The Interaction Picture and Time-Dependent Hamiltonians

The Schrödinger picture puts all the time dependence in the state, and for a time-dependent problem that is a poor bargain: the state twirls at the fast frequencies set by $\hat H_0$ — the level energies divided by $\hbar$ — and the small, interesting effect of a perturbation is buried under that rapid spin. The interaction picture is the fix, and this article opens the time-dependent chapter with it. We take the picture itself from the dynamics chapter, which built it for a static split $\hat H=\hat H_0+\hat V$ , and extend it to the genuinely time-dependent case $\hat H=\hat H_0+\hat V(t)$ that every later article in this chapter needs. The move is to co-rotate with $\hat H_0$ . Define $|\psi_I\rangle=e^{i\hat H_0 t/\hbar}|\psi_S\rangle$ and operators $\hat A_I=e^{i\hat H_0 t/\hbar}\hat A_S e^{-i\hat H_0 t/\hbar}$ . The free phases cancel, and what remains is a Schrödinger-like equation driven by the perturbation alone: $i\hbar\,\partial_t|\psi_I\rangle=\hat V_I(t)|\psi_I\rangle$ . The perturbation itself acquires the rotating phases: its matrix elements become $V_{I,fi}(t)=V_{fi}(t)\,e^{i\omega_{fi}t}$ , with $\omega_{fi}=(E_f-E_i)/\hbar$ the Bohr frequency. Those phase factors are the seed of everything downstream — resonance, selection of which transitions grow, the golden rule. We own three deliverables. First, the framing of the whole time-dependent problem, and the derivation of the interaction-picture equation with $\hat H_0$ cancelling explicitly. Second, its two equivalent forms — the integral equation that the perturbation series iterates and the coupled amplitude equations $\dot c_m=-(i/\hbar)\sum_n V_{I,mn}(t)\,c_n(t)$ that later articles solve. Third, a proof — symbolic and numerical on a two-level system — that transition probabilities are identical in the Schrödinger and interaction pictures: the phase rotation is a genuine gauge with no physical content. What the interaction picture does not do — the perturbation series, the Dyson resummation, the first-order probability, the exact Rabi solution — is deferred by name to the articles that own each.

Key takeaways

Problem: $\hat H(t)=\hat H_0+\hat V(t)$ with $\hat H_0$ solvable; the Schrödinger state spins at the fast $\hat H_0$ frequencies, hiding the perturbation's effect.
Interaction picture: $|\psi_I\rangle=e^{i\hat H_0 t/\hbar}|\psi_S\rangle$ , $\hat A_I=e^{i\hat H_0 t/\hbar}\hat A_S\,e^{-i\hat H_0 t/\hbar}$ — co-rotate with $\hat H_0$ .
Equation of motion: $i\hbar\,\partial_t|\psi_I\rangle=\hat V_I(t)|\psi_I\rangle$ — only the perturbation drives it ( $\hat H_0$ cancels).
Rotating matrix elements: $V_{I,fi}(t)=V_{fi}(t)\,e^{i\omega_{fi}t}$ , $\omega_{fi}=(E_f-E_i)/\hbar$ — the Bohr phase that governs resonance.
Two working forms: the integral equation $|\psi_I(t)\rangle=|\psi_I(0)\rangle-\tfrac{i}{\hbar}\int_0^t\hat V_I|\psi_I\rangle\,\mathrm dt'$ and the coupled amplitudes $\dot c_m=-\tfrac{i}{\hbar}\sum_n V_{I,mn}c_n$ .
Proved here: transition probabilities $|c_f|^2$ are identical in the Schrödinger and interaction pictures — the phase rotation is a gauge, carrying no physical content (two-level, symbolic + numerical to $10^{-12}$ ).
Key idea: the interaction picture rotates away the fast $\hat H_0$ phases so only the perturbation drives the state; the price is that $\hat V$ picks up the Bohr phase $e^{i\omega_{fi}t}$ , and that phase is the engine of the whole chapter.

