Time-Dependent Perturbation Theory

When a quantum system—like an atom, molecule, or nucleus—is subjected to a changing external influence, its state can evolve in surprising ways. Time-dependent perturbation theory provides the essential toolkit for calculating the probability that such a system will make a transition from one quantum state to another. This framework is indispensable for understanding phenomena from the absorption of light to the scattering of particles, bridging the gap between simple static models and the dynamic reality of quantum interactions.

The Framework of a Time-Dependent Perturbation

We begin with a quantum system whose unperturbed Hamiltonian, $H_0$, is time-independent and whose stationary states, ${\psi }_n$, are known exactly. These states satisfy $H_0{\psi }_n=E_n{\psi }_n$. The full Hamiltonian is then written as $H=H_0+V(t)$, where $V(t)$ is a time-dependent perturbation that is "switched on" at some time. The central goal is to solve the time-dependent Schrödinger equation, $i\hslash \frac{\partial }{\partial t}\Psi (t)=[H_0+V(t)]\Psi (t)$, for the state vector $\Psi (t)$.

Since the unperturbed states form a complete set, we expand the full time-dependent state as a superposition: $$\Psi (t)=\sum _n{c}_n(t){\psi }_n{e}^{-i{E}_{n}t/\hslash }.$$ Here, $c_n(t)$ are the time-dependent probability amplitudes. The factor $e^{-i{E}_{n}t/\hslash }$ accounts for the natural time evolution of each stationary state, so any change in the coefficients $c_n(t)$ is due solely to the perturbation $V(t)$. Substituting this expansion into the Schrödinger equation leads to a set of coupled differential equations for the amplitudes: $$i\hslash \frac{d{c}_m(t)}{dt}=\sum _n{V}_{mn}(t)e^{i{\omega }_{mn}t}{c}_n(t),$$ where $V_{mn}(t)=⟨{\psi }_m|V(t)|{\psi }_n⟩$ is the matrix element of the perturbation and ${\omega }_{mn}=(E_m-E_n)/\hslash$ is the Bohr frequency.

First-Order Theory and the Transition Probability

The equations are generally unsolvable exactly. First-order time-dependent perturbation theory applies when $V(t)$ is weak. We assume the system starts in a definite initial state $i$ at $t=0$, so $c_i(0)=1$ and all other $c_n(0)=0$. To first order, we approximate $c_n(t)\approx 0$ for $n\ne i$ on the right-hand side of the equation. This decouples the equations, yielding for a final state $f\ne i$: $$c_f^{(1)}(t)=-\frac{i}{\hslash }{\int }_0^{t}d{t}^{\prime }{V}_{fi}(t^{\prime })e^{i{\omega }_{fi}{t}^{\prime }}.$$ The transition probability from state $i$ to state $f$ is then $P_{i\to f}(t)=|c_f^{(1)}(t){|}^2$. This result is powerful: the probability amplitude is essentially the Fourier transform of the perturbation, evaluated at the Bohr frequency connecting the two states.

Fermi's Golden Rule for a Constant Perturbation

A crucial case is a perturbation that is constant after being switched on at $t=0$: $V(t)=V$ for $t>0$. For this constant perturbation, the first-order amplitude becomes: $$c_f^{(1)}(t)=-\frac{i}{\hslash }{V}_{fi}{\int }_0^{t}d{t}^{\prime }{e}^{i{\omega }_{fi}{t}^{\prime }}=-\frac{V_{fi}}{\hslash {\omega }_{fi}}\left(e^{i{\omega }_{fi}t}-1\right).$$ The corresponding probability oscillates in time: $P_{i\to f}(t)=\frac{4|V_{fi}{|}^2}{{\hslash }^2}\frac{{\sin }^2({\omega }_{fi}t/2)}{{\omega }_{fi}^2}$. This describes a reversible transfer of probability, typical of a coherent drive between two discrete states.

The more common and practically important scenario involves transitions from a discrete initial state into a continuum of closely spaced final states (e.g., an atom ionizing into a continuum of free-electron states). We must sum $P_{i\to f}(t)$ over all final states in an energy range. Replacing the sum with an integral over final state density $\rho (E_f)$, and taking the long-time limit, leads to Fermi's golden rule for a constant perturbation: $${\Gamma }_{i\to [f]}=\frac{d}{dt}{P}_{i\to [f]}=\frac{2\pi }{\hslash }|V_{fi}{|}^2\rho (E_f){|}_{E_f=E_i}.$$ This states that the transition rate $\Gamma$ is constant in time, proportional to the square of the matrix element and the density of states at energy conservation ($E_f=E_i$). The "golden rule" is the workhorse for calculating rates of irreversible processes like scattering and decay.

