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,

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

where $\rho(E_f)$ is the density of final states at the energy selected by energy conservation.

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 $E_f=E_i$. In distribution form,

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

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 $T$:

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

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. A mature calculation states the state normalization and the included degeneracy factors.

Validity Conditions

Fermi’s golden rule requires:

• weak coupling, so first-order amplitudes are enough,
• many available final states or an effectively continuous spectrum,
• times long compared with microscopic oscillation times,
• times short enough that the initial state is not substantially depleted,
• slowly varying matrix elements and density of states over the energy window $\hbar/T$.

The last two conditions can compete. The golden-rule regime lives between transient short-time behavior and late-time depletion or recurrence effects.

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.

Common Mistakes

• Applying the golden rule to a single isolated final state.
• Forgetting the density of states.
• Treating the delta function as exact at finite time.
• Ignoring depletion of the initial state at late times.
• Using a rate formula when the perturbation is strong enough to produce coherent oscillations.
• Hiding normalization conventions inside $\rho(E)$ without stating them.

Cross-Links

• Time-Dependent Perturbation Theory and Transitions
• First-Order Transition Probability
• Interaction Picture
• Small Parameters and Error Estimates
• Density of States in Transition Rates
• Selection Rules in Transition Rates
• Linear Response Preview
• Transition Rates in Light–Matter Interaction
• Spectroscopy connects the continuum rate limit to line positions, strengths, widths, and measured signals.
• Transition Rates develops the spectroscopy-facing channel, degeneracy, density-of-states, and detector conventions.
• Inelastic Scattering Preview
• Symmetry Constraints on Hamiltonians
• Two-Level System
• Path Integrals

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. Laloë, Quantum Mechanics, Wiley, 1977.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.

Exercises

1. Starting from

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

derive the factor $2\pi/\hbar$ in the golden rule.

Solution

The probability contains

$$\frac{\lvert V_{fi}\rvert^2}{\hbar^2} \frac{4\sin^2(\omega_{fi}T/2)}{\omega_{fi}^2}.$$

Using the long-time limit gives

$$\frac{\lvert V_{fi}\rvert^2}{\hbar^2} 2\pi T\,\delta(\omega_{fi}).$$

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

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

After integrating over final energies with density $\rho(E_f)$, the probability is

$$P(T) = \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \rho(E_i)T.$$

Thus

$$\Gamma = \frac{P(T)}{T} = \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \rho(E_i).$$

2. Name two reasons the golden rule can fail even if the perturbation matrix element is small.

Solution

It can fail if the final spectrum is not continuous or dense enough, because then a rate description is inappropriate. It can also fail at very long times when the initial state is depleted or recurrence/coherence effects become important. Other failures include rapidly varying density of states, strong resonant two-level dynamics, and invalid normalization conventions.