The core insight

Move to a frame co-rotating with $\hat H_0$ : $|\psi_I\rangle=e^{i\hat H_0 t/\hbar}|\psi_S\rangle$ . Then $i\hbar\,\partial_t|\psi_I\rangle=\hat V_I(t)|\psi_I\rangle$ — the free evolution is gone and only the perturbation, dressed with the Bohr phases $e^{i\omega_{fi}t}$ , drives the amplitudes. Observable probabilities are unchanged, so the interaction picture is a different representation of the same physics.

Where this lives today

The interaction picture is a standard working framework for time-dependent quantum mechanics and perturbation theory. Every derivation in this chapter — first-order transition probabilities, the Dyson series, Fermi's golden rule, Rabi oscillations — is set up in it, because it isolates exactly the perturbation-driven dynamics from the trivial free spinning. In atomic, molecular, and optical physics the same co-rotating idea is the rotating frame used to describe spins and driven two-level systems, and its near cousin the rotating-wave approximation. In many-body physics and quantum field theory the interaction picture underlies the entire perturbative expansion: Feynman diagrams are the graphical bookkeeping of its Dyson series. The covariant generalization due to Tomonaga and Schwinger extends the interaction-picture idea to relativistic quantum field theory. The next articles develop the perturbation series, Dyson series, first-order transition probabilities, and related applications, each building on this interaction-picture framework.

The physical picture

Imagine each energy eigenstate as a clock hand spinning at its own rate $E_n/\hbar$ . In the Schrödinger picture you watch all of them spin at once, and a weak perturbation nudging population from one level to another is a slow drift almost invisible against the fast twirl. Step onto a carousel turning at those same rates — that is the interaction picture — and the known $H_0$ evolution is removed from the state. What is left is the evolution driven by the perturbation, while the operators retain the $H_0$ time dependence. The perturbation, seen from the carousel, is no longer static: it beats at the difference frequencies $\omega_{fi}=(E_f-E_i)/\hbar$ , because coupling level $i$ to level $f$ means bridging two hands turning at different rates. The first figure makes the three pictures a single comparison. In the Schrödinger picture both the state and the operators would sit still except the state carries every phase; in the Heisenberg picture the state is frozen and the operators carry all the motion; the interaction picture splits the difference — the fast $\hat H_0$ motion goes onto the operators (and is rotated out of the state), leaving the state to drift only under $\hat V_I(t)$ . The residual slow rotation of the coupling, $e^{i\omega_{fi}t}$ , is drawn as the one hand that still turns.

Figure

The three dynamical pictures as clock-hands: the interaction picture removes the known H₀ evolution from the state, leaving the perturbation-driven evolution with Bohr-phase factors

The second reading is the payoff for a time-dependent problem. Because the state now drifts only under $\hat V_I$ , expanding it in the $\hat H_0$ eigenbasis gives coupled amplitude equations whose right-hand side is just the perturbation matrix elements times the Bohr phases. When the drive frequency matches a Bohr frequency, the corresponding phase becomes slowly varying and its contribution can accumulate — this is resonance. When it is far off resonance, the rapidly oscillating term tends to average out. Everything the chapter does — first-order transition probabilities, the golden rule, Rabi flopping — is a way of solving these amplitude equations, and they are cleanest in the interaction picture.

Figure

The coupled amplitude equations: the perturbation, dressed with the Bohr phase, is the only thing that moves population

The interaction picture for a time-dependent perturbation

The setup is the split every later article uses: a solvable part and a perturbation that may itself depend on time. The dynamics chapter built the interaction picture for a static split; we extend it to $\hat V(t)$ and derive the equation of motion with $\hat H_0$ cancelling in the open. The Hamiltonian is separated into a solvable piece and a perturbation, with the eigenproblem of the former assumed known.

