Fermi Golden Rule

Purpose

Fermi’s golden rule converts a weak first-order transition amplitude into a rate when the final spectrum is continuous or sufficiently dense. In a state-sum convention,

$$\Gamma_i = \frac{2\pi}{\hbar} \sum_f \left\lvert V_{fi}\right\rvert^2 \delta(E_f-E_i),$$

where

$$V_{fi} = \langle f\rvert V\lvert i\rangle.$$

If final states are counted by an energy density $\rho_f(E)$ and the matrix element is smooth near resonance,

$$\Gamma_i = \frac{2\pi}{\hbar} \left\lvert V_{fi}\right\rvert^2 \rho_f(E_i).$$

The delta function is a long-time representation of a finite-width energy window. It is not exact at finite observation time. The canonical derivation and physical discussion are at Fermi’s Golden Rule.

At a glance

Task	Formula or rule
State-sum rate	$\Gamma_i=(2\pi/\hbar)\sum_f\lvert V_{fi}\rvert^2\delta(E_f-E_i)$
Density-of-states form	$\Gamma_i=(2\pi/\hbar)\lvert V(E_i)\rvert^2\rho_f(E_i)$
Absorption from $Fe^{-i\omega t}$	$\delta(E_f-E_i-\hbar\omega)$
Emission from $F^\dagger e^{i\omega t}$	$\delta(E_f-E_i+\hbar\omega)$
Total rate	$\Gamma_i=\sum_a\Gamma_{i\to a}$
Branching fraction	$B_a=\Gamma_{i\to a}/\Gamma_i$
Lifetime	$\tau=1/\Gamma_i$ in the constant-rate approximation
Finite-time energy resolution	$\Delta E\sim\hbar/T$

Finite-time starting point

Suppose a time-independent perturbation $V$ is switched on at $t=0$ and held for time $T$. First-order interaction-picture perturbation theory gives

$$\begin{aligned} c_f^{(1)}(T) &= -\frac{i}{\hbar} V_{fi} \int_0^T e^{i\omega_{fi}t},dt \\ &= -\frac{i}{\hbar} V_{fi} e^{i\omega_{fi}T/2} \frac{2\sin(\omega_{fi}T/2)}{\omega_{fi}}, \end{aligned}$$

where

$$\omega_{fi} = \frac{E_f-E_i}{\hbar}.$$

For one discrete final state,

$$P_{i\to f}^{(1)}(T) = \frac{4\lvert V_{fi}\rvert^2}{\hbar^2} \frac{ \sin^2(\omega_{fi}T/2) }{ \omega_{fi}^2 }.$$

This probability is oscillatory and has a central energy width of order $\hbar/T$. It is not itself an irreversible rate. A rate emerges only after summing over many final states whose spacing is small compared with the finite-time energy window.

For a general weak pulse $V(t)$,

$$c_f^{(1)} = -\frac{i}{\hbar} \int dt\, e^{i\omega_{fi}t} V_{fi}(t).$$

The transition spectrum is the squared Fourier transform of the pulse envelope. The delta-function rule is the special long-duration limit.

Long-time continuum limit

As a distribution,

$$\frac{4\sin^2(\omega T/2)}{\omega^2} \longrightarrow 2\pi T\,\delta(\omega)$$

when integrated against a function that varies slowly over the peak. Since

$$\delta(\omega_{fi}) = \hbar\delta(E_f-E_i),$$

the probability per unit time becomes

$$\frac{P_i(T)}{T} \longrightarrow \frac{2\pi}{\hbar} \sum_f \lvert V_{fi}\rvert^2 \delta(E_f-E_i).$$

Energy conservation therefore emerges from phase accumulation over a long observation time. A shorter pulse has a broader spectral window; replacing it by an exact delta function loses that information.

Density of states and normalization

The density $\rho_f(E)$ counts final states per unit energy after all other labels included in the chosen channel have been summed or integrated. With box-normalized states,

$$\sum_f \longrightarrow \int dE\,\rho_f(E),$$

and

$$\Gamma_i = \frac{2\pi}{\hbar} \int dE\, \rho_f(E) \lvert V_{fi}(E)\rvert^2 \delta(E-E_i).$$

The units check is

$$\left[ \frac{2\pi}{\hbar} \lvert V\rvert^2 \rho(E) \right] = \mathsf T^{-1}.$$

For continuum states normalized directly to $\delta(E-E')$, the density factor may be absorbed into the state normalization and matrix element. Do not append a separate $\rho(E)$ without checking how the continuum ket is normalized.

Degeneracy factors belong in either the explicit final-state sum or the density of states, not both. Initial degeneracy is handled differently: average over unresolved, incoherently populated initial states only when the preparation warrants that average.

Use Density of States in Transition Rates for energy-shell Jacobians, volume cancellation, and normalization examples.

