Time-Dependent Perturbation Theory

Recall prompts

Attempt these from memory before reading.

Write Fermi’s Golden Rule. What does each factor mean?
Why does a density of final states appear at all?
What is the selection rule for electric dipole transitions, and where does it come from?
Does a static perturbation cause transitions between non-degenerate states?
What did Einstein have to assume to get the $A$ coefficient?

Check your answers against the text. Prompt 4 is the one that reveals what “time-dependent” is doing in the title: a constant perturbation cannot cause transitions at all.

Motivation: energy conservation is a long-time limit

Time-independent perturbation theory asks how energy levels shift. Time-dependent perturbation theory asks a different question: how does a state change? The two are not variants of one method, because the answer to the second is about rates, and rates need something to be conserved.

A constant perturbation causes no transitions. If $\hat{V}$ does not depend on time, then in the interaction picture the amplitude for $|i\rangle \to |f\rangle$ is

c_f(t) = -\frac{i}{\hbar}V_{fi}\int_0^t e^{i\omega_{fi}t'}\,\mathrm{d}t' = -\frac{i}{\hbar}V_{fi}\,e^{i\omega_{fi}t/2}\, \frac{2\sin(\omega_{fi}t/2)}{\omega_{fi}},

which is peaked at $\omega_{fi} = 0$ and oscillates elsewhere. For long times, $\left|\frac{\sin(\omega t/2)}{\omega/2}\right|^2 \to 2\pi t\,\delta(\omega)$ : the probability grows linearly with $t$ only at $\omega_{fi} = 0$ , and vanishes elsewhere. That linear growth is what “a rate” means, and it is why the Golden Rule has a $t$ in its derivation and no $t$ in its answer.

Energy conservation is not assumed; it emerges. The $\delta(\omega_{fi})$ above is the statement that a transition only happens when the perturbation supplies exactly the energy difference. For a finite time the peak has width $\sim 1/t$ , so energy conservation holds only to within $\hbar/t$ — and short pulses really do drive nominally off-resonant transitions. That is not an approximation to correct; it is physics.

The density of states is what makes the rate finite. For a discrete final state the probability oscillates forever. Summing over a continuum replaces the square of a sinc by a smooth density $\rho(E_f)$ , and that is the whole difference between a Rabi oscillation and a decay.

12.1 Fermi’s Golden Rule

For a time-dependent perturbation $\hat{V}(t) = \hat{V}\,e^{-i\omega t}$ applied to an initial state $|i\rangle$ , the transition rate to a continuum of final states $|f\rangle$ is:

\Gamma_{i \to f} = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2\rho(E_f)

Where $\rho(E_f)$ is the density of final states at energy $E_f = E_i + \hbar\omega$ .

Derivation. Using first-order time-dependent perturbation theory, the transition amplitude to state $|f\rangle$ is:

c_f(t) = -\frac{i}{\hbar}\int_0^t \langle f|\hat{V}|i\rangle\,e^{i\omega_{fi}t'}\, dt'

For a sinusoidal perturbation at frequency $\omega$ :

|c_f|^2 = \frac{|\langle f|\hat{V}|i\rangle|^2}{\hbar^2}\frac{\sin^2[(\omega_{fi} - \omega)t/2]}{(\omega_{fi} - \omega)^2/4}

In the long-time limit, $\sin^2(xt)/x^2 \to 2\pi t\,\delta(x)$ Giving:

\frac{|c_f|^2}{t} = \frac{2\pi}{\hbar^2}|\langle f|\hat{V}|i\rangle|^2\,\delta(E_f - E_i - \hbar\omega)

Summing over all final states with density $\rho(E_f)$ :

\Gamma = \int \frac{d|c_f|^2}{dt}\,\rho(E_f)\,dE_f = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2\rho(E_f) \quad \blacksquare

12.2 Selection Rules for Electric Dipole Transitions

The electric dipole matrix element:

\langle f|\hat{\mathbf{d}}|i\rangle = -e\langle f|\mathbf{r}|i\rangle

For hydrogen-like atoms, the selection rules are:

