5.2.1 Interaction Picture

5.2.1 Interaction Picture#

Prompts

  • What physical regime motivates the interaction picture, and why is it useful to factor out the background evolution generated by \(\hat{H}_0\)?

  • What is the relationship among \(\hat{U}(t)\), \(\hat{U}_0(t)\), and \(\hat{U}_{\mathcal{I}}(t)\), and why is solving for the full operator generally harder than evolving a single state?

  • Why does \(\hat{H}_0\) cancel in the interaction-picture state equation, leaving \(\mathrm{i}\hbar\,\partial_t\vert\psi\rangle_{\mathcal{I}}=\hat{V}_{\mathcal{I}}\vert\psi\rangle_{\mathcal{I}}\)?

  • In matrix elements of \(\hat{V}_{\mathcal{I}}\), why do Bohr-frequency phases \(\mathrm{e}^{\mathrm{i}\omega_{mn}t}\) appear, and what is the physical meaning of fast versus slow phase winding?

  • How does the equation \(\mathrm{i}\hbar\,\partial_t\hat{U}_{\mathcal{I}}=\hat{V}_{\mathcal{I}}\hat{U}_{\mathcal{I}}\) lead to a time-ordered expansion, and what scattering-history picture does that expansion represent?

Lecture Notes#

Overview#

Time-dependent perturbation theory begins with a Hamiltonian split \(\hat{H}(t)=\hat{H}_0+\hat{V}(t)\). The reference part \(\hat{H}_0\) is taken to be solvable in the sense that its evolution operator \(\hat{U}_0(t)\) is known, while \(\hat{V}(t)\) is treated as a small correction whose influence must be added on top of that baseline. The object to construct is the full propagator \(\hat{U}(t)\), ideally as a controlled expansion in the strength of \(\hat{V}\) rather than by solving the fully coupled problem in one shot.

The interaction picture implements that idea by a change of frame: transform with \(\hat{U}_0^{\dagger}(t)\) so the description comoves with the unperturbed evolution and the remaining time dependence is generated only by \(\hat{V}(t)\), after it is carried along into the new frame. The same split appears whether one tracks a single state or the unitary operator itself, and it is the usual starting point for such expansions.

Problem Setup#

The standard application scenario for time-dependent perturbation theory is a solvable baseline Hamiltonian plus a weak time-dependent drive.

Time-Dependent Perturbation Problem

Take a system with a time-independent reference Hamiltonian \(\hat{H}_0\) and add a perturbation \(\hat{V}(t)\). In the eigenbasis of \(\hat{H}_0\),

\[\begin{split} \begin{split} \hat{H}_0 &= \sum_n \vert n\rangle E_n \langle n\vert,\\ \hat{V}(t) &= \sum_{m,n} \vert m\rangle V_{mn}(t) \langle n\vert. \end{split} \end{split}\]

The full Hamiltonian is

\[ \hat{H}(t)=\hat{H}_0+\hat{V}(t). \]

Assume \(V_{mn}(t)\) is small compared with the characteristic scales set by \(\hat{H}_0\).

Our dynamical target is the unitary evolution operator \(\hat{U}(t)\) generated by \(\hat{H}(t)\).

Physical picture: \(\hat{H}_0\) drives fast background phase evolution between levels, while \(\hat{V}(t)\) couples levels and moves probability; the interaction picture is a co-moving frame that subtracts the \(\hat{H}_0\) part so the dynamics driven by \(\hat{V}\) is easier to isolate.

Three pictures

Picture

States

Operators

Typical use

Schrödinger

\(\mathrm{i}\hbar\partial_t\vert\psi\rangle_{\mathcal{S}}=\hat{H}\vert\psi\rangle_{\mathcal{S}}\)

often fixed

Default

Heisenberg

fixed

\(\hat{O}_{\mathcal{H}}(t)=\hat{U}^{\dagger}(t)\hat{O}\hat{U}(t)\)

Correlators

Interaction

\(\mathrm{i}\hbar\partial_t\vert\psi\rangle_{\mathcal{I}}=\hat{V}_{\mathcal{I}}\vert\psi\rangle_{\mathcal{I}}\)

\(\hat{O}_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{O}\hat{U}_0\)

Weak \(\hat{V}\) on solvable \(\hat{H}_0\)

Expectation values agree if \(\vert\psi\rangle\) and \(\hat{O}\) are transformed together. The interaction picture moves most \(\hat{H}_0\) evolution into definitions so the remaining Schrödinger-like equation involves only \(\hat{V}_{\mathcal{I}}\).

