varying \( |V_{fi}|^2 \) and \( \rho \) out of the integral at \( E_f=E_i \), change variables \( \mathrm{d}E_f = \hbar\,\mathrm{d}\omega_{fi} \), and use \( \int 4\sin^2(\omega t/2)/\omega^2\,\mathrm{d}\omega = 2\pi t \). The probability now grows linearly in \( t \). C

\[ \Gamma_{i\to f} = \frac{\mathrm{d}}{\mathrm{d}t}\,P_{i\to\{f\}}(t) = \frac{2\pi}{\hbar}\,|V_{fi}|^2\,\rho(E_i) \]

Because \( P \propto t \), the transition rate is the constant time-derivative — a genuine, time-independent decay rate. Writing \( M_{fi}\equiv V_{fi}=\langle f|\hat V|i\rangle \) gives the Golden Rule. B

Result

\[ \boxed{\;\Gamma_{i\to f} = \frac{2\pi}{\hbar}\,\big|\langle f|\hat V|i\rangle\big|^{2}\,\rho(E_f)\;}\qquad (E_f=E_i) \]

Reading. The decay rate factorises into two independent pieces. The matrix element \( |M_{fi}|^2 \) measures how strongly the perturbation couples the initial state to a final state — the "how hard you push." The density of states \( \rho(E_f) \) counts how many final states are energetically available to receive the transition — the "how much room there is to go." Energy is conserved (\( E_f=E_i \)) not by fiat but as the large-time limit of the sinc-squared resonance. A big rate needs both a strong coupling and a plentiful continuum.

Units check. \( [\hat V] \) is energy \( = \mathrm{J} \); \( [\rho] = \mathrm{J^{-1}} \) (states per joule); \( [\hbar] = \mathrm{J\,s} \). Then \( \dfrac{1}{[\hbar]}\,[\hat V]^2\,[\rho] = \dfrac{1}{\mathrm{J\,s}}\cdot \mathrm{J^2}\cdot \mathrm{J^{-1}} = \dfrac{1}{\mathrm s} = \mathrm{s^{-1}} \), a rate. The dimensionless \( 2\pi \) does not affect units. Correct.

Limiting cases

Short times (\( t \ll \hbar/\Delta E \)): before the sinc resolves into a delta, \( P\propto t^2 \) not \( t \); there is no constant rate (quantum Zeno regime).
Harmonic perturbation \( \hat V(t)=\hat V_0\cos\omega t \): the same steps give \( \Gamma = \dfrac{2\pi}{\hbar}\,\tfrac14|V_0|^2\,\rho(E_i\pm\hbar\omega) \), the energy-conservation delta shifting to \( E_f = E_i\pm\hbar\omega \) — absorption and stimulated emission.
Discrete final state (\( \rho\to\delta \)-like, single level): the continuum step fails and one recovers two-level Rabi oscillations \( P=\sin^2(|V_{fi}|t/\hbar) \).
Degenerate final states: \( \rho(E_f) \) simply counts the multiplicity available at \( E_i \); the rate scales linearly with that number.
Vanishing matrix element (\( M_{fi}=0 \), selection rule): first-order rate is zero; the leading transition must be sought at second order or via higher multipoles.

Breaks when

Strong coupling / long times. Once the initial state depletes appreciably (\( P \) approaching order unity), first-order theory is invalid: the linear-in-\( t \) growth would exceed 1. The correct behaviour is exponential decay \( P_i(t)=e^{-\Gamma t} \) (Weisskopf–Wigner), of which the Golden Rule gives only the initial slope \( \Gamma \).
Discrete or sparse final spectrum. If final states are separated by more than \( \hbar/t \), the sum cannot be turned into an integral over a smooth \( \rho \); the amplitude instead oscillates (Rabi flopping) and there is no monotonic decay.
Rapidly varying \( M_{fi} \) or \( \rho(E) \) near resonance. If either changes sharply across the width \( \hbar/t \), it cannot be pulled outside the integral; the factorised form and even energy conservation acquire corrections (line shifts and asymmetric line shapes).
Very short times. In the Zeno window \( P\propto t^2 \); repeated measurement can freeze the transition, contradicting a constant rate.