Applications: Radiation, Selection Rules, and Scattering

A paramount application is the absorption and emission of radiation. Treating the electromagnetic field as a classical sinusoidal perturbation, $V(t)\propto e^{-i\omega t}+e^{+i\omega t}$, leads to Fermi's golden rule for harmonic perturbations. The $e^{-i\omega t}$ term drives absorption ($E_f=E_i+\hslash \omega$), while $e^{+i\omega t}$ drives stimulated emission ($E_f=E_i-\hslash \omega$). The matrix element $V_{fi}$ involves the dipole operator, $⟨f|\mathbf{r}|i⟩$, giving rise to electric dipole selection rules: $\Delta l=\pm 1$ and $\Delta m=0,\pm 1$ for atomic transitions. These rules determine which spectral lines are allowed or forbidden.

The same formalism describes scattering processes. In particle scattering, the perturbation might represent a static potential $V(\mathbf{r})$. The initial state is a plane wave (momentum ${\mathbf{p}}_i$), and the final state is another plane wave (${\mathbf{p}}_f$) within a continuum. Applying Fermi's golden rule yields the transition rate into a solid angle $d\Omega$, which is directly related to the differential scattering cross-section, $d\sigma /d\Omega$. The matrix element $V_{fi}$ is simply the Fourier transform of the scattering potential, connecting the formalism directly to the Born approximation in scattering theory.

The Adiabatic and Sudden Approximations

Not all perturbations are harmonic or constant. Two fundamental limiting behaviors describe how a system responds to changes in its Hamiltonian.

The adiabatic approximation applies when the perturbation $V(t)$ varies infinitely slowly compared to the natural time scales of the system (given by inverse Bohr frequencies, $1/{\omega }_{mn}$). In this limit, a system starting in an eigenstate of $H_0$ will remain in the instantaneous eigenstate of the full, time-varying Hamiltonian $H(t)$. There is no transition between the original eigenstates; the system adapts smoothly. The geometric phase (Berry's phase) is a profound consequence of adiabatic evolution.

In stark contrast, the sudden approximation applies when the perturbation changes almost instantaneously—faster than any system response time. Here, the state vector $\Psi$ has no time to evolve during the change. Immediately after a sudden change of the Hamiltonian from $H_0$ to $H^{\prime }$, the wavefunction remains the same, but it must now be expanded in the new eigenstates of $H^{\prime }$. The probability to be found in a new eigenstate ${\varphi }_n^{\prime }$ is $|⟨{\varphi }_n^{\prime }|\Psi ⟩{|}^2$. This is useful for modeling processes like beta decay or the sudden switching on of a strong field.

Common Pitfalls

1. Misapplying Fermi's Golden Rule: The golden rule formula $\frac{2\pi }{\hslash }|V_{fi}{|}^2\rho (E_f)$ is only valid for a transition into a continuum of states leading to a constant rate. Applying it to a transition between two discrete states (which yields an oscillating probability) is incorrect. Always check the density of final states.
2. Ignoring Energy Conservation: In the first-order amplitude $c_f^{(1)}(t)$, the integral is largest when the perturbation has frequency components matching ${\omega }_{fi}$. A common error is to forget that for harmonic perturbations, energy conservation ($E_f=E_i\pm \hslash \omega$) emerges from this resonance condition in the long-time limit.
3. Confusing Adiabatic and Sudden Limits: These are opposite extremes. A perturbation that is "fast" or "slow" is relative to the system's internal dynamics ($\hslash /|E_f-E_i|$). Misidentifying the regime leads to wrong predictions about transition probabilities. For example, quickly changing a parameter in a two-level system does not guarantee a sudden approximation; if the level spacing is large, the change may still be adiabatic.
4. Overlooking Selection Rules: When calculating matrix elements $V_{fi}$ for processes like photon absorption, failing to consider the symmetry of the states and the perturbation can lead to predicting a non-zero rate for a transition that is actually forbidden by angular momentum or parity conservation. Always analyze the integrals' symmetry properties first.

Summary

• Time-dependent perturbation theory solves for transition probabilities by expanding the state in known, unperturbed eigenstates and treating the time-varying interaction $V(t)$ as a small driver of change.
• Fermi's golden rule, $\Gamma =\frac{2\pi }{\hslash }|V_{fi}{|}^2\rho (E_f)$, provides a constant transition rate for irreversible processes from a discrete state into a continuum of final states, underpinning calculations in spectroscopy and scattering.
• The interaction of light with matter is modeled as a harmonic perturbation, leading to the concepts of absorption and stimulated emission and yielding selection rules based on the matrix elements of the dipole operator.
• The adiabatic approximation describes the system smoothly following a slowly changing Hamiltonian, while the sudden approximation applies when the Hamiltonian changes too quickly for the state to respond, leaving the wavefunction momentarily unchanged.