5.2.1 Interaction Picture#

Prompts

What is the interaction picture? How does it separate fast free evolution (\(H_0\)) from slow perturbative dynamics (\(V(t)\))?
Derive the equation of motion in the interaction picture. Why is it simpler than the full Schrödinger equation?
What is the Dyson series and why must its terms be time-ordered? Explain physically what each order represents.
How does the retarded Green’s function \(G^+(E) = 1/(E - H_0 + \mathrm{i}\epsilon)\) encode the spectrum of \(H_0\), and what is the role of \(\mathrm{i}\epsilon\)?

Lecture Notes#

Overview#

The interaction picture is a choice of quantum dynamics representation that separates the Hamiltonian into a fast, solvable part (\(H_0\)) and a slow, perturbative part (\(V(t)\)). By factoring out the rapid oscillations from \(H_0\), we expose the essential slow dynamics driven by \(V(t)\), making perturbation theory tractable. This section develops the formalism: how to transform states and operators, derive the equation of motion, solve it via the Dyson series (time-ordered exponential), and introduce Green’s functions and Feynman diagrams as organizing tools.

The Interaction Picture#

Three Pictures of Quantum Mechanics#

State evolution can be described in different pictures:

Picture	Evolution of States	Evolution of Operators	Use
Schrödinger	\(\mathrm{i}\hbar d\vert\psi_S\rangle/dt = H\vert\psi_S\rangle\)	Static: \(O_S(t) = O(0)\)	General; states change, operators fixed
Heisenberg	Static: \(\vert\psi_H\rangle = \text{const}\)	\(O_H(t) = U^\dagger O_S U\)	Classical-like; dynamics in observables
Interaction	Driven by \(V_I(t)\)	Driven by \(H_0\)	Separates free propagation from interactions

Definition and Transformations#

In the interaction picture, we separate the Hamiltonian into two parts:

Free/unperturbed part \(H_0\): evolution is rapid but exactly solvable
Interaction part \(V(t)\): slow compared to \(H_0\) (weak coupling regime)

Interaction Picture Transformations

States transform as:

(86)#\[ \vert\psi_I(t)\rangle = \mathrm{e}^{\mathrm{i}H_0 t/\hbar} \vert\psi_S(t)\rangle \]

Operators transform as:

(87)#\[ O_I(t) = \mathrm{e}^{\mathrm{i}H_0 t/\hbar} O_S \mathrm{e}^{-\mathrm{i}H_0 t/\hbar} \]

The interaction-picture perturbation:

(88)#\[ V_I(t) = \mathrm{e}^{\mathrm{i}H_0 t/\hbar} V(t) \mathrm{e}^{-\mathrm{i}H_0 t/\hbar} \]

Physical meaning: Multiplying by \(\mathrm{e}^{\mathrm{i}H_0 t/\hbar}\) factors out the rapid oscillations from \(H_0\), leaving only the slow dynamics driven by \(V(t)\).

Equation of Motion in the Interaction Picture#

The interaction-picture state obeys a simpler equation:

(89)#\[ \mathrm{i}\hbar \frac{\mathrm{d}\vert\psi_I\rangle}{\mathrm{d}t} = V_I(t) \vert\psi_I\rangle \]

Derivation: Interaction Picture Equation

Start with the Schrödinger equation:

\[ \mathrm{i}\hbar \frac{\mathrm{d}\vert\psi_S\rangle}{\mathrm{d}t} = [H_0 + V(t)]\vert\psi_S\rangle \]

Transform to the interaction picture using \(\vert\psi_S\rangle = \mathrm{e}^{-\mathrm{i}H_0 t/\hbar}\vert\psi_I\rangle\), expand the left side via the product rule:

\[ H_0 \mathrm{e}^{-\mathrm{i}H_0 t/\hbar}\vert\psi_I\rangle + \mathrm{i}\hbar \mathrm{e}^{-\mathrm{i}H_0 t/\hbar}\frac{\mathrm{d}\vert\psi_I\rangle}{\mathrm{d}t} = [H_0 + V(t)]\mathrm{e}^{-\mathrm{i}H_0 t/\hbar}\vert\psi_I\rangle \]

