Time-Dependent Perturbation Theory and Molecular Transitions

Every absorption and emission spectrum recorded in a chemistry laboratory — UV-Vis of a conjugated dye, infrared of a carbonyl stretch, NMR of a proton environment, microwave rotational spectra of interstellar molecules detected by radio telescopes — is a direct experimental manifestation of the $P_{i\to k}(t)$ machinery derived here, converted (via the long-time limit) into steady absorption cross-sections and line intensities. Laser design depends on stimulated emission rates computed from exactly this formalism, extended to include population inversion and cavity feedback. Time-resolved spectroscopy — pump-probe experiments watching a photoexcited molecule relax on femtosecond-to-picosecond timescales — directly measures the short-time, not-yet-saturated regime of $P_{i\to k}(t)$, where the quadratic-in-time growth near resonance and the Rabi-flopping behavior at very strong coupling both become experimentally visible, for instance in Rabi oscillations of a two-level quantum dot or a nitrogen-vacancy center driven by a resonant laser used in emerging quantum-information technologies. Magnetic resonance imaging and NMR spectrometers apply the identical perturbative logic to nuclear spin transitions driven by radiofrequency pulses, where the resonance condition $\omega=\omega_{ki}$ becomes the Larmor frequency condition central to every pulse sequence. In atmospheric and astrochemistry, the rotational and vibrational transition probabilities computed from this framework underlie the interpretation of infrared absorption by greenhouse gases and the identification of interstellar molecules from their microwave and millimeter-wave rotational fingerprints.

Ask the tutor: How does taking the long-time limit of this resonance lineshape turn a growing probability into the constant rate constant used in Fermi's Golden Rule?

Consider a single molecule described, in the absence of any external field, by the time-independent Hamiltonian $\hat{H}_0$, with known orthonormal eigenstates $\psi_n$ and energies $E_n$:

If the molecule starts in eigenstate $\psi_i$, the full time-dependent wavefunction under $\hat{H}_0$ alone is simply $\Psi(t) = \psi_i e^{-iE_i t/\hbar}$. Every observable probability, $|\langle \psi_n|\Psi(t)\rangle|^2$, is time-independent: the population sits in $\psi_i$ forever. This is the defining property of a stationary state, and it is why an isolated, field-free molecule never spontaneously jumps between eigenstates of its own unperturbed Hamiltonian. Now switch on light. An oscillating electromagnetic wave couples to the molecule's charges — chiefly through the interaction of its oscillating electric field with the molecular dipole moment — and this coupling is explicitly time-dependent because the field itself oscillates as $\cos\omega t$. The total Hamiltonian becomes

where $\hat{H}'(t)$ is small compared to the energy gaps of $\hat{H}_0$ (weak-field regime, true for essentially any non-laser or moderate-intensity source) and carries the explicit oscillation at frequency $\omega$. Because $\hat{H}(t)$ no longer commutes with itself at different times in the way a static operator would, and because $\hat{H}_0$'s eigenstates are no longer eigenstates of the full Hamiltonian, the wavefunction can now genuinely evolve between $\hat{H}_0$-eigenstates. That evolution — the leakage of probability amplitude from $\psi_i$ into some other $\psi_k$ — is exactly what a spectroscopic transition is. The natural strategy is to keep using the $\hat{H}_0$ eigenstates as a basis (they are complete, and they are the states an experiment actually distinguishes — e.g., different vibrational or electronic levels) but let the expansion coefficients become functions of time:

At $t=0$, if the system starts purely in state $i$, then $c_i(0)=1$ and $c_n(0)=0$ for all $n\ne i$. The entire physical content of "how light drives a transition" is the question: how does $c_k(t)$ grow away from zero for $k\ne i$, and how large does $|c_k(t)|^2$ — the probability of finding the system in state $k$ — become?

Two physical facts already visible in this picture deserve emphasis before any algebra. First, transitions are resonant: the drive frequency $\omega$ must roughly match the level spacing $(E_k-E_i)/\hbar$, or the amplitude sloshes back and forth without net transfer. Second, not every pair of states is connected — whether amplitude can flow from $i$ to $k$ at all depends on whether the matrix element $\langle k|\hat{H}'|i\rangle$ is nonzero, which is a question about the symmetry and shape of the wavefunctions $\psi_i,\psi_k$ and the perturbing operator, not about energy alone. This is the microscopic origin of every spectroscopic selection rule.

