Time-Dependent Perturbation Theory

Derivation

Time-Dependent Perturbation Theory

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Statement

For a system with time-independent Hamiltonian \(\hat{H}_0\) subject to a time-dependent perturbation \(\hat{V}(t)\) switched on at \(t_0\), with the system prepared in an eigenstate \(|i\rangle\) of \(\hat{H}_0\), the first-order amplitude to be found in a different eigenstate \(|f\rangle\) at time \(t\) is \(\displaystyle c_f^{(1)}(t) = -\frac{i}{\hbar}\int_{t_0}^{t} \langle f|\hat{V}(t')|i\rangle\, e^{i\omega_{fi}t'}\,dt'\), where \(\omega_{fi} = (E_f - E_i)/\hbar\).

Why it matters

Almost every process in which a quantum system exchanges energy with a driving field — absorption and emission of light, spectroscopic line intensities, scattering rates, and the decay of excited states — is computed at leading order from this single amplitude. It converts an abstract coupling \(\hat V(t)\) into a number \(|c_f^{(1)}|^2\) that is a measurable transition probability.

It is also the gateway to Fermi's golden rule: taking a sinusoidal perturbation and a continuum of final states turns this integral into a constant transition rate, the workhorse of atomic, molecular, and condensed-matter physics.

Assumptions

The unperturbed problem is solved exactly.Without a complete known eigenbasis \(\{|n\rangle\}\) of \(\hat H_0\) with \(\hat H_0|n\rangle = E_n|n\rangle\), there is no fixed set of "levels" to make transitions between and the expansion below has no meaning.

The perturbation is weak over the interval.If \(\hat V\) is not small, first order is a poor approximation: population leaves \(|i\rangle\) appreciably, \(c_i(t)\) departs from 1, and the neglected higher-order terms (and back-reaction on the initial state) dominate.

The initial state is a single eigenstate \(|i\rangle\) at \(t_0\).For a superposition or mixed initial state the linear amplitudes must be summed coherently (or the density matrix propagated); using one \(|i\rangle\) alone then misses interference between initial components.

The spectrum and states are those of \(\hat H_0\) alone (no adiabatic dressing).If instead one wants transitions between the instantaneous eigenstates of the full slowly varying Hamiltonian, this "sudden-basis" expansion is the wrong frame and Berry/adiabatic corrections are dropped.

Convergence of the Dyson series is assumed at the order kept.Near a resonance the effective expansion parameter is \(|V_{fi}|t/\hbar\), which can grow without bound; truncating at first order then breaks even for "small" \(\hat V\) once \(t\) is large.

Derivation

\[ i\hbar\,\frac{\partial}{\partial t}|\psi(t)\rangle = \left(\hat H_0 + \hat V(t)\right)|\psi(t)\rangle \]

Start from the exact time-dependent Schrödinger equation with the split \(\hat H = \hat H_0 + \hat V(t)\); this is the postulate we perturb. A

\[ |\psi(t)\rangle = \sum_n c_n(t)\, e^{-iE_n t/\hbar}\,|n\rangle \]

Expand in the complete unperturbed basis, factoring out the free phase \(e^{-iE_n t/\hbar}\) so that \(c_n\) would be constant if \(\hat V=0\). Legal because \(\{|n\rangle\}\) is complete. A

\[ i\hbar\sum_n \dot c_n\, e^{-iE_n t/\hbar}|n\rangle = \sum_n c_n\, e^{-iE_n t/\hbar}\,\hat V(t)|n\rangle \]

Insert the expansion into step 1. The \(\hat H_0\) term cancels the \(E_n\) produced by \(\partial_t\) acting on the free phase, leaving only the \(\hat V\) coupling. A

\[ i\hbar\,\dot c_f = \sum_n c_n(t)\, \langle f|\hat V(t)|n\rangle\, e^{i\omega_{fn} t}, \qquad \omega_{fn}\equiv\frac{E_f - E_n}{\hbar} \]

Project onto \(\langle f|\) and use orthonormality \(\langle f|n\rangle=\delta_{fn}\). This is an exact set of coupled ODEs for the coefficients — the interaction-picture equations of motion. B

\[ \hat V(t)\to \lambda\hat V(t),\qquad c_n(t) = c_n^{(0)} + \lambda\, c_n^{(1)}(t) + \lambda^2 c_n^{(2)}(t)+\cdots \]