The \(H_0\) terms cancel; multiply both sides by \(\mathrm{e}^{\mathrm{i}H_0 t/\hbar}\):

\[ \mathrm{i}\hbar \frac{\mathrm{d}\vert\psi_I\rangle}{\mathrm{d}t} = \mathrm{e}^{\mathrm{i}H_0 t/\hbar} V(t) \mathrm{e}^{-\mathrm{i}H_0 t/\hbar}\vert\psi_I\rangle = V_I(t)\vert\psi_I\rangle\]

Physical content: Expanding \(V_I(t)\) in the energy eigenbasis of \(H_0\) yields oscillatory factors \(\mathrm{e}^{\mathrm{i}\omega_{nm}t}\) with transition frequencies \(\omega_{nm} = (E_n^{(0)} - E_m^{(0)})/\hbar\). Resonance (\(\omega_{nm} \approx 0\)) dramatically enhances coupling between nearly degenerate states.

The Evolution Operator and Dyson Series#

Formal Solution#

The interaction-picture evolution operator \(U_I(t)\) satisfies \(\vert\psi_I(t)\rangle = U_I(t) \vert\psi_I(0)\rangle\) with the formal solution:

Time-Ordered Exponential and Dyson Series

(90)#\[ U_I(t) = \mathcal{T} \exp\!\left(-\frac{\mathrm{i}}{\hbar} \int_0^t \mathrm{d}t' \, V_I(t')\right) \]

Expanding order by order gives the Dyson series:

(91)#\[ U_I(t) = \hat{I} - \frac{\mathrm{i}}{\hbar} \int_0^t \mathrm{d}t_1\, V_I(t_1) + \left(-\frac{\mathrm{i}}{\hbar}\right)^2 \int_0^t \mathrm{d}t_1 \int_0^{t_1} \mathrm{d}t_2\, V_I(t_1) V_I(t_2) + \cdots \]

Each term represents a process with a definite number of interactions, all time-ordered (\(t_1 > t_2 > \cdots\)).

Derivation: Dyson Series

From the equation of motion \(\mathrm{i}\hbar \partial_t U_I = V_I(t) U_I\), \(U_I(0) = \hat{I}\), integrate formally:

\[ U_I(t) = \hat{I} + \frac{1}{\mathrm{i}\hbar} \int_0^t \mathrm{d}t_1 \, V_I(t_1) U_I(t_1) \]

Iterate by substituting \(U_I(t_1)\) back into the integral:

\[ U_I(t) = \hat{I} + \frac{1}{\mathrm{i}\hbar}\int_0^t V_I(t_1)\mathrm{d}t_1 + \left(\frac{1}{\mathrm{i}\hbar}\right)^2 \int_0^t\mathrm{d}t_1 \int_0^{t_1}\mathrm{d}t_2\, V_I(t_1) V_I(t_2) + \cdots \]

The integral limits \(t > t_1 > t_2 > \cdots\) enforce time ordering automatically.

Time Ordering and Causality#

Why Time Ordering Matters

Operators at different times do not commute in general: \([V_I(t_1), V_I(t_2)] \neq 0\). The order in which interactions occur is physical. Time ordering enforces the natural causal sequence: later times act to the left.

The full Schrödinger-picture evolution factorizes as:

(92)#\[ U_S(t) = \mathrm{e}^{-\mathrm{i}H_0 t/\hbar} U_I(t) \]

The first factor handles rapid \(H_0\) oscillations; \(U_I(t)\) captures the slow interactions via the Dyson series.

Transition Amplitudes#

The amplitude to go from state \(\vert i\rangle\) to \(\vert f\rangle\) at time \(t\) is:

(93)#\[ \mathcal{A}_{i\to f}(t) = \langle f \vert U_I(t) \vert i \rangle \]

To first order:

(94)#\[ \mathcal{A}_{i\to f}^{(1)}(t) = -\frac{\mathrm{i}}{\hbar} \int_0^t \mathrm{d}t_1\, \mathrm{e}^{\mathrm{i}\omega_{fi}t_1} \langle f \vert V(t_1) \vert i \rangle \]

where \(\omega_{fi} = (E_f^{(0)} - E_i^{(0)})/\hbar\) is the transition frequency.

