5.2.2 Fermi’s Golden Rule#

Prompts

What does the sinc function tell us about transition probabilities, and how does energy conservation emerge in the long-time limit?
What is Fermi’s golden rule? Why does the transition rate depend on both the coupling strength and the density of final states?
How do selection rules arise from symmetry, and why are some transitions forbidden at leading order?
What is the adiabatic theorem? Under what conditions does a system remain in its instantaneous eigenstate as the Hamiltonian changes?
What is the Berry phase, and why is it geometric rather than dynamical?

Lecture Notes#

Overview#

Fermi’s golden rule distills quantum transitions to a single formula: when a system couples weakly to a dense spectrum of final states, the transition rate is proportional to the squared matrix element times the density of states. This rule underlies atomic decay, scattering, and spectroscopy—it is the bridge from formal perturbation theory to measurable rates.

Transition Amplitude#

Consider a system initially in eigenstate \(\vert i\rangle\) of \(\hat{H}_0\) with energy \(E_i^{(0)}\). The transition amplitude to reach state \(\vert f\rangle\) (energy \(E_f^{(0)}\)) at time \(t\) is \(c_{i\to f}(t) = \langle f \vert \hat{U}_I(t) \vert i \rangle\), so the transition probability is:

(96)#\[P_{i\to f}(t) = \vert c_{i\to f}(t)\vert^2\]

To first order in the Dyson series:

First-Order Transition Probability

(97)#\[P_{i\to f}^{(1)}(t) = \frac{1}{\hbar^2} \left\vert \int_0^t \mathrm{d}t_1\, \mathrm{e}^{\mathrm{i}\omega_{fi} t_1} V_{fi}(t_1) \right\vert^2\]

where \(V_{fi}(t) = \langle f \vert \hat{V}(t) \vert i \rangle\) and \(\omega_{fi} = (E_f^{(0)} - E_i^{(0)})/\hbar\) is the transition frequency.

Sinc Function and Energy Conservation#

For a constant perturbation \(\hat{V}(t) = \hat{V}_0\) switched on at \(t = 0\):

(98)#\[P_{i\to f}^{(1)}(t) = \frac{\vert V_{fi}\vert^2}{\hbar^2} \frac{\sin^2(\omega_{fi} t/2)}{(\omega_{fi}/2)^2}\]

Derivation: Sinc Formula

Evaluate the integral for constant \(\hat{V}_0\):

\[\int_0^t \mathrm{d}t_1\, \mathrm{e}^{\mathrm{i}\omega_{fi}t_1} = \frac{\mathrm{e}^{\mathrm{i}\omega_{fi}t} - 1}{\mathrm{i}\omega_{fi}}\]

Taking the modulus squared:

\[\left\vert\frac{\mathrm{e}^{\mathrm{i}\omega_{fi}t} - 1}{\mathrm{i}\omega_{fi}}\right\vert^2 = \frac{2 - 2\cos(\omega_{fi}t)}{\omega_{fi}^2} = \frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2}\]

On-Resonance vs Off-Resonance

On resonance (\(\omega_{fi} \to 0\), i.e. \(E_f = E_i\)): \(P \approx \vert V_{fi}\vert^2 t^2/\hbar^2\) — probability grows quadratically.
Off-resonance (\(\vert\omega_{fi}\vert \gg 1/t\)): the sinc oscillates rapidly and averages to a small value \(\propto 1/\omega_{fi}^2\), suppressing the transition.
The main lobe has width \(\Delta\omega \sim 2\pi/t\): longer observation sharpens energy resolution.

The sinc function encodes the time–energy uncertainty relation: transitions are confined to a frequency window \(\Delta\omega \sim 2\pi/t\) around resonance. In the limit \(t \to \infty\), this window narrows to a delta function enforcing strict energy conservation.

Fermi’s Golden Rule#

In real systems the final state is usually part of a continuum (scattering momenta, photon modes, band states). We sum over all accessible final states weighted by their density:

\[P_{i \to \text{continuum}}(t) = \int \mathrm{d}E_f \, \rho(E_f)\, P_{i\to f}^{(1)}(t)\]

In the long-time limit, the sinc-squared becomes a delta function and the transition rate (probability per unit time) becomes constant:

Fermi’s Golden Rule

(99)#\[\Gamma_{i\to f} = \frac{2\pi}{\hbar} \vert\langle f \vert \hat{V} \vert i \rangle\vert^2 \, \rho(E_i)\]

Three factors determine the rate:

Coupling strength \(\vert\langle f \vert \hat{V} \vert i \rangle\vert^2\): matrix element squared
Density of final states \(\rho(E_i)\): number of available targets at the transition energy
Energy conservation: only resonant final states (\(E_f = E_i\)) contribute