Introduce a bookkeeping parameter \(\lambda\) and expand each coefficient as a power series in it; equate powers of \(\lambda\) order by order. Legal as a formal perturbative (Dyson) expansion. C

\[ c_n^{(0)}(t) = \delta_{ni} \quad(\text{constant}),\qquad i\hbar\,\dot c_f^{(1)} = \sum_n c_n^{(0)}\,\langle f|\hat V(t)|n\rangle\, e^{i\omega_{fn}t} \]

Zeroth order (\(\lambda^0\)): with \(\hat V\) absent the coefficients are constant, fixed by the initial condition \(c_n(t_0)=\delta_{ni}\). First order (\(\lambda^1\)): the right side uses the zeroth-order coefficients. B

\[ i\hbar\,\dot c_f^{(1)}(t) = \langle f|\hat V(t)|i\rangle\, e^{i\omega_{fi}t} \]

The sum collapses because \(c_n^{(0)}=\delta_{ni}\) selects the single term \(n=i\); only the matrix element out of the initial state survives at this order. A

\[ c_f^{(1)}(t) = -\frac{i}{\hbar}\int_{t_0}^{t}\langle f|\hat V(t')|i\rangle\, e^{i\omega_{fi}t'}\,dt' \]

Integrate step 7 from \(t_0\) to \(t\) using \(c_f^{(1)}(t_0)=0\) for \(f\neq i\) (all first-order population starts at zero). The \(1/i\hbar\) becomes \(-i/\hbar\). Rearranged in symbols before any numbers are inserted. A

Result

\[ c_f^{(1)}(t) = -\frac{i}{\hbar}\int_{t_0}^{t}\langle f|\hat V(t')|i\rangle\, e^{i\omega_{fi}t'}\,dt', \qquad \omega_{fi}=\frac{E_f-E_i}{\hbar} \]

Reading. The amplitude to end up in \(|f\rangle\) is the coupling matrix element \(\langle f|\hat V|i\rangle\) accumulated over time, but each instant is weighted by the phase \(e^{i\omega_{fi}t'}\) at the Bohr transition frequency \(\omega_{fi}\). Fourier-like, the transition is efficient only when \(\hat V(t)\) contains frequency components near \(\omega_{fi}\) — this is the origin of resonance. The transition probability is \(P_{i\to f}(t)=|c_f^{(1)}(t)|^2\).

Units check. \(\langle f|\hat V|i\rangle\) has units of energy (J); the integral over \(dt'\) gives J·s; dividing by \(\hbar\) (J·s) yields a dimensionless amplitude, as required for a probability amplitude. The phase \(\omega_{fi}t'\) is (rad/s)(s) = dimensionless. Consistent.

Limiting cases

Constant perturbation \(\hat V\) switched on at \(t_0=0\): the integral gives \(c_f^{(1)}=-\frac{V_{fi}}{\hbar}\,\frac{e^{i\omega_{fi}t}-1}{\omega_{fi}}\), so \(P_{i\to f}=\frac{|V_{fi}|^2}{\hbar^2}\,\frac{\sin^2(\omega_{fi}t/2)}{(\omega_{fi}/2)^2}\), the familiar \(\mathrm{sinc}^2\) diffraction pattern in energy.
Exact degeneracy \(\omega_{fi}\to 0\): the phase is unity, \(c_f^{(1)}=-\frac{i}{\hbar}\int V_{fi}\,dt'\); probability grows as \(t^2\) — the resonant limit before saturation.
Harmonic drive \(\hat V(t)=\hat W\cos\omega t\): the integrand has phases \(e^{i(\omega_{fi}\pm\omega)t'}\); resonance at \(\omega\approx\omega_{fi}\) (absorption) and \(\omega\approx-\omega_{fi}\) (stimulated emission), with the counter-rotating term small (rotating-wave regime).
Continuum of final states: summing \(P_{i\to f}\) with density of states \(\rho(E_f)\) and taking \(t\to\infty\) gives the constant rate \(\Gamma_{i\to f}=\frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)\) — Fermi's golden rule.