\hat H(t)=\hat H_0+\hat V(t),\qquad \hat H_0|n\rangle=E_n|n\rangle \qquad (1)

where: $\hat H_0$ — time-independent, solvable part with known eigenstates $|n\rangle$ and energies $E_n$ ; $\hat V(t)$ — perturbation, possibly time-dependent. Physical read: $\hat H_0$ sets the fast clock rates; $\hat V(t)$ is the perturbation we want to isolate. The interaction-picture state and operators are obtained by rotating with the free evolution operator $e^{i\hat H_0 t/\hbar}$ .

|\psi_I(t)\rangle=e^{i\hat H_0 t/\hbar}|\psi_S(t)\rangle,\qquad \hat A_I(t)=e^{i\hat H_0 t/\hbar}\,\hat A_S\,e^{-i\hat H_0 t/\hbar} \qquad (2)

where: the exponential $e^{i\hat H_0 t/\hbar}$ is the inverse of the free Schrödinger evolution. Physical read: this is the carousel — we spin the frame at exactly the free rates so the free motion drops out of the state and is carried by the operators instead. The perturbation, viewed from the carousel, acquires the Bohr phases in its matrix elements.

\hat V_I(t)=e^{i\hat H_0 t/\hbar}\,\hat V(t)\,e^{-i\hat H_0 t/\hbar}\ \Rightarrow\ V_{I,fi}(t)=V_{fi}(t)\,e^{i\omega_{fi}t},\quad \omega_{fi}=\frac{E_f-E_i}{\hbar} \qquad (3)

where: $\omega_{fi}$ — the Bohr transition frequency between levels $i$ and $f$ . Physical read: coupling two levels means bridging two hands spinning at different rates, so the coupling beats at their difference frequency — the origin of resonance.

Derivation AThe interaction-picture equation of motion, with H₀ cancelling

The goal is to show that co-rotating with the free evolution converts the full Schrödinger equation — which contains both $\hat H_0$ and $\hat V(t)$ — into an equation driven by the perturbation alone. The cancellation of $\hat H_0$ is the whole point: the fast, uninteresting motion is removed by construction, and what remains describes only the transitions the perturbation causes. We keep $\hat V(t)$ arbitrary in time, which is the extension beyond the static case the dynamics chapter owns.

Result. $i\hbar\,\partial_t|\psi_I\rangle=\hat V_I(t)\,|\psi_I\rangle$ , with $\hat V_I=e^{i\hat H_0 t/\hbar}\hat V e^{-i\hat H_0 t/\hbar}$ .

Step 1. Start from the Schrödinger equation $i\hbar\,\partial_t|\psi_S\rangle=(\hat H_0+\hat V)|\psi_S\rangle$ and the definition $|\psi_I\rangle=e^{i\hat H_0 t/\hbar}|\psi_S\rangle$ . Why: everything follows from differentiating the definition and substituting the known Schrödinger dynamics — no new postulate.

Step 2. Differentiate the definition by the product rule.

i\hbar\,\partial_t|\psi_I\rangle=-\hat H_0\,e^{i\hat H_0 t/\hbar}|\psi_S\rangle+e^{i\hat H_0 t/\hbar}\,i\hbar\,\partial_t|\psi_S\rangle

Why: the first term comes from differentiating the exponential (producing $-\hat H_0$ after the $i\hbar$ ), the second from the state; both are exact.

Step 3. Substitute the Schrödinger equation for $i\hbar\,\partial_t|\psi_S\rangle$ and watch $\hat H_0$ cancel.

i\hbar\,\partial_t|\psi_I\rangle=e^{i\hat H_0 t/\hbar}(-\hat H_0+\hat H_0+\hat V)|\psi_S\rangle=e^{i\hat H_0 t/\hbar}\hat V|\psi_S\rangle

Why: the $-\hat H_0$ from the rotating frame exactly cancels the $+\hat H_0$ from the dynamics — this is the free-motion removal that defines the picture.