Derivation: Fermi’s Golden Rule

Starting from the sinc formula, sum over final states in a continuum with density \(\rho(E_f)\). Change variables to \(\omega_{fi}\) and use the identity:

\[\lim_{t\to\infty} \frac{t\sin^2(\omega_{fi}t/2)}{(\omega_{fi}/2)^2} = \pi t\,\delta(\omega_{fi})\]

Over the narrow peak, \(\rho\) and \(\vert V_{fi}\vert^2\) are essentially constant at their resonant values. This gives:

\[P_{i\to\text{continuum}}(t) \approx \frac{\pi}{\hbar}\,\vert V_{fi}\vert^2\,\rho(E_i)\,t\]

The transition rate \(\Gamma = \mathrm{d}P/\mathrm{d}t = (2\pi/\hbar)\vert V_{fi}\vert^2\,\rho(E_i)\).

Validity Conditions

Fermi’s golden rule requires: (1) weak coupling (\(\vert V\vert t/\hbar \ll 1\), first-order perturbation theory valid); (2) long observation time (\(t \gg 1/\vert\omega_{fi}\vert\), sinc narrows to delta function); (3) dense or continuous final-state spectrum. When any condition fails, higher-order corrections or non-perturbative methods are needed.

Selection Rules#

The matrix element \(\langle f \vert \hat{V} \vert i \rangle\) may vanish identically when \(\hat{V}\) and the states have incompatible symmetries. These constraints are called selection rules.

Electric Dipole Selection Rules

For radiative transitions with dipole coupling \(\hat{V} = -\boldsymbol{d} \cdot \boldsymbol{E}\) (where \(\boldsymbol{d} = -e\boldsymbol{r}\)):

\(\Delta \ell = \pm 1\) (orbital angular momentum changes by one unit)
\(\Delta m_\ell = 0, \pm 1\) (depends on photon polarization)
\(\Delta S = 0\) (spin unchanged at leading order)

Forbidden transitions (e.g. \(2s \to 1s\), \(\Delta\ell = 0\)) have vanishing dipole matrix elements. They may proceed via higher-order processes (quadrupole, magnetic dipole) at rates suppressed by \(\sim 10^{-5}\), creating metastable states used in lasers and atomic clocks.

Adiabatic Theorem#

When the Hamiltonian changes slowly compared to internal energy gaps, the system remains in its instantaneous eigenstate:

Adiabatic Theorem

If \(\hat{H}(t)\) varies slowly and the system starts in eigenstate \(\vert\psi_n(0)\rangle\), it tracks the instantaneous eigenstate:

\[\vert\psi(t)\rangle = \mathrm{e}^{\mathrm{i}\gamma_n(t)}\,\vert\psi_n(t)\rangle\]

Adiabatic condition: the rate of change must satisfy \(\vert\dot{H}\vert \ll (\Delta E)^2/\hbar\), where \(\Delta E\) is the energy gap.

Derivation: Adiabatic Theorem

Expand the state in the instantaneous eigenbasis: \(\vert\psi(t)\rangle = \sum_n c_n(t)\,\mathrm{e}^{\mathrm{i}\gamma_n^{\text{dyn}}(t)}\,\vert\psi_n(t)\rangle\), where \(\gamma_n^{\text{dyn}} = -\frac{1}{\hbar}\int_0^t E_n(t')\,\mathrm{d}t'\). Substituting into the Schrödinger equation and projecting onto \(\vert\psi_m\rangle\) (\(m \neq n\)):

\[\dot{c}_m = -\sum_{n\neq m} c_n \frac{\langle\psi_m\vert\dot{H}\vert\psi_n\rangle}{E_n - E_m}\,\mathrm{e}^{\mathrm{i}(\gamma_m^{\text{dyn}} - \gamma_n^{\text{dyn}})}\]

When \(\hat{H}\) varies on timescale \(\tau\), the numerator scales as \(1/\tau\) while the phase oscillates at frequency \(\Delta E/\hbar\). For \(\Delta E \cdot \tau/\hbar \gg 1\) the oscillations average to zero, suppressing transitions. The system stays in state \(n\).

Berry phase connection. The phase \(\gamma_n(t)\) acquired during adiabatic evolution has a geometric piece — the Berry phase (§4.2):

\[\gamma_n^{\text{Berry}} = \mathrm{i}\int_0^t \langle\psi_n(t')\vert\dot{\psi}_n(t')\rangle\,\mathrm{d}t'\]

This depends only on the path traced in parameter space, not on the speed of traversal. For a spin-1/2 in a slowly rotating magnetic field, the Berry phase equals half the solid angle swept by the field direction.