Schrödinger Picture#

In the Schrödinger picture, the state \(\vert\psi(t)\rangle_{\mathcal{S}}\) carries the time dependence.

State Equation#

Time-evolution of the state follows from the Schrödinger equation:

(181)#\[ \mathrm{i}\hbar\,\partial_t\vert\psi(t)\rangle_{\mathcal{S}}=\hat{H}(t)\vert\psi(t)\rangle_{\mathcal{S}}, \]

with \(\hat{H}(t)=\hat{H}_0+\hat{V}(t)\).

Propagator Equation#

Introduce the time-evolution operator (or propagator) \(\hat{U}(t)\), which gives

\[\vert\psi(t)\rangle_{\mathcal{S}}=\hat{U}(t)\vert\psi(0)\rangle_{\mathcal{S}}.\]

The equation for \(\hat{U}(t)\) is the Schrödinger equation for the propagator:

(182)#\[ \mathrm{i}\hbar\,\partial_t\hat{U}(t)=\hat{H}(t)\hat{U}(t),\qquad \hat{U}(0)=\hat{I}. \]

Advantage of this language: once \(\hat{U}(t)\) is known, every initial state evolves through the same map \(\vert\psi(0)\rangle_{\mathcal{S}}\mapsto \hat{U}(t)\vert\psi(0)\rangle_{\mathcal{S}}\).

The unperturbed time-evolution operator \(\hat{U}_0\) is

(183)#\[\begin{split} \begin{split} \mathrm{i}\hbar\,\partial_t\hat{U}_0(t)&=\hat{H}_0\hat{U}_0(t)\\ \quad\Rightarrow\quad \hat{U}_0(t)&=\mathrm{e}^{-\mathrm{i}\hat{H}_0 t/\hbar}\\ &=\sum_n \vert n\rangle\,\mathrm{e}^{-\mathrm{i}E_n t/\hbar}\,\langle n\vert.\end{split} \end{split}\]

Goal: compute the deviation of \(\hat{U}(t)\) from \(\hat{U}_0(t)\) as a power series in \(\hat{V}\).

\[\begin{split} \begin{array}{ccc} \hat{H}_0 & \xrightarrow{\;+\hat{V}(t)\;} & \hat{H}(t)\\[4pt] \downarrow & & \downarrow\\[2pt] \hat{U}_0(t) & \xrightarrow{\;+\,?\;} & \hat{U}(t) \end{array} \end{split}\]

The next step is to carry the same background subtraction from individual states to the propagator itself: factor \(\hat{U}(t)\) into the baseline \(\hat{U}_0(t)\) and whatever unitary remains once that \(\hat{H}_0\)-controlled leg is set aside.

Interaction Picture#

The idea is to switch to a co-moving frame defined by \(\hat{U}_0(t)\), so the explicit evolution equation isolates the perturbation.

State Equation#

Co-moving with \(\hat{U}_0(t)\): multiply \(\vert\psi\rangle_{\mathcal{S}}\) by \(\hat{U}_0^{\dagger}(t)\) to strip \(\hat{H}_0\) evolution. Take the interaction-picture state to be

(184)#\[ \vert\psi(t)\rangle_{\mathcal{I}}:=\hat{U}_0^{\dagger}(t)\,\vert\psi(t)\rangle_{\mathcal{S}}. \]

and the interaction-picture perturbation

(185)#\[\begin{split} \begin{split} \hat{V}_{\mathcal{I}}(t)&:=\hat{U}_0^{\dagger}(t)\,\hat{V}(t)\,\hat{U}_0(t)\\ &=\sum_{m,n}\vert m \rangle \mathrm{e}^{\mathrm{i}\omega_{mn}t}\,V_{mn}(t) \langle n \vert, \end{split} \end{split}\]

with \(\omega_{mn}=(E_m-E_n)/\hbar\).