Harmonic perturbations

Write a Hermitian harmonic perturbation as

$$V(t) = Fe^{-i\omega t} + F^\dagger e^{i\omega t}.$$

The $Fe^{-i\omega t}$ component drives transitions satisfying

$$E_f-E_i = \hbar\omega,$$

with rate

$$\Gamma_{i\to f}^{(+)} = \frac{2\pi}{\hbar} \left\lvert \langle f\rvert F\lvert i\rangle \right\rvert^2 \delta(E_f-E_i-\hbar\omega).$$

The $F^\dagger e^{i\omega t}$ component drives

$$E_f-E_i = -\hbar\omega,$$

with

$$\Gamma_{i\to f}^{(-)} = \frac{2\pi}{\hbar} \left\lvert \langle f\rvert F^\dagger\lvert i\rangle \right\rvert^2 \delta(E_f-E_i+\hbar\omega).$$

For $V(t)=V_0\cos\omega t$, each Fourier component is $V_0/2$. The factor $1/2$ must be squared in the rate. Calling one sign “absorption” or “emission” assumes $\omega>0$ and refers to energy gained or lost by the quantum system.

Use Harmonic Perturbations for finite-duration drives and rotating-wave choices.

Channels, differential rates, and branching

If final states have channel label $a$ and continuous variables $\xi$, write

$$d\Gamma_{i\to a} = \frac{2\pi}{\hbar} \left\lvert M_{ai}(\xi) \right\rvert^2 \delta(E_a(\xi)-E_i) d\mu_a(\xi),$$

where $d\mu_a$ is the state-counting measure in the same normalization as the matrix element. The partial and total rates are

$$\Gamma_{i\to a} = \int d\Gamma_{i\to a}, \qquad \Gamma_i = \sum_a\Gamma_{i\to a}.$$

The branching fraction is

$$B_a = \frac{\Gamma_{i\to a}}{\Gamma_i}, \qquad \sum_aB_a=1$$

when the listed channels exhaust the decay. Selection rules can force a partial matrix element to zero, while phase space can suppress a channel even when its matrix element is allowed.

For identical final particles, avoid counting permutations as distinct states when the normalized final-state basis already implements particle symmetry.

Initial mixtures and degeneracies

For an incoherent initial ensemble

$$\rho_i = \sum_r p_r\lvert i_r\rangle\langle i_r\rvert,$$

the observed rate is

$$\overline\Gamma = \sum_r p_r\Gamma_{i_r}.$$

For an equally populated unresolved initial multiplet of degeneracy $g_i$,

$$\overline\Gamma = \frac{1}{g_i} \sum_{r=1}^{g_i} \Gamma_{i_r}.$$

Final states are summed, not averaged. A coherent initial superposition can contain interference terms and must be treated at the amplitude or density- matrix level rather than by averaging separate rates.

Time-scale window

A golden-rule regime requires a nonempty intermediate-time window. Schematically,

$$\tau_{\mathrm{micro}} \ll T \ll \min \left( \Gamma_i^{-1}, t_{\mathrm{rec}} \right).$$

The lower bound ensures that the energy window $\hbar/T$ is narrow compared with the scale on which the matrix element and density of states vary. The upper bounds ensure negligible depletion in first-order perturbation theory and no finite-spectrum recurrence.

At the same time, the window must contain many final levels. If the local level spacing is $\delta E$, one needs roughly

$$\delta E \ll \frac{\hbar}{T}.$$

These requirements explain why simply taking $T\to\infty$ in a finite isolated system is not the physical golden-rule limit.

At very short times, survival probability is generically quadratic rather than exponential. At later times, a constant golden-rule rate can seed the resummed approximation

$$P_i(t) \simeq e^{-\Gamma_i t},$$

but this exponential is not produced by retaining only the first-order transition probability indefinitely.

Lifetime and linewidth

If a single constant total rate controls depletion,

$$\tau = \frac{1}{\Gamma_i}.$$

The associated energy scale is of order

$$\Delta E \sim \hbar\Gamma_i.$$

Exact factors depend on whether a quoted linewidth means full width at half maximum, half width, amplitude decay, or population decay. State the line-shape convention before identifying a numerical linewidth with $\hbar\Gamma$.

Finite lifetimes broaden the ideal delta function. A Lorentzian replacement usually comes from resumming self-energy or damping effects, not from the bare golden-rule formula by itself.

Selection rules and many-body factors

The matrix element carries the operator’s symmetry selection rules. A vanishing $V_{fi}$ removes the leading process even if the density of states is large. An allowed matrix element can still be small because of radial overlap, polarization, destructive interference, or weak coupling.

In many-body systems, occupation factors can modify the available phase space: Pauli blocking suppresses occupied fermionic final states, while bosonic stimulation can enhance already occupied modes. These are additions to the state sum, not automatic factors in the single-particle rule.

Use Selection Rules in Transition Rates for the symmetry-to-rate workflow.

