Time-Dependent Perturbation Theory

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 $\rho(E_f)$:

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

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.

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.

1. 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.
2. 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.
3. 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.
4. 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.

• 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.

• 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.

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.

flowchart TD
    A[12_Time Dependent Perturbation Theory] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

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.

• 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

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.