Failure modes

Forgetting the density of states. Writing \( \Gamma = \frac{2\pi}{\hbar}|M|^2 \) alone — dimensionally wrong (units of energy, not rate) and physically meaningless without the available phase space.
Using \( \rho \) at the wrong energy. \( \rho \) must be evaluated at \( E_f=E_i \) (or \( E_i\pm\hbar\omega \) for a harmonic drive), not at some arbitrary or average energy.
Squaring after summing. Computing \( \big|\sum_f V_{fi}\big|^2 \) instead of \( \sum_f |V_{fi}|^2 \). Transitions to distinct final states are independent channels; probabilities add, not amplitudes.
Double-counting the factor of 4 for a cosine drive. For \( \hat V=\hat V_0\cos\omega t \) the effective coupling is \( V_0/2 \), giving \( |V_0|^2/4 \); students often drop the \( 1/4 \) or the \( 1/2 \).
Treating \( t^2 \) growth as the rate. Reading the short-time \( P\propto t^2 \) as evidence of a \( t \)-dependent rate, rather than waiting for the linear regime that defines \( \Gamma \).
Confusing \( |i\rangle,|f\rangle \) energies. Using \( \omega_{fi}=(E_i-E_f)/\hbar \) with the wrong sign, which flips absorption and emission.

Discussion

The most striking feature of the derivation is the appearance of a Dirac delta enforcing energy conservation. It is not imposed by hand: it emerges as the \( t\to\infty \) limit of the sinc-squared resonance, whose width \( \Delta E \sim \hbar/t \) shrinks with elapsed time. This is precisely a time–energy uncertainty statement: to resolve energy conservation to precision \( \Delta E \) one must wait a time \( t\sim\hbar/\Delta E \). For times shorter than this the "final energy" is genuinely fuzzy and transitions to nominally non-conserving states are allowed — the origin of virtual processes and of natural linewidth.

Equally important is the transition from \( t^2 \) to \( t \). A single isolated final state receives probability quadratically in time, the hallmark of coherent, reversible evolution. It is only the sum over a continuum of final states, each dephasing at its own frequency \( \omega_{fi} \), that converts coherent \( t^2 \) growth into incoherent linear-in-\( t \) accumulation — and hence a constant rate. Irreversibility here is not fundamental; it is destructive interference among infinitely many reversible channels. Recurrences would eventually return probability to \( |i\rangle \), but for a true continuum the Poincaré recurrence time is effectively infinite.

The Golden Rule is only the leading term of a fuller story. Summing the geometric series of self-energy corrections (Weisskopf–Wigner / resolvent methods) replaces the linear growth with genuine exponential decay \( e^{-\Gamma t} \) and produces a Lorentzian line of full width \( \hbar\Gamma \), together with a small energy shift (the analogue of the Lamb shift). The Golden Rule \( \Gamma \) is the imaginary part of the second-order self-energy \( \Sigma(E) \) evaluated on-shell, \( \Gamma = -\frac{2}{\hbar}\,\mathrm{Im}\,\Sigma(E_i) \), while its real part gives the level shift. Seen this way, the rule is the pole structure of the exact propagator, linearised.

Common misconceptions. The rule does not say energy is exactly conserved at all times — only in the long-time limit; on short timescales the energy window is \( \sim\hbar/t \) wide. It is not restricted to "golden" or special situations — the name is Fermi's own ironic label for a formula he found so useful. And \( \Gamma \) is the initial decay rate, i.e. the slope at \( t=0 \) of the true exponential; it is not literally the constant probability-per-second forever, since \( P_i \) must eventually curve over to preserve \( P_i\le 1 \).

Worked examples

Estimate the spontaneous-emission rate of the hydrogen \( 2p\to 1s \) transition treating the electromagnetic vacuum as the continuum of photon final states.

\[ \Gamma = \frac{\omega_0^3\,|\langle 1s|\,e\,\hat{\mathbf r}\,|2p\rangle|^2}{3\pi\varepsilon_0\hbar c^3} \]

This is Fermi's rule specialised to the dipole coupling \( \hat V = -\hat{\mathbf d}\cdot\hat{\mathbf E} \) with \( \rho \) the photon density of states; a standard reduction. Symbols first. B

\[ \hbar\omega_0 = 10.2\ \mathrm{eV} \;\Rightarrow\; \omega_0 = \frac{10.2\times1.602\times10^{-19}}{1.055\times10^{-34}} = 1.55\times10^{16}\ \mathrm{s^{-1}} \]