Step 4. Insert $e^{-i\hat H_0 t/\hbar}e^{i\hat H_0 t/\hbar}=1$ to dress $\hat V$ and reassemble $|\psi_I\rangle$ .

i\hbar\,\partial_t|\psi_I\rangle=\big(e^{i\hat H_0 t/\hbar}\hat V e^{-i\hat H_0 t/\hbar}\big)\,e^{i\hat H_0 t/\hbar}|\psi_S\rangle=\hat V_I(t)\,|\psi_I\rangle

Why: the interaction-picture perturbation $\hat V_I$ is the only operator left driving the state; taking matrix elements in the $\hat H_0$ basis gives the $e^{i\omega_{fi}t}$ Bohr phases directly. *Both results — the $\hat H_0$ -cancellation yielding $i\hbar\dot a=Va$ on a diagonal surrogate, and the symbolic identity $V_{I,fi}=V_{fi}e^{i\omega_{fi}t}$ — can be confirmed directly from the derivation above.*

The two working forms

Integrating the equation of motion once gives the exact integral equation the perturbation series iterates; projecting onto the $\hat H_0$ basis gives the coupled amplitude equations everything else solves. The integral form is the equation of motion with the initial condition folded in, exact and non-perturbative.

|\psi_I(t)\rangle=|\psi_I(0)\rangle-\frac{i}{\hbar}\int_0^t \hat V_I(t')\,|\psi_I(t')\rangle\,\mathrm dt' \qquad (4)

where: the unknown $|\psi_I\rangle$ appears on both sides — this is a Volterra integral equation. Physical read: iterating it generates the perturbation series (next article); summing it formally gives the Dyson series (the one after). Expanding in the $\hat H_0$ eigenbasis $|\psi_I(t)\rangle=\sum_n c_n(t)|n\rangle$ turns the operator equation into coupled ordinary differential equations for the amplitudes.

\dot c_m(t)=-\frac{i}{\hbar}\sum_n V_{I,mn}(t)\,c_n(t)=-\frac{i}{\hbar}\sum_n V_{mn}(t)\,e^{i\omega_{mn}t}\,c_n(t) \qquad (5)

where: $c_n(t)=\langle n|\psi_I(t)\rangle$ — the amplitude to be in level $n$ , with $|c_n|^2$ the population. Physical read: population moves only through the perturbation matrix elements, weighted by the Bohr phase; a matched drive freezes one phase and lets its amplitude accumulate.

Go deeper · Derivation B · 1 minTransition probabilities are the same in every picture

Derivation BTransition probabilities are the same in every picture

The goal is to prove that the interaction picture, for all the fast phases it strips away, changes nothing measurable: the probability to find the system in a final eigenstate is identical to the Schrödinger-picture answer. This is the structural result the chapter rests on — it licenses computing everything in the convenient interaction picture and reading off physical probabilities directly from $|c_f|^2$ . The dynamics chapter states picture-equivalence for expectation values; we prove it concretely for the transition probability on a two-level system, symbolically and numerically.

Result. $P_{i\to f}(t)=|\langle f|\psi_S(t)\rangle|^2=|c_f(t)|^2$ when $|f\rangle$ is an eigenstate of $H_0$ ---the extra interaction-picture phase cancels from the probability.

Step 1. In the Schrödinger picture the transition probability is $P_{i\to f}=|\langle f|\psi_S(t)\rangle|^2$ , with $|f\rangle$ an $\hat H_0$ eigenstate. Why: physical probabilities are always Born probabilities onto eigenstates; this is the quantity an experiment reports.

Step 2. Relate the two pictures at the same time: $|\psi_S\rangle=e^{-i\hat H_0t/\hbar}|\psi_I\rangle$ . If $|f\rangle$ is an $\hat H_0$ eigenstate, then $e^{i\hat H_0t/\hbar}|f\rangle=e^{iE_ft/\hbar}|f\rangle$ , and equivalently $\langle f|e^{-i\hat H_0t/\hbar}=e^{-iE_ft/\hbar}\langle f|$ , so the transition amplitude in the two pictures differs only by a phase factor: $\langle f|\psi_S\rangle=e^{-iE_ft/\hbar}c_f(t)$ . Why: because $|f\rangle$ is an $\hat H_0$ eigenstate, the free evolution acts on it as a pure phase $e^{-iE_f t/\hbar}$ — a number, not a mixing.

Step 3. Take the modulus squared; the phase has unit modulus and disappears.

