Fermi's Golden Rule

Fermi’s golden rule is the standard transition-rate formula for weak perturbations into a continuum or effectively dense set of final states. In its simplest form,

$$\Gamma_{i\to\mathcal C} = \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \rho(E_f),$$

Here $\mathcal C$ is a continuum channel, $\rho(E_f)$ counts its box-normalized final states per energy, and the matrix element is representative of those states on the selected energy shell. A single isolated final state has a probability, not a constant golden-rule rate.

Required background. First-Order Transition Probability gives the finite-time integral. Density of States explains continuum normalization and channel counting.

The rule is not just “first-order perturbation theory.” It also assumes a time-scale hierarchy and a continuum of final states.

From Probability to Rate

For a constant perturbation switched on for time $T$, first-order perturbation theory gives

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

where

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

For a single discrete final state, this is a finite-time probability. To obtain a rate, sum or integrate over many final states.

Continuum Limit

Suppose final states are labeled by energy and have density $\rho(E_f)$. The total transition probability is schematically

$$P_i(T) = \int dE_f\, \rho(E_f) \frac{4\lvert V_{fi}\rvert^2}{\hbar^2} \frac{\sin^2[(E_f-E_i)T/(2\hbar)]} {[(E_f-E_i)/\hbar]^2}.$$

For long times, the sinc-squared factor becomes sharply peaked around $E_f=E_i$. The precise distributional statement is that the following normalized kernel tends to a delta function when integrated against a smooth test function:

$$\frac{1}{T}\frac{4\sin^2(\omega T/2)}{\omega^2} \longrightarrow 2\pi\,\delta(\omega).$$

Since

$$\delta(\omega) = \hbar\,\delta(E_f-E_i),$$

the probability grows approximately linearly:

$$P_i(T) \approx \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \rho(E_i)\,T.$$

The rate is the coefficient of $T$:

$$\Gamma \simeq \frac{dP_i}{dT} \simeq \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \rho(E_i).$$

Energy Conservation

Energy conservation in the golden rule is not imposed by hand at the beginning. It emerges from the long-time phase integral. The longer the perturbation acts, the narrower the energy window:

$$\Delta E\sim\frac{\hbar}{T}.$$

In the long-time continuum limit, the window is represented by a delta function.

This is why short pulses have broad spectral response, while long weak drives select nearly energy-conserving transitions.

Density of States

The density of final states converts a probability per final state into a total rate. Its definition depends on normalization conventions and on which quantum numbers are included.

The normalization dictionary, energy-shell Jacobians, volume cancellation, channel counting, and finite-time continuum window are developed in Density of States in Transition Rates. Here the density enters only as one ingredient of the golden-rule limit.

If several final channels are available, the rate is a sum or integral:

$$\Gamma_i = \frac{2\pi}{\hbar} \sum_\alpha \lvert V_{\alpha i}\rvert^2 \delta(E_\alpha-E_i),$$

or, after replacing a dense sum by an integral,

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

The notation hides important convention choices. Specify the state normalization and included degeneracy factors; an energy-normalized matrix element already contains the density Jacobian.

The Golden-Rule Time Window

Let $E_{\mathrm{var}}$ be the energy scale over which the weighted density $\rho(E)|V(E)|^2$ changes, and let $\delta E$ be the spacing of effectively distinct final levels in a large finite system. A useful interval is

$$\frac{\hbar}{E_{\mathrm{var}}}\ll T \ll\min\!\left(\frac{\hbar}{\delta E},\frac1\Gamma\right).$$

The lower bound permits a local energy-shell approximation. The first upper bound keeps many levels within the resolution $\hbar/T$; the second keeps depletion small. A sufficiently weak coupling and dense, smooth spectrum can separate these scales. A threshold or narrow spectral feature can destroy that separation even when individual matrix elements are small.

The notation $T\to\infty$ used to identify the kernel does not authorize taking infinite time in the truncated probability $P\simeq\Gamma T$. Exponential decay $1-e^{-\Gamma T}$ requires a further Markov or resonance resummation, whose assumptions must be checked separately.

Worked Example: A Continuum with Finite Bandwidth

The crossover from a quadratic probability to a rate can be calculated without replacing the finite-time kernel by a delta function. Consider a normalized initial state coupled to orthogonal final states with weighted density

$$\mathcal S_i(E)=\rho(E)|V(E)|^2 =\begin{cases} S_0,& |E-E_i|<B,\\ 0,& |E-E_i|>B, \end{cases} \qquad B>0.$$

Here $S_0$ has units of energy, and $2B$ is the width of the accessible band. The initial state belongs to a separate channel with its energy in the middle of this band. Take no diagonal perturbation or couplings among the final states. This deliberately simple model isolates the effects of bandwidth and depletion; it is not a universal density of states near a physical threshold.

Writing $\epsilon=E-E_i$ and $x=BT/\hbar$, the probability obtained from first-order amplitudes is

$$\begin{aligned} P_{\mathrm{band}}^{(1)}(T) &=4S_0\int_{-B}^{B} \frac{\sin^2(\epsilon T/2\hbar)}{\epsilon^2}\,d\epsilon\\ &=\frac{4S_0T}{\hbar} \left[\operatorname{Si}(x)-\frac{1-\cos x}{x}\right], \qquad \operatorname{Si}(x)=\int_0^x\frac{\sin u}{u}\,du. \end{aligned}$$

The second line follows by integrating $(1-\cos u)/u^2$ by parts. It is an exact evaluation of the leading perturbative probability, not an exact solution of the coupled dynamics.