Discussion

Fermi’s golden rule predicts \(P(t) \approx 1 - \Gamma t\) at long times (exponential decay). But at very short times, \(P(t) \approx 1 - (Vt/\hbar)^2\) (quadratic decay from the sinc function).

At what timescale does the crossover from quadratic to linear occur?
How does this relate to the quantum Zeno effect, where frequent measurements suppress decay?
Can you design an experiment to observe both regimes?

Summary#

First-order transition probability: \(P \propto \sin^2(\omega_{fi}t/2)/\omega_{fi}^2\); enhanced on resonance, suppressed off-resonance.
Fermi’s golden rule: \(\Gamma = (2\pi/\hbar)\vert\langle f\vert\hat{V}\vert i\rangle\vert^2\,\rho(E_i)\) — constant rate from the long-time sinc→delta limit with energy conservation.
Selection rules: symmetry forces \(\langle f\vert\hat{V}\vert i\rangle = 0\) for certain transitions; forbidden transitions are suppressed, not eliminated.
Adiabatic theorem: slow Hamiltonian changes keep the system in its instantaneous eigenstate, accumulating a geometric Berry phase.

Homework#

1. Starting from the first-order transition probability for constant perturbation \(P_{i\to f}^{(1)}(t) = \vert V_{fi}\vert^2 \sin^2(\omega_{fi}t/2) / [\hbar^2(\omega_{fi}/2)^2]\), show that for short times \(\vert\omega_{fi}\vert t \ll 1\) the probability grows quadratically as \(P \approx \vert V_{fi}\vert^2 t^2/\hbar^2\).

2. Verify the delta-function identity: \(\lim_{t\to\infty} t\sin^2(\omega_{fi}t/2)/(\omega_{fi}/2)^2 = \pi t\,\delta(\omega_{fi})\). (Hint: show that the function is sharply peaked at \(\omega_{fi}=0\) with width \(\sim 2\pi/t\) and height \(\sim t\), and integrate to check normalization.)

3. For elastic scattering of a non-relativistic particle in 3D, the density of final plane-wave states at energy \(E\) is \(\rho(E) = \frac{Vm}{2\pi^2\hbar^3}\sqrt{2mE}\). (a) Derive this from the momentum-space density \(V/(2\pi\hbar)^3\) by converting to energy. (b) Use Fermi’s golden rule to write the total scattering rate and explain why it increases with \(\rho(E)\).

4. Light of frequency \(\omega_L\) shines on an atom with transition frequency \(\omega_0\). Define the detuning \(\Delta = \omega_L - \omega_0\). (a) Write the transition probability in terms of the sinc function and \(\Delta\). (b) Explain why transitions are suppressed when \(\vert\Delta\vert \tau \gg 2\pi\) for pulse duration \(\tau\).

5. (a) State the electric dipole selection rules (\(\Delta\ell\), \(\Delta m_\ell\), \(\Delta S\)). (b) Is \(2s \to 1s\) allowed? Is \(2p,\,m_\ell=1 \to 1s,\,m_\ell=0\) allowed for \(z\)-polarized light (\(\Delta m_\ell = 0\))? (c) Explain why forbidden transitions create metastable states.

6. The photon density of states is \(\rho_\gamma(\omega) = V\omega^2/(\pi^2 c^3)\). Use Fermi’s golden rule with dipole coupling \(\hat{V} = -\boldsymbol{d}\cdot\boldsymbol{E}\) and matrix element \(d_0\) to show the spontaneous decay rate scales as \(\Gamma \propto \omega_0^3 d_0^2\). Define the lifetime \(\tau = 1/\Gamma\) and explain why higher-frequency transitions decay faster.

7. A system starts in eigenstate \(\vert n\rangle\) of \(\hat{H}(0)\). The Hamiltonian changes on timescale \(\tau\) with energy gap \(\Delta E\). (a) State the adiabatic condition relating \(\tau\), \(\Delta E\), and \(\hbar\). (b) If \(\Delta E = 1\) eV, estimate the minimum \(\tau\) for adiabatic evolution. (c) What geometric quantity does the adiabatic phase encode?

8. Consider a spin-1/2 in a magnetic field \(\boldsymbol{B}(t)\) that rotates slowly around the \(z\)-axis, completing one full cycle. If the spin starts aligned with \(\boldsymbol{B}\), it adiabatically follows the field. Show that the Berry phase accumulated is \(\gamma = -\Omega/2\) where \(\Omega\) is the solid angle swept by \(\boldsymbol{B}\). For rotation in the equatorial plane, what is \(\gamma\)?