The starting point is the exact time-dependent Schrödinger equation with the full Hamiltonian:

where: $\Psi(t)$ is the exact molecular wavefunction, $\hat{H}_0$ the field-free (time-independent) Hamiltonian with known eigenstates, and $\hat{H}'(t)$ the explicit perturbation due to the field. Substituting the expansion $\Psi(t)=\sum_n c_n(t)\psi_n e^{-iE_nt/\hbar}$ and projecting onto $\psi_k$ gives the exact coupled equations for the coefficients:

where: $H'_{kn}(t) \equiv \langle \psi_k|\hat{H}'(t)|\psi_n\rangle$ is the time-dependent perturbation matrix element between eigenstates $k$ and $n$, and $\omega_{kn} \equiv (E_k-E_n)/\hbar$ is the Bohr transition (angular) frequency between them. Reading: this equation is exact — no approximation has yet been made — but it is an infinite set of coupled ODEs, one for every state $k$, each driven by all the other coefficients simultaneously. It cannot generally be solved in closed form; the entire point of perturbation theory is to truncate this ladder. At first order in the perturbation strength, assume the system starts entirely in state $i$ and stays there to a very good approximation (the perturbation is weak, so only a small amplitude ever leaks out). Then set $c_i(t)\approx 1$ and $c_{n\ne i}(t)\approx 0$ on the right-hand side, which decouples the equation for any other state $k$:

where: $c_k^{(1)}(t)$ is the first-order amplitude in state $k$, and $H'_{ki}(t) = \langle \psi_k|\hat{H}'(t)|\psi_i\rangle$ is the single matrix element connecting the initial state to the final state $k$. Integrating from $0$ to $t$, with $c_k^{(1)}(0)=0$:

Reading: this is the master formula of first-order time-dependent perturbation theory. Everything about whether and how strongly a transition $i\to k$ occurs is packed into this single time integral of the matrix element against the oscillating Bohr phase factor. For a light-induced transition, the perturbation is an oscillating field, most simply written as

where: $\hat{V}$ is the time-independent operator part of the perturbation (for the electric-dipole interaction, $\hat{V}$ is built from the dipole operator and field amplitude, made explicit when computing the Transition Dipole Moment), and $\omega$ is the angular frequency of the light. Substituting and integrating produces two terms with denominators $(\omega_{ki}+\omega)$ and $(\omega_{ki}-\omega)$; near resonance for absorption ($E_k>E_i$, so $\omega_{ki}>0$) the second term dominates because its denominator nearly vanishes, and the first (anti-resonant, or "counter-rotating") term is dropped — the rotating wave approximation. This yields

where: $V_{ki}=\langle \psi_k|\hat{V}|\psi_i\rangle$. The transition probability is $P_{i\to k}(t) = |c_k^{(1)}(t)|^2$, and using the identity $|e^{ix}-1|^2 = 4\sin^2(x/2)$ gives the central result:

where: $|V_{ki}|^2$ is the squared coupling matrix element (the "strength" factor), and the $\sin^2(x)/x^2$-type function is the resonance lineshape in frequency, whose width narrows and height grows as $t$ increases. Reading: at exact resonance ($\omega=\omega_{ki}$) this expression's apparent $0/0$ resolves (by l'Hôpital or Taylor expansion) to $P_{i\to k}(t) = |V_{ki}|^2 t^2/4\hbar^2$ — probability grows quadratically in time right at resonance, while off-resonance it oscillates and stays small. As $t\to\infty$, the function $\sin^2[(\omega_{ki}-\omega)t/2]/(\omega_{ki}-\omega)^2$ becomes sharply peaked at $\omega=\omega_{ki}$ and its integral over $\omega$ grows linearly in $t$ — this is precisely the limit that produces a constant transition rate rather than an ever-growing probability, the transition to Fermi's Golden Rule and Transition Rates.

Example 1 — Foundational: a two-level system driven off-resonance. Take a model two-level system (e.g., a simplified spin or a truncated electronic two-state model) with $E_i=0$ and $E_k = 3.20\times10^{-20}\ \text{J}$, so the Bohr frequency is

Suppose the driving light has angular frequency $\omega = 3.000\times10^{14}\ \text{s}^{-1}$ (deliberately off-resonance, not tuned exactly to $\omega_{ki}$) and the coupling matrix element is $V_{ki} = 4.50\times10^{-24}\ \text{J}$. At $t = 2.00\times10^{-12}\ \text{s}$ (2 picoseconds), compute $P_{i\to k}(t)$. First the detuning:

Then

Now assemble:

Compute piece by piece:

So after 2 ps, only about 1 part in $10^5$ of the population has transferred — small, consistent with the perturbative assumption $c_i\approx1$ staying valid, and small because the light is detuned from resonance by roughly 1% of $\omega_{ki}$, which is already enough to strongly suppress transfer at this timescale.