Breaks when

Strong coupling / long times near resonance. When \(|V_{fi}|t/\hbar \sim 1\) the first-order probability approaches or exceeds 1 (a nonsense value); population genuinely oscillates (Rabi flopping) and the full two-level or higher-order treatment is required.
Non-perturbative or non-analytic switching. A perturbation comparable to level spacings, or one that mixes states so strongly that \(\hat H_0\) is no longer a good starting point, invalidates the Dyson expansion entirely; the "unperturbed" basis stops labelling anything physical.
Degenerate initial manifold. If \(|i\rangle\) shares its energy with other states that \(\hat V\) connects, the sum in step 6 cannot be collapsed to a single term; one must first diagonalise \(\hat V\) within the degenerate subspace (degenerate-perturbation logic) before applying this formula.

Failure modes

Sign/factor slip on \(1/i\hbar\): writing \(+\frac{i}{\hbar}\) instead of \(-\frac{i}{\hbar}\). It does not affect \(|c_f|^2\) but corrupts any interference calculation between amplitudes.
Wrong phase sign: using \(e^{-i\omega_{fi}t'}\). The correct interaction-picture factor for \(\langle f|\dots|i\rangle\) is \(e^{+i\omega_{fi}t'}\) with \(\omega_{fi}=(E_f-E_i)/\hbar\); flipping it swaps absorption and emission resonances.
Forgetting the free phase in step 2, i.e. expanding \(|\psi\rangle=\sum c_n|n\rangle\) without \(e^{-iE_n t/\hbar}\). Then \(c_n\) is not slowly varying and the tidy first-order integral does not emerge.
Using \(V_{fi}(t)\) at the final time only rather than integrating it. The amplitude is a time integral; evaluating the matrix element at one instant discards the resonance weighting.
Applying it to a degenerate \(|i\rangle\) without first resolving the degeneracy, producing a divergent or ill-defined \(1/\omega_{fi}\).
Squaring too early: computing \(\sum_f|c_f|^2\) from incoherently added contributions when the physical measurement is coherent, losing interference terms.

Discussion

The structure of the result is best seen in the interaction picture, where states evolve only under \(\hat V_I(t)=e^{i\hat H_0 t/\hbar}\hat V(t)e^{-i\hat H_0 t/\hbar}\). The exact evolution operator is the time-ordered Dyson series \(\hat U_I(t)=\mathcal T\exp\!\left(-\frac{i}{\hbar}\int \hat V_I\,dt'\right)\), and \(c_f^{(1)}\) is simply its first term sandwiched as \(\langle f|\hat U_I^{(1)}|i\rangle\). Everything higher — two-photon processes, virtual intermediate states — lives in the later terms.

The phase factor \(e^{i\omega_{fi}t'}\) is the entire physics of resonance. Because the amplitude is essentially the Fourier transform of the matrix element evaluated at the Bohr frequency, a transition proceeds only when the drive supplies (or removes) exactly the energy gap \(E_f-E_i=\hbar\omega_{fi}\). Energy conservation is not imposed by hand; it emerges dynamically as constructive phase accumulation, sharpening into a delta function as \(t\to\infty\).

Connecting to the threads of this unit: the energy thread appears through \(\omega_{fi}\) enforcing the level gap; the waves thread through the Fourier/interference reading of the integral; and the chance thread through \(P=|c_f^{(1)}|^2\), the Born-rule probability that ties this abstract amplitude to a measured count rate. It is the same weak-coupling logic as nondegenerate stationary perturbation theory, now unrolled in time rather than energy.

A subtle point is the interplay of the truncation error with the long-time limit. First order is a series in the dimensionless \(|V_{fi}|t/\hbar\), not merely in \(|V_{fi}|\); the golden-rule rate is extracted precisely in the window where \(t\) is long enough for the \(\mathrm{sinc}^2\) to be sharp yet short enough that \(P\ll 1\). This double limit — often glossed over — is why the golden rule is a rate (linear in \(t\)) even though the underlying amplitude squared for a single state is quadratic in \(t\) on resonance; the linearity comes only after integrating over the continuum density of states.

Common misconceptions. The formula does not say the transition "happens at" the final time — the amplitude is built up continuously and phases from all earlier instants interfere. Nor does it require \(E_f=E_i\) exactly at finite \(t\): off-resonant transitions have real, nonzero amplitude (the wings of the \(\mathrm{sinc}^2\)), and strict energy conservation is only recovered as \(t\to\infty\).