Convert the \( 2p\!-\!1s \) energy gap to angular frequency. A

\[ |\langle 1s|e\,\hat{\mathbf r}|2p\rangle|^2 \approx (0.74\,e\,a_0)^2,\quad a_0=5.29\times10^{-11}\ \mathrm m,\ e=1.602\times10^{-19}\ \mathrm C \]

Standard hydrogen dipole matrix element for \( 2p\to1s \) (radial integral \( \approx 0.74\,a_0 \), angular factors absorbed). B

\[ \Gamma = \frac{(1.55\times10^{16})^3\,(0.74\cdot1.602\times10^{-19}\cdot5.29\times10^{-11})^2}{3\pi\,(8.854\times10^{-12})(1.055\times10^{-34})(3\times10^8)^3} \approx 6\times10^{8}\ \mathrm{s^{-1}} \]

Insert numbers with SI units throughout. A

\[ \Gamma \approx 6\times10^{8}\ \mathrm{s^{-1}} \;\Rightarrow\; \tau = 1/\Gamma \approx 1.6\ \mathrm{ns} \]

Reading. The computed lifetime matches the measured \( 2p \) lifetime of \( 1.6\ \mathrm{ns} \) — a triumph of the Golden Rule plus the photon density of states.

Units check. \( \mathrm{s^{-3}\cdot C^2 m^2}\big/\big(\mathrm{F\,m^{-1}\cdot J\,s\cdot m^3 s^{-3}}\big) \) reduces to \( \mathrm{s^{-1}} \). Correct.

A particle in a 1D infinite well of width \( L=1.0\ \mathrm{nm} \), initially in the ground state, is subjected at \( t=0 \) to a weak constant perturbation with matrix element \( |M_{fi}| = 2.0\times10^{-3}\ \mathrm{eV} \) coupling it to a band of final states of density \( \rho = 5.0\ \mathrm{states/eV} \). Find the transition rate.

\[ \Gamma = \frac{2\pi}{\hbar}\,|M_{fi}|^2\,\rho \]

Direct application of the Golden Rule. Symbols before numbers. A

\[ |M_{fi}|^2 = (2.0\times10^{-3}\ \mathrm{eV})^2 = 4.0\times10^{-6}\ \mathrm{eV^2}, \quad \rho = 5.0\ \mathrm{eV^{-1}} \]

Assemble the squared matrix element and density of states in consistent eV units. A

\[ \Gamma = \frac{2\pi}{\hbar}\,(4.0\times10^{-6}\ \mathrm{eV^2})(5.0\ \mathrm{eV^{-1}}) = \frac{2\pi}{\hbar}\,(2.0\times10^{-5}\ \mathrm{eV}) \]

Multiply; note \( \mathrm{eV^2\cdot eV^{-1}=eV} \), leaving one factor of energy to divide by \( \hbar \). A

\[ \frac{2.0\times10^{-5}\ \mathrm{eV}}{\hbar} = \frac{2.0\times10^{-5}\times1.602\times10^{-19}\ \mathrm J}{1.055\times10^{-34}\ \mathrm{J\,s}} = 3.04\times10^{10}\ \mathrm{s^{-1}} \]

Convert eV to joules and divide by \( \hbar \) in SI. A

\[ \Gamma = 2\pi\times 3.04\times10^{10}\ \mathrm{s^{-1}} \approx 1.9\times10^{11}\ \mathrm{s^{-1}} \]

Reading. The state decays with lifetime \( \tau=1/\Gamma\approx 5.2\ \mathrm{ps} \). The well width \( L \) entered only through \( M_{fi} \) and \( \rho \), which were given; the rate itself needs just those two numbers.

Units check. \( \mathrm{eV^2\cdot eV^{-1}/(J\,s)} \to \mathrm{eV/(J\,s)} \to \mathrm{s^{-1}} \) after the eV\(\to\)J conversion. Correct.