Limiting check: if instead the light were tuned exactly on resonance ($\omega=\omega_{ki}$, $\Delta=0$), the formula's $0/0$ resolves to $P_{i\to k}(t) = |V_{ki}|^2t^2/4\hbar^2$. Plugging the same numbers, $P = (2.025\times10^{-47})(2.00\times10^{-12})^2/[4(1.1122\times10^{-68})] = (2.025\times10^{-47})(4.00\times10^{-24})/(4.449\times10^{-68}) = 1.821\times10^{6}$ — a value far exceeding 1, signaling (correctly) that first-order theory has already broken down at resonance by 2 ps for this coupling strength, and a shorter time or Golden Rule treatment would be needed. This contrast is exactly why resonance is special: off-resonance the probability is capped and oscillatory, on-resonance it grows without the bound built into the perturbative approximation.

Example 2 — Applied: rotational transition driven by microwave radiation. Consider a polar diatomic molecule such as carbon monoxide, CO, with rotational constant $B = 1.9225\ \text{cm}^{-1}$ (a real spectroscopic value), being driven from the rotational ground state $J=0$ to $J=1$ by microwave radiation. The transition energy is

Compute $hc$ first in convenient units: $hc = 1.9864\times10^{-23}\ \text{J cm}$. Then

The resonant angular frequency is

corresponding to a frequency $\nu = \omega_{ki}/2\pi = 1.153\times10^{11}\ \text{Hz} = 115.3\ \text{GHz}$ — squarely in the microwave/millimeter-wave region, matching real rotational spectroscopy instrumentation. Now suppose the microwave source is tuned exactly to this resonance and the coupling matrix element (built from the transition dipole moment and field strength, computed properly in the treatment of the Transition Dipole Moment) works out to $V_{ki} = 2.10\times10^{-27}\ \text{J}$, a realistic order of magnitude for a modest field driving a rotational transition with CO's permanent dipole of 0.122 D. At resonance,

Evaluate the prefactor:

For $P_{i\to k}$ to reach, say, $0.010$ (1% transferred — still safely within first-order validity), solve

This tells us that even a modest microwave field drives a measurable population transfer on a nanosecond timescale — consistent with why rotational spectroscopy is done with continuous-wave or pulsed microwave sources rather than requiring intense lasers: the coupling, while weak in absolute terms, easily builds appreciable probability over realistic interaction times because rotational transitions are so low in energy that even a modest dipole coupling accumulates significant phase over nanoseconds.

Example 3 — Exam-level: comparing two nearly-degenerate final states and the resonance-width argument. A molecule has an initial state $i$ and two nearby excited states, $k_1$ and $k_2$, separated in energy by only $\delta E = 8.0\times10^{-24}\ \text{J}$ (a small splitting, e.g., from a minor perturbation like spin–orbit coupling), with Bohr frequencies

Compute the shift:

Light is tuned exactly to $\omega = \omega_{k_1i} = 5.000\times10^{13}\ \text{s}^{-1}$, driving state $k_1$ resonantly ($\Delta_1=0$) while state $k_2$ is off-resonance by $\Delta_2 = \omega_{k_2i}-\omega = 7.586\times10^{10}\ \text{s}^{-1}$. Both states have equal coupling strength $V_{ki}=3.0\times10^{-25}\ \text{J}$. At $t = 5.0\times10^{-11}\ \text{s}$, compare $P_{i\to k_1}$ and $P_{i\to k_2}$. For $k_1$ (resonant): $P_{i\to k_1}(t) = |V_{ki}|^2t^2/4\hbar^2$.

For $k_2$ (off-resonant), first find $\Delta_2 t/2$:

Compute each factor:

So $P_{i\to k_1}/P_{i\to k_2} \approx 5.058\times10^{-3}/1.260\times10^{-3} \approx 4.0$: the exactly resonant channel wins by about a factor of four over a channel detuned by roughly $0.15\%$ of $\omega_{k_1i}$, even though the coupling strengths were made identical — demonstrating quantitatively how sharp the resonance selectivity is, and why closely-spaced levels can still be spectroscopically resolved when a narrow-linewidth source is used. As $t$ grows further, this ratio grows even more sharply in favor of the resonant channel (the resonant probability grows as $t^2$ initially while the off-resonant one stays bounded and oscillatory), the microscopic root of increasingly fine spectral resolution with longer interaction/coherence times.

The mathematical apparatus of time-dependent perturbation theory is due to Paul Dirac, who introduced it in 1927 explicitly to explain how atoms absorb and emit radiation — the birth of quantum electrodynamics and, simultaneously, of quantitative theoretical spectroscopy. Dirac's method of variation of constants (exactly the $c_n(t)$ expansion developed here) let him derive Einstein's phenomenological absorption and stimulated-emission coefficients from first principles, closing a loop Einstein had opened purely thermodynamically in 1917.