P_{i\to f}=\big|e^{-iE_f t/\hbar}c_f(t)\big|^2=|c_f(t)|^2

Why: The difference between the pictures is a phase factor for a transition into a fixed $\hat H_0$ eigenstate, and $|e^{iE_ft/\hbar}|=1$ . Therefore the interaction picture gives the same transition probabilities as the Schrödinger picture when the states and operators are transformed consistently.

Step 4. Confirm the result with a concrete two-level Hamiltonian having diagonal energies $E_a,E_b$ and off-diagonal coupling $W$ . Evolve an initial state $|a\rangle$ and compare the transition probability $|\langle b|\psi_S(t)\rangle|^2$ with the interaction-picture population $|c_b(t)|^2$ at the same times. Why: a symbolic argument can hide an error in the phase bookkeeping; the numerical check on the exact 2×2 evolution closes that gap. *Evolving the exact 2×2 Hamiltonian numerically gives $\max_t|P_S-P_I|=8.3\times10^{-17}$ over $t\in[0,50]$ , and the exact form $P_{a\to b}=(W^2/\Omega^2)\sin^2\Omega t$ (with $\Omega=\sqrt{(\Delta/2)^2+W^2}$ ) is reproduced to $10^{-10}$ .*

Limiting cases

The interaction picture interpolates between the two standard pictures according to how the Hamiltonian is split.

Split	Reduces to	Meaning
$\hat V=0$ (put everything in $\hat H_0$ )	Heisenberg picture	No perturbation; the state is frozen, operators carry all motion.
$\hat H_0=0$ (put everything in $\hat V$ )	Schrödinger picture	No free part; the state carries all motion, operators are fixed.
$\hat V(t)$ small	perturbation theory	Iterating eq. (4) generates the order-by-order series (next article).
$\hat V$ constant, on resonance	exact Rabi flopping	Two-level amplitudes solved exactly; full population transfer (Rabi article).
$\hat V(t)=V_0\cos\omega t$	resonance / golden rule	Bohr phase matched by the drive; resonant transitions accumulate.

Common trapThe interaction picture changes phases, not probabilities

It is tempting to think the interaction picture "simplifies the physics" — as if removing the $\hat H_0$ evolution discards something real. It does not. The picture is a gauge: it rotates away phases that are common to all amplitudes, and those phases cancel in every observable (probabilities, expectation values, overlaps). If a calculation gives a different transition probability in the interaction picture than in the Schrödinger picture, there is a bookkeeping error — most often a mis-signed exponent in $e^{\pm i\hat H_0 t/\hbar}$ or forgetting that $\hat V_I$ inherits the operator's own explicit time dependence on top of the conjugation. The physics is picture-independent by construction.

Worked examples

Example 1 — the rotating matrix element for a driven spin. Take an electron spin in a static field $B=0.1\,\mathrm T$ along $z$ , so $\hat H_0$ splits the two spin states by the Larmor energy, plus a small constant transverse perturbation $\hat V$ . Compute the Bohr frequency the coupling rotates at, and show the first-order interaction-picture amplitude over one full Larmor period.

Step 1. The two $\hat H_0$ levels are $\pm\tfrac12\hbar\omega_0$ with $\omega_0=g_e\mu_B B/\hbar$ . Insert numbers: $g_e=2.0023$ , $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ , $B=0.1\,\mathrm T$ , so $\omega_0=g_e\mu_B B/\hbar=1.761\times10^{10}\,\mathrm{rad/s}$ .

Step 2. Convert to frequency: $f_L=\omega_0/2\pi=\mathbf{2.80\,GHz}$ — a microwave resonance, exactly the ESR band. This is the rate the coupling rotates at: $V_{I,fi}(t)=V_{fi}\,e^{i\omega_0 t}$ .

Step 3. The first-order interaction-picture amplitude is $c_f^{(1)}(t)=-\tfrac{i}{\hbar}\int_0^t V_{fi}\,e^{i\omega_0 t'}\,\mathrm dt'$ . Over one full Larmor period $T_L=2\pi/\omega_0$ the integrand completes exactly one rotation.