When the rule fails

• A single isolated resonant final state produces coherent oscillations rather than irreversible linear growth.
• Strong coupling invalidates first-order amplitudes and can split or dress the levels.
• Very short times retain the full pulse spectrum and quadratic survival law.
• Late times show depletion, backaction, memory, or recurrence.
• A rapidly varying density of states, threshold, band edge, or narrow resonance defeats the smooth-continuum approximation.
• Structured environments can produce non-Markovian decay.
• An on-shell scattering or decay process may require a resummed $T$-matrix rather than the bare perturbation $V$.

Calculation workflow

1. Specify the time dependence and extract its Fourier components.
2. State the normalization of initial and final states.
3. Compute the operator matrix element and apply selection rules.
4. Construct the final-state measure or density without double-counting degeneracies.
5. impose the correct energy delta function for the chosen Fourier component.
6. Sum or integrate all final channels and average only over genuinely unresolved initial mixtures.
7. Check units and the intermediate-time hierarchy.
8. Verify that depletion, coherent recurrence, and structure in the continuum remain negligible on the time scale used.

Common mistakes

• Applying the rate formula to one isolated final level.
• Treating the finite-time energy window as an exact delta function.
• Mixing a density of states with continuum-normalized matrix elements that already contain it.
• Summing a degeneracy in $\rho(E)$ and multiplying by it again.
• Averaging over final states instead of summing them.
• Forgetting the $1/2$ Fourier amplitude in a cosine drive.
• Reversing the absorption and emission delta-function signs.
• Extending first-order linear probability growth past substantial depletion.
• Equating “allowed” with “large.”
• Using a bare $V_{fi}$ where repeated scattering requires an on-shell $T$-matrix.

Canonical links

• First-Order Transition Probability gives the finite-time amplitude before the continuum limit.
• Transition Rates in Light–Matter Interaction applies harmonic driving, polarization, and photon density of states.
• Linear Response Preview connects weak driving to response functions.
• Inelastic Scattering Preview replaces the bare perturbation by channel scattering amplitudes when needed.
• Two-Level System shows the coherent discrete-state alternative.

References

• E. Fermi, Nuclear Physics, University of Chicago Press, 1950.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.

Exercises

1. Starting from the finite-time probability, recover the factor $2\pi/\hbar$ in the continuum rate.

Solution

Use

$$P_{i\to f}^{(1)}(T) = \frac{\lvert V_{fi}\rvert^2}{\hbar^2} \frac{4\sin^2(\omega_{fi}T/2)}{\omega_{fi}^2}.$$

In the long-time distribution limit,

$$\frac{4\sin^2(\omega T/2)}{\omega^2} \longrightarrow 2\pi T\delta(\omega).$$

Since $\delta(\omega_{fi})=\hbar\delta(E_f-E_i)$,

$$\frac{P_i(T)}{T} = \frac{2\pi}{\hbar} \sum_f \lvert V_{fi}\rvert^2 \delta(E_f-E_i).$$

2. A perturbation is $V(t)=V_0\cos\omega t$. Write the golden-rule absorption rate from $\lvert i\rangle$ to a higher-energy state $\lvert f\rangle$.

Solution

Decompose

$$V(t) = \frac{V_0}{2}e^{-i\omega t} + \frac{V_0}{2}e^{i\omega t}.$$

Absorption by the system uses the $e^{-i\omega t}$ component and satisfies $E_f-E_i=\hbar\omega$. Therefore

$$\Gamma_{i\to f}^{\mathrm{abs}} = \frac{2\pi}{\hbar} \frac{ \left\lvert \langle f\rvert V_0\lvert i\rangle \right\rvert^2 }{4} \delta(E_f-E_i-\hbar\omega).$$

The factor $1/4$ is the square of the Fourier-component factor $1/2$.

3. An initial multiplet has three equally populated unresolved states. Their total rates are $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$. Two disjoint final channels have partial rates $\Gamma_{rA}$ and $\Gamma_{rB}$ for each initial state $r$. Write the observed total rate and channel-$A$ branching fraction.

Solution

For each initial state,

$$\Gamma_r = \Gamma_{rA}+\Gamma_{rB}.$$

Equal incoherent populations give

$$\overline\Gamma = \frac13 \sum_{r=1}^{3} \left( \Gamma_{rA}+\Gamma_{rB} \right).$$

The ensemble-averaged partial rate into $A$ is

$$\overline\Gamma_A = \frac13 \sum_{r=1}^{3}\Gamma_{rA}.$$

Therefore

$$B_A = \frac{\overline\Gamma_A}{\overline\Gamma}.$$

Final channels are summed; only the incoherent initial preparation is averaged.

4. Explain why the inequalities $\tau_{\mathrm{micro}}\ll T\ll\Gamma^{-1}$ can both be necessary.

Solution

The lower inequality makes the finite-time energy peak narrow enough that the matrix element and density of states are approximately constant across it. It supports the delta-function and energy-conservation approximation.

The upper inequality keeps the transition probability small:

$$\Gamma T\ll1.$$

Then first-order perturbation theory can neglect depletion of the initial state. The golden-rule window exists only when the weak coupling makes $\Gamma^{-1}$ much longer than the microscopic correlation time.