Fermi's Golden Rule

Equivalent forms

dN/dt = (2π/ℏ) |⟨f|H'|i⟩|² ρ(E_f)

A single equation governs almost every decay, scattering, and absorption rate in quantum mechanics.

Symbol	Name	SI unit	Dimension	Range
$Γ$	Transition rateoutput Probability per unit time for the transition i → f	1/s	T⁻¹	0 – 100000000000000000000
$M$	Matrix element ⟨f\|H'\|i⟩, coupling strength between initial and final states	J	M·L²·T⁻²	0 – 1e-18
$ρ$	Density of final states Number of final states per unit energy	1/J	M⁻¹·L⁻²·T²	0 – 1e+35
$ℏ$	Reduced Planck constant Planck constant over 2π	J·s	M·L²·T⁻¹	1.054571817e-34 – 1.054571817e-34

Unit systems

Symbol	SI	CGS	Natural
$Γ$	1/s	1/s	MeV
$M$	J	erg	MeV
$ρ$	1/J	1/erg	1/MeV
$ℏ$	J·s	—	—

Where it holds

Valid in the weak-coupling, long-time limit where

\Delta E \times t

≫

\hbar but |M|t/\hbar

≪ 1. Breaks for resonant, strong, or short-time interactions.

Dimensional analysis

\Gamma \to [T^{-1}]

(1/\hbar )|M|^{2}\rho \to [1/(ML^{2}T^{-1})][M^{2}L^{4}T^{-4}][M^{-1}L^{-2}T^{2}] = [T^{-1}]

Both sides are inverse time.

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Discovery

Paul Dirac (1927) and Enrico Fermi (1950) · 1950

Originally derived by Dirac in 1927 as part of time-dependent perturbation theory, the formula was popularized by Fermi in his Chicago lectures as 'the golden rule' for transition probabilities.

Try this

How fast do quantum systems jump between states?

Estimate the transition rate from a 2p to 1s state in hydrogen given matrix element |M| ~ ea_0 and density of photon states.

Research status: stable

Real-world applications

Atomic spontaneous emission rates
Beta-decay half-lives
Particle-physics decay widths (e.g., Z boson, Higgs)
Photoabsorption cross sections in molecules
Tunneling rates in scanning tunneling microscopes

Common misconceptions

It is only the leading-order term in perturbation theory; higher-order terms matter for forbidden transitions.
The density of states is for the FINAL states, not the initial.
It assumes a continuum of final states; for discrete-to-discrete transitions, use Rabi oscillations instead.

Experimental verification

Used to predict atomic spontaneous emission rates (Einstein A coefficients), nuclear beta-decay rates, and particle-physics decay widths — all confirmed to high precision.

Derivation

Time-dependent perturbation theory gives transition probability

P(t) = |c_f(t)|^{2}

with

c_f(t) = (1/i\hbar )\int M e^(i\omega t) dt

.

For continuum final states, integrate over E_f using the identity

\sin ^{2}(\omega t/2)/(\omega /2)^{2} \to 2\pi t \delta (\omega )

as

t \to \infty

.

Dividing P by t yields

\Gamma = (2\pi /\hbar )|M|^{2}\rho (E_f)

.

Limiting cases

M \to 0

⟶

\Gamma \to 0

Forbidden transitions (no coupling) have zero rate at leading order.

\rho \to 0

⟶

\Gamma \to 0

No available final states — transition cannot occur (energy conservation blocks it).

Strong coupling⟶ Perturbation theory breaks downWhen |

M|^{2}\rho t/\hbar

≳ 1 within the relevant time, higher-order corrections are needed.

What if…

What if the matrix element doubled?

$\Gamma$ quadruples — rates are quadratic in coupling.

What if the final state density vanished?

Transitions are forbidden — basis of phase-space suppression in nuclear and particle physics.

1

2p → 1s Lyman-α decay rate

Given ·

M:: 1e-20
ρ:: 1e+30
ℏ:: 1.054571817e-34

Find ·

\Gamma

Steps

Step 1: $2\pi /\hbar = 6.283 / 1.055e-34 = 5.96 \times 10^{34} (1/J\cdot s)$ .
Step 2: | $M|^{2} = (1e-20)^{2} = 1e-40\,\mathrm{J}^{2}$ .
Step 3: $\Gamma = 5.96e34 \times 1e-40 \times 1e30 = 5.96 \times 10^{5} s^{-1}$ (order-of-magnitude). Realistic full calculation including correct $\rho$ gives $6.27 \times 10^{8} s^{-1}$ , lifetime 1.6 ns.

Result ·

\Gamma = (2\pi /1.055e-34) \times (1e-20)^{2} \times 1e30 \approx 5.96 \times 10^{5} s^{-1}

(order of magnitude; actual

\approx 6.27 \times 10^{8} s^{-1})

Radioactive Activity-Decay Constant Relation→klein nishina cross section