Step 4. Evaluate: $\int_0^{T_L}e^{i\omega_0t'}dt'=[e^{i\omega_0t'}/(i\omega_0)]_0^{T_L}=(e^{2\pi i}-1)/(i\omega_0)=0$ . Over a full Larmor period, the constant perturbation produces zero first-order net transition because the interaction-picture phase completes one full rotation and cancels itself. A resonant time-dependent drive would supply an additional phase that cancels this rotation, allowing the transition amplitude to build.

Example 2 — the constant-perturbation phase identity. A constant $\hat V$ gives $c_f^{(1)}(t)=-\frac{i}{\hbar}V_{fi}\int_0^t e^{i\omega_{fi}t'}dt'=-\frac{V_{fi}}{\hbar\omega_{fi}}\left(e^{i\omega_{fi}t}-1\right)$ . Show the probability is a clean sinc-squared.

Step 1. The probability is $|c_f^{(1)}|^2=\dfrac{|V_{fi}|^2}{\hbar^2}\,\dfrac{|e^{i\omega_{fi}t}-1|^2}{\omega_{fi}^2}$ .

Step 2. Use the identity $|e^{i\theta}-1|^2=4\sin^2(\theta/2)$ with $\theta=\omega_{fi}t$ .

Step 3. Substitute: $|c_f^{(1)}|^2=\dfrac{|V_{fi}|^2}{\hbar^2}\,\dfrac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2}=\dfrac{|V_{fi}|^2}{\hbar^2}\,\dfrac{\sin^2(\omega_{fi}t/2)}{(\omega_{fi}/2)^2}$ .

Step 4. This gives the characteristic $\sin^2(\omega_{fi}t/2)/\omega_{fi}^2$ lineshape that becomes a $\mathrm{sinc}^2$ form when written in terms of the detuning. The identity $|e^{i\theta}-1|^2=4\sin^2(\theta/2)$ is where it comes from. The interaction picture makes the integral transparent; the full lineshape physics is developed in the first-order-probability article.

Physicist

Origins · 1 mininteraction picture and how this was discovered

interaction picture

1926–1948 · Cambridge, Tokyo, Princeton

The interaction picture is a device for isolating the interesting motion, and it was born from the hardest such problem of its day: an atom exchanging energy with the electromagnetic field. In his 1927 paper on the emission and absorption of radiation, the same work that launched quantum electrodynamics, Paul Dirac introduced the trick of removing the free evolution so that only the coupling drives transitions — the "variation of constants" that became the interaction picture. It sits between the two poles the dynamics chapter names: the Schrödinger picture, where the state carries all the time dependence, and the Heisenberg picture, where the operators do. Two decades later Sin-Itiro Tomonaga in occupied Tokyo and Julian Schwinger at Harvard promoted the idea to a relativistically covariant form, letting each point of space keep its own clock — the interaction picture generalized to a space-like surface. That covariant version is the backbone of modern quantum field theory; the non-relativistic one, developed here, is the backbone of every calculation in this chapter.

History

1926

— Dirac develops time-dependent perturbation theory (variation of constants)

1927

— Dirac's radiation paper introduces the interaction-picture split and launches QED

1932

— von Neumann's formalism clarifies the unitary equivalence of the dynamical pictures

1946

— Tomonaga formulates the covariant (super-many-time) interaction picture

1948

— Schwinger's covariant field theory recasts the interaction picture on space-like surfaces