For $x\ll1$, expand the bracket as $x/2-x^3/72+O(x^5)$:

$$P_{\mathrm{band}}^{(1)}(T) =\frac{2BS_0T^2}{\hbar^2} \left[1-\frac{x^2}{36}+O(x^4)\right].$$

The coefficient $2BS_0=\int\mathcal S_i(E)\,dE$ is the initial Hamiltonian variance in this model. At these short times, transitions throughout the band add to the quadratic escape probability.

For $x\gg1$, use $\operatorname{Si}(x)=\pi/2-\cos x/x+O(x^{-2})$. Then

$$\begin{aligned} P_{\mathrm{band}}^{(1)}(T) &=\Gamma T-\frac{4S_0}{B} +O\!\left(\frac{S_0}{Bx}\right), &\Gamma&=\frac{2\pi S_0}{\hbar},\\ \frac{dP_{\mathrm{band}}^{(1)}}{dT} &=\frac{4S_0}{\hbar}\operatorname{Si}(x) \longrightarrow\Gamma. \end{aligned}$$

The band edges cause a transient correction and oscillations of the instantaneous slope. A useful constant rate emerges after $\hbar/B$, while depletion still requires $\Gamma T\ll1$. These conditions need $S_0/B\ll1/(2\pi)$. If a finite box replaces the continuum by levels spaced by $\delta E$, one must also retain $T\ll\hbar/\delta E$; merely making each box-normalized coupling small does not remove that upper limit. This example makes the long-time, small-probability conditions in Zwiebach’s continuum derivation (§4.3.1) explicit.

Harmonic Absorption and Emission

For $V(t)=We^{-i\omega t}+W^\dagger e^{i\omega t}$ with $\omega>0$, resolved frequency components give

$$\begin{aligned} \Gamma_i^{\mathrm{abs}}&=\frac{2\pi}{\hbar}\sum_f|W_{fi}|^2 \delta(E_f-E_i-\hbar\omega),\\ \Gamma_i^{\mathrm{stim}}&=\frac{2\pi}{\hbar}\sum_f|(W^\dagger)_{fi}|^2 \delta(E_f-E_i+\hbar\omega). \end{aligned}$$

The sums represent a continuum limit. For a real cosine drive $A\cos\omega t$, use $W=A/2$; inserting the peak operator $A$ directly would overestimate either rate by four. A prescribed classical field produces stimulated transitions, not vacuum spontaneous emission.

Applications

The golden rule appears in absorption, emission, decay, tunneling into a continuum, scattering previews, and transition rates induced by weak noise or weak driving.

In spectroscopy, matrix elements encode selection rules and coupling strength. The density of states encodes how many final states are available. Both are needed.

In scattering, golden-rule logic foreshadows the relation between transition probabilities, flux, and cross sections. The full scattering formalism has additional normalization and boundary-condition structure.

Inelastic Scattering Preview makes that bridge explicit for channel-changing collisions and shows when the bare matrix element must be replaced by the on-shell $T$-matrix.

Bridge to Quantum Field Theory

In quantum field theory, decay rates and cross sections are computed from transition amplitudes with relativistic normalization and phase space. The golden-rule structure survives, but the density of states becomes Lorentz-invariant phase space and matrix elements are produced by field-theoretic interactions.

This page gives the nonrelativistic quantum-mechanical origin of that logic.

Exercises

Find a useful observation time

For the finite-band model, let $S_0/B=10^{-4}$ and $\delta E/B=10^{-3}$. Express the three time-scale conditions in terms of $x=BT/\hbar$. At $x=50$, compare the finite-band probability with $\Gamma T$. Would $x=5000$ improve the controlled approximation?

Solution

The conditions are

$$1\ll x\ll\min\!\left(10^3,\frac{10^4}{2\pi}\right).$$

At $x=50$,

$$P_{\mathrm{band}}^{(1)}\simeq0.0310183, \qquad \Gamma T\simeq0.0314159.$$

The finite-band result is about $1.27\%$ below the rate estimate, and the leading probability is about $3.1\%$. These numbers quantify the bandwidth correction and the depletion scale separately; they do not bound all higher-order corrections. At $x=5000$, the finite-size condition fails and $\Gamma T=\pi>1$. The delta-kernel approximation alone cannot justify that time.

Put the initial energy at a sharp band edge

Instead take $\mathcal S_i(E)=S_0$ for $0<E-E_i<B$, with zero weight elsewhere. Evaluate the finite-time probability and its large-$x$ slope. Why is substituting an assigned value of $\mathcal S_i(E_i)$ into the usual rule insufficient?

Solution

The finite-time kernel is even in $E-E_i$. Only one half of the symmetric band remains, so

$$P_{\mathrm{edge}}^{(1)}(T) =\frac12P_{\mathrm{band}}^{(1)}(T), \qquad \frac{dP_{\mathrm{edge}}^{(1)}}{dT} \longrightarrow\frac{\pi S_0}{\hbar}.$$

Changing the density’s value at the single endpoint cannot change the finite-time integral. The symmetric kernel samples both sides of the discontinuity, giving half the interior-band rate. The smoothness assumption fails exactly on the shell; the finite-time integral resolves the ambiguity. This factor of one half belongs to this abrupt flat-band model, not to every threshold, where the weighted density may instead vanish or diverge. The large-$x$ statement still concerns the leading perturbative probability before depletion becomes significant.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• E. Fermi, Nuclear Physics, University of Chicago Press, 1950.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• 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.
• B. Zwiebach, “Time Dependent Perturbation Theory,” §4.3.1, Quantum Physics III, MIT OpenCourseWare 8.06, Spring 2018, pp. 90–94. Continuum integration and the simultaneous long-time and small-probability conditions.