$\Delta l = \pm 1$ (parity change required)
$\Delta m = 0, \pm 1$ (for $z$ , $x \pm iy$ polarisation respectively)
$\Delta n$ unrestricted

The transition rate for $2p \to 1s$ in hydrogen:

A_{2p \to 1s} = \frac{\omega^3}{3\pi\varepsilon_0\hbar c^3}|\langle 1s|e\mathbf{r}|2p\rangle|^2

With $|\langle 1s|z|2p, m=0\rangle| = \frac{2^7\sqrt{2}}{3^5}a_0$ This gives $A_{2p \to 1s} \approx 6.3 \times 10^8$ s $^{-1}$ Corresponding to a lifetime $\tau \approx 1.6$ ns.

12.3 Spontaneous Emission and Einstein Coefficients

The Einstein $A$ coefficient (spontaneous emission rate) is related to the $B$ coefficient (stimulated emission/absorption):

A_{21} = \frac{\hbar\omega^3}{\pi^2 c^3}B_{21}

This relation, derived by Einstein in 1917 using thermodynamic arguments (detailed balance in a blackbody radiation field), was one of the first indications that spontaneous emission requires quantum electrodynamics.

Worked Example 11.1: Selection Rules for Hydrogen

Consider the transition $3d \to 1s$ in hydrogen. Is this an allowed E1 transition?

The matrix element involves the integral $\langle n'l'm'|\mathbf{r}|nlm\rangle = \langle 1,0,0|r_q|3,2,m\rangle$ where $r_q$ is a spherical tensor component.

By the Wigner—Eckart theorem and parity selection rules:

$\Delta l = 0 - 2 = -2 \neq \pm 1$ : forbidden for E1

The $3d \to 1s$ transition can proceed via:

E2 (electric quadrupole): $\Delta l = 0, \pm 2$ Rate $\sim \alpha(kR)^2$ times slower than E1
M1 (magnetic dipole): requires $\Delta l = 0$ Not applicable here
Two-photon decay: $3d \to 2p \to 1s$ (two successive E1 transitions)

The $3d \to 2p$ transition ( $\Delta l = -1$ ) is E1-allowed and dominates, with $A_{3d \to 2p} \sim 6.4 \times 10^7$ s $^{-1}$ .

Key Relationships

Formula	Name	Application
$\Gamma_{i\to f} = \frac{2\pi}{\hbar}\\|\langle f\\|\hat{V}\\|i\rangle\\|^2\rho(E_f)$	Fermi’s Golden Rule	Transition rate to continuum
$\Delta l = \pm 1$ , $\Delta m = 0, \pm 1$	E1 selection rules	Electric dipole transitions
$A_{21} = \frac{\hbar\omega^3}{\pi^2 c^3}B_{21}$	Einstein relation	Connects spontaneous/stimulated rates
$\\|c_f\\|^2 = \frac{\\|\langle f\\|\hat{V}\\|i\rangle\\|^2}{\hbar^2}\frac{\sin^2[(\omega_{fi}-\omega)t/2]}{(\omega_{fi}-\omega)^2/4}$	Rabi formula	Transition probability for discrete states

Common Pitfalls

Fermi’s Golden Rule requires a continuum: For transitions between discrete states (e.g., two bound states in an atom), the Rabi formula must be used. The density of states $\rho(E_f)$ in FGR ensures energy conservation is satisfied with finite probability.
First-order perturbation fails for strong fields: When the Rabi frequency $\Omega = |\langle f|\hat{V}|i\rangle|/\hbar$ is comparable to the detuning, higher-order effects (AC Stark shift, Rabi oscillations) become important.
Selection rules are not absolute: Forbidden transitions can proceed via higher multipole orders (E2, M1) or multiphoton processes, albeit at slower rates. A transition is truly forbidden only when all contributing channels vanish.
The rotating wave approximation (RWA): Dropping counter-rotating terms requires $|\omega - \omega_{fi}| \ll \omega + \omega_{fi}$ . For very strong fields or ultrastrong coupling, the RWA breaks down.