Worked examples

\[ \hat V(t)=\hat W\,e^{-t/\tau}\ \ (t>0),\qquad c_f^{(1)}(\infty)=-\frac{i}{\hbar}\int_0^{\infty} W_{fi}\,e^{-t'/\tau}\,e^{i\omega_{fi}t'}\,dt' \]

Exponentially decaying kick switched on at \(t=0\). Set up the standard integral before numbers. A

\[ c_f^{(1)}(\infty) = -\frac{i}{\hbar}\,W_{fi}\,\frac{1}{\,1/\tau - i\omega_{fi}\,}, \qquad P_{i\to f}=\frac{|W_{fi}|^2}{\hbar^2}\,\frac{\tau^2}{1+\omega_{fi}^2\tau^2} \]

Evaluate \(\int_0^\infty e^{-(1/\tau - i\omega_{fi})t'}dt' = (1/\tau - i\omega_{fi})^{-1}\); take modulus squared. B

\[ W_{fi}=1.0\times10^{-3}\ \text{eV}=1.60\times10^{-22}\ \text{J},\ \ \tau=1.0\times10^{-14}\ \text{s},\ \ \omega_{fi}=2.0\times10^{14}\ \text{s}^{-1} \]

Insert numbers. \(\hbar=1.055\times10^{-34}\ \text{J·s}\); \(\omega_{fi}\tau = 2.0\), so \(1+\omega_{fi}^2\tau^2 = 5.0\). A

\[ P_{i\to f}=\frac{(1.60\times10^{-22})^2}{(1.055\times10^{-34})^2}\cdot\frac{(10^{-14})^2}{5.0}\approx 4.6\times10^{-4} \]

Reading. A weak, fast kick transfers about \(0.05\%\) of the population; detuning (\(\omega_{fi}\tau=2\)) suppresses it fivefold relative to resonance. Well within first order (\(P\ll1\)). Units check. \((\text{J})^2/(\text{J·s})^2\times(\text{s})^2\) is dimensionless.

\[ \hat V(t)=\hat W\cos(\omega t)\ \ (0<t<T),\qquad c_f^{(1)}=-\frac{i}{2\hbar}W_{fi}\int_0^{T}\!\left(e^{i(\omega_{fi}+\omega)t'}+e^{i(\omega_{fi}-\omega)t'}\right)dt' \]

Harmonic drive; write \(\cos\omega t=\tfrac12(e^{i\omega t}+e^{-i\omega t})\). A

\[ \text{near resonance }\omega\approx\omega_{fi}:\quad c_f^{(1)}\approx -\frac{i}{2\hbar}W_{fi}\,\frac{e^{i(\omega_{fi}-\omega)T}-1}{i(\omega_{fi}-\omega)},\quad P=\frac{|W_{fi}|^2}{\hbar^2}\,\frac{\sin^2\!\big(\tfrac{(\omega_{fi}-\omega)T}{2}\big)}{(\omega_{fi}-\omega)^2} \]

Drop the fast counter-rotating term (rotating-wave approximation); keep the near-resonant one. B

\[ W_{fi}=2.0\times10^{-24}\ \text{J},\ \ T=1.0\times10^{-9}\ \text{s},\ \ \Delta\equiv\omega_{fi}-\omega = 1.0\times10^{9}\ \text{s}^{-1} \]

Insert numbers. \(\Delta T/2 = 0.50\), \(\sin^2(0.50)=0.2298\), \(\Delta^2=1.0\times10^{18}\). A

\[ P_{i\to f}=\frac{(2.0\times10^{-24})^2}{(1.055\times10^{-34})^2}\cdot\frac{0.2298}{1.0\times10^{18}}\approx 8.3\times10^{-2} \]

Reading. Near resonance the population reaches \(\sim8\%\) in a nanosecond — approaching the edge of validity; at exact resonance \(P\to\frac{|W_{fi}|^2T^2}{4\hbar^2}\) grows as \(T^2\). Units check. \((\text{J})^2/(\text{J·s})^2\times(\text{s}^{-1})^{-2}\) is dimensionless.