Green’s Functions and Feynman Diagrams#

Retarded Green’s Function#

The retarded Green’s function in energy representation is:

Retarded Green’s Function

(95)#\[ G^+(E) = \frac{1}{E - H_0 + \mathrm{i}\epsilon} \]

where \(\epsilon \to 0^+\). The imaginary part enforces causality (retarded boundary conditions). In time domain: \(G^+(t,t') = -\mathrm{i}\theta(t-t')\,\mathrm{e}^{-\mathrm{i}H_0(t-t')/\hbar}\).

The poles of \(G^+(E)\) encode the spectrum of \(H_0\): bound states appear as poles on the real axis.

Note: Connection to Path Integrals

The retarded propagator \(G^+(t,t')\) is the inverse of the free kinetic operator in the path integral (§3). The pole structure of \(G^+(E)\) directly reveals the energy eigenvalues of \(H_0\).

Feynman Diagrams#

The Dyson series has a natural graphical representation:

Lines represent free propagation: \(G_0^+(t,t')\) or \(\mathrm{e}^{-\mathrm{i}H_0(t-t')/\hbar}\)
Vertices represent interactions: insertions of \(V_I(t)\)
Time flows left-to-right (later times to the left)

Each order in the Dyson expansion corresponds to a diagram with a definite number of vertices. This graphical language, developed systematically in quantum field theory, organizes the perturbative expansion and reveals which processes contribute at each order.

Discussion: When Does Perturbation Theory Break Down?

The Dyson series is a formal expansion, but it does not always converge.

What physical condition must hold for the first-order term to dominate? Express the condition in terms of \(V\), \(\hbar\), and the relevant timescale.
Can you construct an example where resummation of all orders changes the qualitative physics (e.g., bound state formation)?
How does the interaction picture help identify which terms are “fast” (oscillating, averaging out) vs. “slow” (contributing to real transitions)?
What is the role of the \(\mathrm{i}\epsilon\) prescription in \(G^+(E)\)? What goes wrong if \(\epsilon = 0\)?

Summary#

Interaction picture: Rotates with free evolution \(H_0\); exposes slow dynamics of \(V_I(t)\) and removes rapid oscillations.
Time-ordered exponential and Dyson series: \(U_I(t) = \mathcal{T}\exp(-\mathrm{i}\int V_I\,\mathrm{d}t/\hbar)\); each order is a causal sequence of interaction events.
Transition amplitude: \(\mathcal{A}_{i\to f}^{(1)} = -(i/\hbar)\int \mathrm{e}^{\mathrm{i}\omega_{fi}t'}\langle f\vert V\vert i\rangle \mathrm{d}t'\); resonance (\(\omega_{fi}\approx 0\)) enhances transitions.
Green’s function: Retarded propagator \(G^+(E) = (E - H_0 + \mathrm{i}\epsilon)^{-1}\) encodes spectrum and causality; Feynman diagrams visualize each order of the Dyson series.

Homework#

1. Transformations between pictures

A state \(|\psi_S(t)\rangle\) evolves in the Schrödinger picture under Hamiltonian \(H = H_0 + V(t)\).

(a) Write the transformation relating \(|\psi_I(t)\rangle\) to \(|\psi_S(t)\rangle\). Why does the interaction picture factor out the \(H_0\) evolution?

(b) Similarly, relate an observable \(O_I(t)\) to \(O_S\). What does this transformation physically represent?

(c) Show that if \(|\psi_I(t)\rangle\) obeys \(\mathrm{i}\hbar d|\psi_I\rangle/dt = \lambda V_I(t)|\psi_I\rangle\), then \(|\psi_S(t)\rangle\) obeys the full Schrödinger equation.

2. The interaction-picture perturbation

The unperturbed Hamiltonian \(H_0\) has eigenvalues \(\{E_n^{(0)}\}\) and eigenstates \(\{|n\rangle\}\). A time-dependent perturbation \(V(t)\) is applied.