Derivation: matrix elements and Bohr frequencies

Using

\[\begin{split} \begin{split} \hat{U}_0(t) &= \sum_n \vert n\rangle\,\mathrm{e}^{-\mathrm{i}E_n t/\hbar}\,\langle n\vert,\\ \hat{U}_0^{\dagger}(t) &= \sum_m \vert m\rangle\,\mathrm{e}^{+\mathrm{i}E_m t/\hbar}\,\langle m\vert. \end{split} \end{split}\]

we get

\[\begin{split} \begin{split} \langle m\vert\hat{V}_{\mathcal{I}}(t)\vert n\rangle &= \langle m\vert\hat{U}_0^{\dagger}(t)\hat{V}(t)\hat{U}_0(t)\vert n\rangle\\ &= \mathrm{e}^{+\mathrm{i}E_m t/\hbar}\,\langle m\vert\hat{V}(t)\vert n\rangle\,\mathrm{e}^{-\mathrm{i}E_n t/\hbar}\\ &= \mathrm{e}^{\mathrm{i}(E_m-E_n)t/\hbar}\,V_{mn}(t). \end{split} \end{split}\]

Hence

(186)#\[\langle m\vert\hat{V}_{\mathcal{I}}(t)\vert n\rangle =\mathrm{e}^{\mathrm{i}\omega_{mn}t}\,V_{mn}(t)\;, \qquad \omega_{mn}=\frac{E_m-E_n}{\hbar}. \]

Then the state equation becomes:

Interaction-picture Schrödinger equation (states)

In the interaction picture, the state evolution is generated by the perturbation \(\hat{V}_{\mathcal{I}}(t)\) only.

(187)#\[ \mathrm{i}\hbar\,\partial_t\vert\psi(t)\rangle_{\mathcal{I}}=\hat{V}_{\mathcal{I}}(t)\vert\psi(t)\rangle_{\mathcal{I}} \]

Derivation: state equation in the interaction picture

Start from Eq. (184) and write

\[ \vert\psi\rangle_{\mathcal{S}}=\hat{U}_0\vert\psi\rangle_{\mathcal{I}}. \]

Differentiate in time:

\[ \partial_t\vert\psi\rangle_{\mathcal{S}}=(\partial_t\hat{U}_0)\vert\psi\rangle_{\mathcal{I}}+\hat{U}_0\,\partial_t\vert\psi\rangle_{\mathcal{I}}. \]

Multiply through by \(\mathrm{i}\hbar\):

\[ \mathrm{i}\hbar\,\partial_t\vert\psi\rangle_{\mathcal{S}}=\mathrm{i}\hbar\,(\partial_t\hat{U}_0)\vert\psi\rangle_{\mathcal{I}}+\mathrm{i}\hbar\,\hat{U}_0\,\partial_t\vert\psi\rangle_{\mathcal{I}}. \]

Substitute the Schrödinger equations on both sides — on the LHS use Eq. (181) together with \(\vert\psi\rangle_{\mathcal{S}}=\hat{U}_0\vert\psi\rangle_{\mathcal{I}}\), and in the first RHS term use Eq. (183):

\[ (\hat{H}_0+\hat{V})\hat{U}_0\vert\psi\rangle_{\mathcal{I}}=\hat{H}_0\hat{U}_0\vert\psi\rangle_{\mathcal{I}}+\mathrm{i}\hbar\,\hat{U}_0\,\partial_t\vert\psi\rangle_{\mathcal{I}}. \]

The two \(\hat{H}_0\hat{U}_0\vert\psi\rangle_{\mathcal{I}}\) terms cancel, leaving

\[ \hat{V}\hat{U}_0\vert\psi\rangle_{\mathcal{I}}=\mathrm{i}\hbar\,\hat{U}_0\,\partial_t\vert\psi\rangle_{\mathcal{I}}. \]

Multiply by \(\hat{U}_0^{\dagger}\):

\[ \mathrm{i}\hbar\,\partial_t\vert\psi\rangle_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{V}\hat{U}_0\vert\psi\rangle_{\mathcal{I}}=\hat{V}_{\mathcal{I}}(t)\vert\psi\rangle_{\mathcal{I}}. \]