Applications

Laser cooling: Doppler cooling relies on repeated absorption—spontaneous emission cycles, with each cycle reducing atomic momentum by $\hbar k$ .
Optical pumping: Polarised light selectively populates magnetic sublevels via dipole selection rules, preparing spin-polarised atomic samples.
Quantum optics: Spontaneous emission is the fundamental source of decoherence in quantum optical systems, limiting qubit lifetimes in cavity QED.
Spectroscopy: Fermi’s Golden Rule underlies the interpretation of absorption spectra, photoemission, and inelastic scattering cross sections.
Photovoltaics: The solar cell efficiency limit (Shockley—Queisser) derives from detailed balance between absorption and spontaneous emission.

Connections to Other Topics

Quantum electrodynamics: Spontaneous emission is not predicted by non-relativistic quantum mechanics alone, it requires coupling to the quantised electromagnetic field vacuum.
Scattering theory: The $T$ -matrix formalism generalises FGR to higher orders. The optical theorem ( $\sigma_{\text{tot}} = (4\pi/k)\,\text{Im}\,f(0)$ ) relates the forward scattering amplitude to the total cross section.
Solid-state physics: Fermi’s Golden Rule describes electron—phonon scattering, carrier relaxation, and exciton decay in semiconductors.

Additional Worked Example: Photoionisation Rate

Problem. A hydrogen atom in the ground state is illuminated by monochromatic light at $\lambda = 50$ nm, well above the ionisation threshold (13.6 eV). Estimate the photoionisation rate using Fermi’s Golden Rule.

Solution. The dipole matrix element for $1s \to$ continuum is approximately $\langle \epsilon_p | e\mathbf{r} | 1s \rangle \sim ea_0$ . The density of continuum states at photoelectron energy $\epsilon = \hbar\omega - 13.6$ eV:

k = \sqrt{2m\epsilon}/\hbar, \quad \rho(\epsilon) = \frac{V m k}{(2\pi)^3 \hbar^2} d\Omega

The photoionisation cross section near threshold is $\sigma \approx 6.3 \times 10^{-18}$ cm $^2$ for hydrogen. At intensity $I = 10^{12}$ W/m $^2$ , the rate $\Gamma = \sigma I / (\hbar\omega) \approx 10^{14}$ s $^{-1}$ , complete ionisation occurs within femtoseconds for intense fields.

Summary Table: Transition Types

Transition	Selection Rules	Rate	Typical Timescale
Electric dipole (E1)	$\Delta l = \pm 1$ , $\Delta m = 0,\pm 1$	$A \sim 10^7$ — $10^9$ s $^{-1}$	ns
Magnetic dipole (M1)	$\Delta l = 0$ , $\Delta s = \pm 1$ (spin-flip)	$A \sim 10^3$ s $^{-1}$	$\mu$ s—ms
Electric quadrupole (E2)	$\Delta l = 0, \pm 2$	$A \sim 10^{-3}$ — $10$ s $^{-1}$	ms—s
Two-photon	No parity constraint	$A \propto I^2$	Depends on intensity
Forbidden (all channels)	None allowed	$A = 0$	Metastable if no decay path

Intuition

Time-dependent perturbation theory treats external influences as small disturbances to a quantum system. When a system is subjected to a time-varying field, transitions between energy levels become possible. The transition rate depends on how much the perturbation overlaps with the initial and final states, measured by matrix elements. Fermi’s golden rule packages this into a simple formula: the transition rate is proportional to the square of the matrix element times the density of final states. This explains atomic transitions, photon absorption, and spontaneous emission, forming the basis of spectroscopy and laser physics.