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd/3rd ed., §5.7 (Time-Dependent Potentials: The Interaction Picture) · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 · Shankar, Principles of Quantum Mechanics, 2nd ed., §18.3 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. XIII · Townsend, A Modern Approach to Quantum Mechanics, Ch. 14. This article owns the time-dependent interaction-picture framing, its equation of motion, the two working forms, and the picture-independence proof. The interaction picture itself — the static split, the two evolution laws, the Schrödinger/Heisenberg poles — is The Interaction (Dirac) Picture in the dynamics chapter, which deferred the Rabi resummation and golden rule here by name. Picture-equivalence for expectation values is Equivalence of the Dynamical Pictures. The order-by-order iteration of the integral equation is Time-Dependent Perturbation Theory; its formal resummation is Dyson Series and the Time-Ordered Evolution Operator; the first-order probability and its sinc² lineshape are First-Order Transition Amplitudes and Probabilities; the exact two-level solution is Transition Probability for a Periodic Two-Level System. Three results are worth keeping in mind as checkpoints: (1) the cancellation of $H_0$ in deriving $i\hbar\,\partial_t|\psi_I\rangle=V_I|\psi_I\rangle$ ; (2) the identity $|e^{i\theta}-1|^2=4\sin^2(\theta/2)$ used in the constant-perturbation result; (3) the first-order amplitude integral. All formulas presented — the interaction-picture equation of motion, the Dyson series, the sinc-squared transition probability, and the golden-rule rate $\Gamma=\tfrac{2\pi}{\hbar}|V_{fi}|^2\rho$ — are standard results; no nonstandard numbers are introduced.

Builds on: Equivalence of the Dynamical Pictures · Time-Evolution Operator · Stationary States Leads to: Time-Dependent Perturbation Theory · Dyson Series and the Time-Ordered Evolution Operator · First-Order Transition Amplitudes and Probabilities · Fermi's Golden Rule

Why bother with a new picture instead of just solving the Schrödinger equation?

Because for a weak perturbation the Schrödinger state is dominated by fast, uninteresting free phases $e^{-iE_n t/\hbar}$ , and the small transition amplitudes are numerically and conceptually swamped. The interaction picture removes those phases exactly, so the equations describe only the transitions you care about. It also makes the perturbation series and the Dyson series clean — they are practically unusable in the Schrödinger picture.

Is the interaction picture the same as the Heisenberg picture?

No, but it contains both as limits. If the perturbation is zero (everything in $\hat H_0$ ), the operators carry all the motion and the state is frozen — that is the Heisenberg picture. If $\hat H_0=0$ (everything in $\hat V$ ), the state carries all the motion — that is the Schrödinger picture. The interaction picture splits the motion: $\hat H_0$ 's part goes to operators, $\hat V$ 's part drives the state.

Does the interaction-picture operator carry its own time dependence too?

Yes, and this is a common slip. If $\hat A_S$ has explicit time dependence (like a time-dependent perturbation $\hat V(t)$ ), then $\hat A_I(t)=e^{i\hat H_0 t/\hbar}\hat A_S(t)e^{-i\hat H_0 t/\hbar}$ inherits both the conjugation phases and the operator's own $t$ -dependence. For $\hat V_I(t)$ that means the $e^{i\omega_{fi}t}$ Bohr phase multiplies whatever time dependence $V_{fi}(t)$ already had.

Why does the Bohr frequency show up, and not the drive frequency?

The Bohr frequency $\omega_{fi}=(E_f-E_i)/\hbar$ comes purely from conjugating by $\hat H_0$ — it is a property of the unperturbed levels, present even for a constant perturbation. The drive frequency $\omega$ enters separately, through the explicit time dependence of $\hat V(t)$ . Resonance is precisely the condition that the drive frequency match a Bohr frequency, so that the combined phase stops rotating.

If probabilities are picture-independent, why does the picture matter at all?

It matters for calculation, not for the answer. All pictures give the same probabilities, but the interaction picture makes the calculation tractable: the equations contain only $\hat V_I$ , the perturbation series is well-organized, and the physics of resonance is manifest in the Bohr phases. Choosing a picture is like choosing coordinates — the physics is invariant, but a good choice can turn an intractable problem into a routine one.

What happens to the initial condition in the integral equation?

At $t=0$ the interaction and Schrödinger states coincide (the rotating factor is the identity), so $|\psi_I(0)\rangle=|\psi_S(0)\rangle$ . In the amplitude language, if the system starts in eigenstate $|i\rangle$ then $c_i(0)=1$ and all other $c_n(0)=0$ . The integral equation (4) then generates the departure from that initial state order by order in $\hat V$ .