Problems

(A) A constant perturbation \(\hat V\) with \(V_{fi}=1.0\times10^{-23}\) J is switched on for the resonant case \(\omega_{fi}=0\) for \(t=2.0\times10^{-12}\) s. Find \(P_{i\to f}\).
Solution
For \(\omega_{fi}=0\), \(c_f^{(1)}=-\frac{i}{\hbar}V_{fi}t\), so \(P=\frac{|V_{fi}|^2 t^2}{\hbar^2}=\frac{(10^{-23})^2(2.0\times10^{-12})^2}{(1.055\times10^{-34})^2}=\frac{(10^{-46})(4.0\times10^{-24})}{1.113\times10^{-68}}\approx 3.6\times10^{-2}\). About 3.6%.
(A) State the transition probability for a constant perturbation switched on at \(t=0\) with nonzero \(\omega_{fi}\), and identify the first zero of \(P(t)\) as a function of \(t\).
Solution
\(P=\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2(\omega_{fi}t/2)}{(\omega_{fi}/2)^2}=\frac{4|V_{fi}|^2}{\hbar^2\omega_{fi}^2}\sin^2(\omega_{fi}t/2)\). Zeros occur when \(\omega_{fi}t/2=n\pi\), i.e. \(t=2\pi n/\omega_{fi}\); the first nonzero zero is at \(t=2\pi/\omega_{fi}\).
(B) For the harmonic drive worked example, at exact resonance \(\Delta=0\) with \(W_{fi}=2.0\times10^{-24}\) J, find the time \(T\) at which the first-order probability reaches \(P=0.10\), and comment on validity.
Solution
At resonance \(P=\frac{|W_{fi}|^2T^2}{4\hbar^2}\). Set \(=0.10\): \(T=\frac{2\hbar}{|W_{fi}|}\sqrt{0.10}=\frac{2(1.055\times10^{-34})}{2.0\times10^{-24}}(0.3162)=\ (1.055\times10^{-10})(0.3162)\approx 3.3\times10^{-11}\) s. Since \(P=0.1\) is not \(\ll1\), first order is becoming unreliable; the exact Rabi result should be used for a precise value.
(B) A perturbation \(\hat V(t)=\hat W\,e^{-t/\tau}\) acts for \(t\in(0,\infty)\). Show that the on-resonance (\(\omega_{fi}=0\)) probability is \(P=\frac{|W_{fi}|^2\tau^2}{\hbar^2}\) and evaluate it for \(W_{fi}=5.0\times10^{-23}\) J, \(\tau=1.0\times10^{-13}\) s.
Solution
From the worked result \(P=\frac{|W_{fi}|^2}{\hbar^2}\frac{\tau^2}{1+\omega_{fi}^2\tau^2}\); at \(\omega_{fi}=0\) this is \(\frac{|W_{fi}|^2\tau^2}{\hbar^2}\). Numerically \(=\frac{(5.0\times10^{-23})^2(10^{-13})^2}{(1.055\times10^{-34})^2}=\frac{(2.5\times10^{-45})(10^{-26})}{1.113\times10^{-68}}\approx 2.2\times10^{-3}\). About 0.22%.
(C) Derive Fermi's golden rule for a constant perturbation by integrating \(P_{i\to f}(t)\) over a continuum of final states with density \(\rho(E_f)\), using \(\lim_{t\to\infty}\frac{\sin^2(\omega t/2)}{\pi t(\omega/2)^2}=\delta(\omega)\).
Solution
Total probability to any final state near \(E_f\): \(P=\sum_f\frac{4|V_{fi}|^2}{\hbar^2\omega_{fi}^2}\sin^2(\omega_{fi}t/2)=\int dE_f\,\rho(E_f)\frac{4|V_{fi}|^2}{\hbar^2\omega_{fi}^2}\sin^2(\omega_{fi}t/2)\). Write \(\omega_{fi}=(E_f-E_i)/\hbar\) so \(dE_f=\hbar\,d\omega_{fi}\). For large \(t\), \(\frac{\sin^2(\omega_{fi}t/2)}{(\omega_{fi}/2)^2}\to 2\pi t\,\delta(\omega_{fi})\). Then \(P=\frac{2\pi t}{\hbar}|V_{fi}|^2\rho(E_f)\), and the rate \(\Gamma=dP/dt=\frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)\), the golden rule, with \(\rho\) evaluated at \(E_f=E_i\).