Counterexamples worth knowing by name

Claim	True?	Counterexample
Fermi’s Golden Rule applies to any transition	no	it requires a continuum of final states; between two bound states the system Rabi-oscillates instead
The transition rate is constant from $t = 0$	no	it is linear at first and only becomes constant after a transient of order $\hbar/\Delta E$
A static perturbation causes transitions	no	$\int e^{i\omega_{fi}t'}\,\mathrm{d}t'$ vanishes for long times unless $\omega_{fi} = 0$ , i.e. unless the states are degenerate
Spontaneous emission follows from the Schrodinger equation	no	a time-dependent potential couples to a classical field, which cannot produce spontaneous emission; Einstein had to postulate the $A$ coefficient from detailed balance
Selection rules are absolute	no	they are selection rules for the dipole approximation; electric quadrupole and magnetic dipole transitions are weaker but real, and “forbidden” lines are observed in astrophysics
Energy conservation is exact in a finite process	no	it holds to within $\Delta E \sim \hbar/t$ ; short pulses drive off-resonant transitions routinely
Stronger perturbation always gives a larger transition rate	no	beyond the perturbative regime the population oscillates (Rabi) and the rate concept stops applying

The fourth row is the most important and is usually glossed. Spontaneous emission cannot be derived from a classical field coupled to a quantum atom — a classical field has no vacuum fluctuations to stimulate it. Einstein got the $A$ coefficient in 1917 by demanding detailed balance with the (derivable) $B$ coefficient and the Planck spectrum, and that argument is still the cleanest route to it.

The last row is the boundary of the method: once the perturbation is strong enough to move appreciable population, “rate” is the wrong concept and the oscillation is the physics.

Where the research frontier is

Strong-field and attosecond physics. Fields intense enough to ionise with many photons, or short enough to resolve electron motion, sit outside perturbation theory entirely. Above-threshold ionisation and high-harmonic generation are the standard phenomena.
Floquet engineering. Periodic driving creates effective Hamiltonians with band structures that the undriven system does not have, which is now used to realise topological phases on demand.
Cavity QED and the Purcell effect. Spontaneous emission is not a fixed property of an atom: putting it in a cavity changes the density of states and so changes the rate. That is the Golden Rule with an engineered $\rho$ .
Landau-Zener transitions. A swept two-level crossing has an exact solution, and it is the building block for adiabatic quantum computing and for control protocols in qubits.
Quantum control. Shaped pulses that drive a system to a chosen target state, with the pulse found by optimisation rather than intuition.

Interleaving problems

Mix these with approximation methods, angular momentum, operators, Series and Limits and Thermodynamics and Statistical Mechanics.

(Series and limits.) Show that $\left|\frac{\sin(\omega t/2)}{\omega/2}\right|^2 \to 2\pi t\,\delta(\omega)$ as $t \to \infty$ in the sense of distributions, by integrating against a test function. Identify where the factor of $2\pi$ comes from.
(Operators.) Show that the dipole matrix element $\langle f|\hat{\mathbf{r}}|i\rangle$ vanishes unless $l_f = l_i \pm 1$ , by writing $\hat{\mathbf{r}}$ in spherical coordinates and using the properties of spherical harmonics. Which quantum number is not constrained, and why?
(Angular momentum.) A photon carries one unit of angular momentum. Show that this is what makes $\Delta l = \pm 1$ a conservation statement rather than merely a selection rule, and explain what happens for $\Delta l = \pm 2$ .
(Thermal physics.) Einstein’s argument: given the $B$ coefficient and the Planck spectrum, derive the $A$ coefficient from detailed balance. Then explain what quantum mechanics adds that classical electrodynamics cannot supply.
(Approximation methods.) A perturbation that is constant in time but switched on suddenly at $t = 0$ is not time-independent in this sense. Show that it causes transitions, and explain how a sudden switch supplies energy.
(Research.) The Landau-Zener formula gives the transition probability for a linearly swept two-level crossing. Derive it, state the adiabatic limit, and explain why the result is exact rather than perturbative.

Cross-References

Wave Functions and the Schrodinger Equation: Time-dependent perturbation theory builds on the Schrodinger equation for time evolution.
Operators and Observables: Matrix elements of operators determine transition rates and selection rules.
Angular Momentum and the Hydrogen Atom: Angular momentum theory provides the selection rules for electric dipole transitions.
Calculus
Linear Algebra
Vector Calculus
Quantum Computing

Mark this page as reviewed