Propagator Equation#

The full map is \(\hat{U}(t)\). Factoring out \(\hat{U}_0(t)\) means following \(\hat{U}(t)\) with \(\hat{U}_0^{\dagger}(t)=\hat{U}_0^{-1}(t)\). Define

(188)#\[ \hat{U}_{\mathcal{I}}(t):=\hat{U}_0^{\dagger}(t)\,\hat{U}(t). \]

so that the interaction-picture states evolve as

\[ \vert\psi(t)\rangle_{\mathcal{I}}=\hat{U}_{\mathcal{I}}(t)\vert\psi(0)\rangle_{\mathcal{I}}. \]

  • Relation: \(\hat{U}(t)=\hat{U}_0(t)\,\hat{U}_{\mathcal{I}}(t)\).

  • If \(\hat{V}=0\), then \(\hat{U}_{\mathcal{I}}=\hat{I}\).

Interaction-picture Schrödinger equation (propagator)

In the interaction picture, the propagator evolution is generated by the perturbation \(\hat{V}_{\mathcal{I}}(t)\) only.

(189)#\[ \mathrm{i}\hbar\,\partial_t\hat{U}_{\mathcal{I}}(t)=\hat{V}_{\mathcal{I}}(t)\hat{U}_{\mathcal{I}}(t),\qquad \hat{U}_{\mathcal{I}}(0)=\hat{I}. \]

Derivation: operator equation for \(\hat{U}_{\mathcal{I}}\)

Differentiate \(\hat{U}_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{U}\):

\[ \partial_t\hat{U}_{\mathcal{I}}=(\partial_t\hat{U}_0^{\dagger})\hat{U}+\hat{U}_0^{\dagger}(\partial_t\hat{U}). \]

Taking the Hermitian conjugate of \(\mathrm{i}\hbar\partial_t\hat{U}_0=\hat{H}_0\hat{U}_0\) (Eq. (183)) and using \(\hat{H}_0^{\dagger}=\hat{H}_0\) gives the equation for \(\hat{U}_0^{\dagger}\):

\[ \mathrm{i}\hbar\,\partial_t\hat{U}_0^{\dagger}=-\hat{U}_0^{\dagger}\hat{H}_0. \]

Multiplying \(\partial_t\hat{U}_{\mathcal{I}}\) by \(\mathrm{i}\hbar\) and substituting this together with \(\mathrm{i}\hbar\partial_t\hat{U}=(\hat{H}_0+\hat{V})\hat{U}\):

\[\begin{split} \begin{split} \mathrm{i}\hbar\,\partial_t\hat{U}_{\mathcal{I}} &=-\hat{U}_0^{\dagger}\hat{H}_0\hat{U}+\hat{U}_0^{\dagger}(\hat{H}_0+\hat{V})\hat{U}\\ &=\hat{U}_0^{\dagger}\hat{V}\hat{U}\\ &=\hat{U}_0^{\dagger}\hat{V}\,\hat{U}_0\hat{U}_0^{\dagger}\,\hat{U}\\ &=(\hat{U}_0^{\dagger}\hat{V}\hat{U}_0)(\hat{U}_0^{\dagger}\hat{U})\\ &=\hat{V}_{\mathcal{I}}\hat{U}_{\mathcal{I}}, \end{split} \end{split}\]

where the third equality inserts \(\hat{U}_0\hat{U}_0^{\dagger}=\hat{I}\) between \(\hat{V}\) and \(\hat{U}\), and the last equality uses the definitions \(\hat{V}_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{V}\hat{U}_0\) (Eq. (185)) and \(\hat{U}_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{U}\) (Eq. (188)).

At \(t=0\), \(\hat{U}_0(0)=\hat{U}(0)=\hat{I}\), so \(\hat{U}_{\mathcal{I}}(0)=\hat{I}\).

Key point: \(\hat{H}_0\) no longer appears explicitly in Eqs. (187) and (189); its effect is encoded in the interaction-picture definitions.

Summary#

  • The interaction picture is a co-moving description that removes explicit background evolution from \(\hat{H}_0\) and isolates the perturbation-driven dynamics.