(a) Write the Schrödinger-picture perturbation \(V(t)\) in the energy eigenbasis:

\[V(t) = \sum_{m,n} V_{mn}(t) |m\rangle\langle n|\]

(b) Transform to the interaction picture. Show that:

\[V_I(t) = \sum_{m,n} V_{mn}(t) \mathrm{e}^{\mathrm{i}\omega_{mn}t} |m\rangle\langle n|\]

where \(\omega_{mn} = (E_m^{(0)} - E_n^{(0)})/\hbar\).

3. The time-evolution operator in the interaction picture

Define \(U_I(t, t_0)\) such that \(|\psi_I(t)\rangle = U_I(t, t_0)|\psi_I(t_0)\rangle\).

(a) Derive the equation of motion for \(U_I(t, t_0)\).

(b) Show that \(U_I\) satisfies the Dyson series (time-ordered exponential):

\[U_I(t, t_0) = \mathcal{T} \exp\left(-\frac{\mathrm{i}}{\hbar}\int_{t_0}^{t} \lambda V_I(t') \mathrm{d}t'\right)\]

(c) Expand the time-ordered exponential to second order in \(\lambda\) and verify that it correctly satisfies the differential equation.

4. Rabi oscillations in a two-level system

A two-level atom with ground state \(|0\rangle\) (energy \(E_0 = 0\)) and excited state \(|1\rangle\) (energy \(E_1 = \hbar\omega_0\)) is driven by a resonant field \(V(t) = \hbar\Omega \sigma^x \cos(\omega_0 t)\), where \(\sigma^x = |0\rangle\langle 1| + |1\rangle\langle 0|\) and \(\Omega\) is the Rabi frequency.

(a) Decompose \(V(t)\) into rotating and counter-rotating terms. Which term dominates near resonance?

(b) In the interaction picture with \(H_0 = \hbar\omega_0 |1\rangle\langle 1|\), show that \(V_I(t) \approx \hbar\Omega \sigma^x\) (rotating-wave approximation).

(c) Write down the interaction-picture Schrödinger equation for the state \(|\psi_I(t)\rangle = a_0(t)|0\rangle + a_1(t)|1\rangle\).

(d) Solve to find the time evolution. Show that the population oscillates as:

\[P_1(t) = \sin^2(\Omega t)\]

5. Relation between pictures: when to use each

Three quantum systems require calculations:

System A: A bound particle in a slowly-varying potential (e.g., gravity slowly turned on).

System B: An atom driven by a monochromatic laser at high intensity.

System C: A free particle subject to a brief, intense impulse (delta-function kick).

For each system, discuss:

Which picture(s) would be most natural for analysis?
What is your reasoning (e.g., separation of timescales, nature of unperturbed dynamics)?
What would be the main computational challenge in the less-suitable picture(s)?

6. Dyson series and time ordering

Consider a perturbation that acts in two distinct time intervals: \(V(t) \neq 0\) for \(0 < t < t_1\) and \(t_1 < t < 2t_1\), but \(V(t) = 0\) at \(t = t_1\) itself.

(a) Compute the first-order perturbation correction to \(U_I(2t_1, 0)\) by hand (integral without time-ordering).

(b) Compare with the time-ordered result from the Dyson series. Where does the difference appear, and why is time ordering essential for causality?

(c) If the two intervals had \(V(t)\) with different operators \(V_1(t)\) and \(V_2(t)\), would they commute? How does time-ordering handle the non-commutativity?

7. Eigenstate-perturbation correspondence

A system starts in an eigenstate of \(H_0\), say \(|\psi_I(0)\rangle = |n\rangle\). It evolves under a perturbation \(V_I(t)\) that couples \(|n\rangle\) to a nearby state \(|m\rangle\).

(a) Show that the interaction-picture amplitude for transition \(n \to m\) is:

\[a_m(t) = -\frac{\mathrm{i}}{\hbar}\int_0^t \mathrm{e}^{\mathrm{i}\omega_{mn}t'} V_{mn}(t') a_n(t') \mathrm{d}t'\]

(b) In the weak-coupling limit where \(a_n(t) \approx 1\) remains nearly constant, integrate to find the transition probability \(P_{n \to m}(t) = |a_m(t)|^2\) to first order in \(V\).

(c) For a constant coupling \(V_{mn} = V_0\), compute \(P_{n \to m}(t)\). Interpret the result in terms of resonance width and Fourier components of the coupling.