Problems

State the two physical quantities that the Golden Rule multiplies together, give the SI units of each, and show that their product with \( 1/\hbar \) has units of \( \mathrm{s^{-1}} \).
Solution
The two quantities are the squared coupling matrix element \( |M_{fi}|^2 \) (units \( \mathrm{J^2} \)) and the density of final states \( \rho(E_f) \) (units \( \mathrm{J^{-1}} \)). With \( [\hbar]=\mathrm{J\,s} \): \( \frac{1}{\mathrm{J\,s}}\cdot\mathrm{J^2}\cdot\mathrm{J^{-1}} = \frac{\mathrm{J}}{\mathrm{J\,s}} = \mathrm{s^{-1}} \). The dimensionless \( 2\pi \) is irrelevant to units, so \( \Gamma \) is a rate.
A perturbation couples \( |i\rangle \) to a continuum with \( |M_{fi}|=1.0\times10^{-4}\ \mathrm{eV} \) and \( \rho=20\ \mathrm{eV^{-1}} \). Compute \( \Gamma \) and the lifetime \( \tau \).
Solution
\( |M_{fi}|^2 = 1.0\times10^{-8}\ \mathrm{eV^2} \). Then \( |M_{fi}|^2\rho = 2.0\times10^{-7}\ \mathrm{eV} \). Convert: \( 2.0\times10^{-7}\times1.602\times10^{-19}=3.20\times10^{-26}\ \mathrm J \). Divide by \( \hbar \): \( 3.20\times10^{-26}/1.055\times10^{-34}=3.04\times10^{8}\ \mathrm{s^{-1}} \). Multiply by \( 2\pi \): \( \Gamma = 1.9\times10^{9}\ \mathrm{s^{-1}} \). Lifetime \( \tau = 1/\Gamma = 5.3\times10^{-10}\ \mathrm s = 0.53\ \mathrm{ns} \).
Show explicitly, starting from \( P_{i\to f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2} \), that for a single final state the short-time probability grows as \( t^2 \). Why does this not contradict a constant decay rate?
Solution
For small argument, \( \sin(\omega_{fi}t/2)\approx \omega_{fi}t/2 \), so \( \frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2}\approx \frac{4(\omega_{fi}t/2)^2}{\omega_{fi}^2}=t^2 \). Hence \( P_{i\to f}\approx \frac{|V_{fi}|^2}{\hbar^2}t^2 \), quadratic in \( t \). This is the coherent, reversible growth to a single state. A constant rate only appears after summing over a continuum: the differing frequencies \( \omega_{fi} \) dephase, converting the coherent \( t^2 \) into incoherent linear-in-\( t \) accumulation. So there is no contradiction — the \( t^2 \) law holds per state, the linear law holds for the total into a continuum.
For a harmonic perturbation \( \hat V(t)=\hat V_0\cos\omega t \), state the modified Golden Rule and explain the sign of the energy shift in the delta function for absorption versus stimulated emission.
Solution
Writing \( \cos\omega t = \tfrac12(e^{i\omega t}+e^{-i\omega t}) \), the integral in step 3 produces two resonances. The result is \( \Gamma = \frac{2\pi}{\hbar}\,\tfrac14|V_{0,fi}|^2\,\rho(E_f) \) with energy conservation \( E_f = E_i + \hbar\omega \) (absorption: system gains a quantum \( \hbar\omega \)) or \( E_f = E_i - \hbar\omega \) (stimulated emission: system loses \( \hbar\omega \)). The factor \( \tfrac14 = (\tfrac12)^2 \) comes from squaring the \( \tfrac12 \) in the cosine decomposition. Which resonance is active depends on whether final states exist above (\( +\hbar\omega \)) or below (\( -\hbar\omega \)) the initial energy.
The Golden Rule predicts \( P_{i\to\{f\}}(t)=\Gamma t \), which exceeds 1 for \( t>1/\Gamma \). Explain the resolution and write the corrected long-time behaviour, identifying what role \( \Gamma \) plays in it.
Solution
The linear growth is only the first-order approximation, valid while the initial state is barely depleted (\( \Gamma t \ll 1 \)). Once depletion matters, back-transitions and higher orders must be included. Resumming (Weisskopf–Wigner) gives exponential decay \( P_i(t)=e^{-\Gamma t} \), so \( P_{i\to\{f\}}(t)=1-e^{-\Gamma t} \), which stays \( \le 1 \). Expanding for small \( t \): \( 1-e^{-\Gamma t}\approx \Gamma t \), recovering the Golden Rule. Thus \( \Gamma \) is the initial slope of the true exponential decay — the instantaneous rate at \( t=0 \), not a literal constant probability-per-second forever. The corresponding spectral line is a Lorentzian of full width \( \hbar\Gamma \).