  • State evolution becomes \(\mathrm{i}\hbar\,\partial_t\vert\psi\rangle_{\mathcal{I}}=\hat{V}_{\mathcal{I}}\vert\psi\rangle_{\mathcal{I}}\), so the generator is only the transformed perturbation.

  • Matrix elements of \(\hat{V}_{\mathcal{I}}\) carry Bohr-frequency phases \(\mathrm{e}^{\mathrm{i}\omega_{mn}t}\), which encode relative phase winding between unperturbed levels.

  • The same split holds at operator level through \(\hat{U}=\hat{U}_0\hat{U}_{\mathcal{I}}\), with \(\hat{U}_{\mathcal{I}}\) governed by \(\mathrm{i}\hbar\,\partial_t\hat{U}_{\mathcal{I}}=\hat{V}_{\mathcal{I}}\hat{U}_{\mathcal{I}}\) and \(\hat{U}_{\mathcal{I}}(0)=\hat{I}\).

  • This framework assumes a solvable reference problem and a weak disturbance, and it sets up the time-ordered expansion developed next.

See Also

Homework#

1. Expectations across pictures. Let \(\vert\psi\rangle_{\mathcal{S}}\) and \(\hat{O}\) be any state and observable in the Schrödinger picture. With \(\vert\psi\rangle_{\mathcal{I}}=\hat{U}_0^{\dagger}\vert\psi\rangle_{\mathcal{S}}\) and \(\hat{O}_{\mathcal{I}}=\hat{U}_0^{\dagger}\hat{O}\hat{U}_0\), show that \({}_{\mathcal{S}}\langle\psi\vert\hat{O}\vert\psi\rangle_{\mathcal{S}}={}_{\mathcal{I}}\langle\psi\vert\hat{O}_{\mathcal{I}}\vert\psi\rangle_{\mathcal{I}}\).

2. Transition frequencies. The unperturbed Hamiltonian \(\hat{H}_0\) has eigenstates \(\{\vert n\rangle\}\) with energies \(\{E_n^{(0)}\}\).

(a) Show that in the interaction picture, the matrix elements of \(\hat{V}_{\mathcal{I}}(t)\) are \(\langle m\vert \hat{V}_{\mathcal{I}}(t)\vert n\rangle = V_{mn}(t)\,\mathrm{e}^{\mathrm{i}\omega_{mn}t}\) where \(\omega_{mn} = (E_m^{(0)} - E_n^{(0)})/\hbar\).

(b) Interpret the oscillatory factors: why do matrix elements between nearly degenerate states (\(\omega_{mn} \approx 0\)) evolve slowly, while those between widely separated states oscillate rapidly?

3. Choosing a picture. Three quantum systems require calculations: (A) a bound particle in a slowly varying potential, (B) an atom driven by a monochromatic laser, (C) a free particle kicked by a brief impulse. For each, state which picture (Schrödinger, Heisenberg, or interaction) is most natural and explain your reasoning.

4. Operator-picture transformation. Define the interaction-picture operator \(\hat{O}_{\mathcal{I}}(t):=\hat{U}_0^{\dagger}(t)\,\hat{O}(t)\,\hat{U}_0(t)\) for any Schrödinger-picture observable \(\hat{O}(t)\) that may carry its own explicit time dependence.

(a) Differentiate the definition and use \(\mathrm{i}\hbar\,\partial_t\hat{U}_0=\hat{H}_0\hat{U}_0\) to show that

\[ \mathrm{i}\hbar\,\partial_t\hat{O}_{\mathcal{I}}(t)=[\hat{O}_{\mathcal{I}}(t),\hat{H}_0]+\mathrm{i}\hbar\,(\partial_t\hat{O})_{\mathcal{I}}(t). \]

(b) Contrast this with the Heisenberg equation of motion. Which Hamiltonian generates operator evolution in each picture, and what dynamics does the interaction-picture state equation Eq. (187) track that the Heisenberg picture instead absorbs into the operators?

5. Two-level system under monochromatic drive. Consider \(\hat{H}_0=\tfrac{\hbar\omega_0}{2}\hat{\sigma}^z\) with eigenstates \(\vert\uparrow\rangle\) (energy \(+\hbar\omega_0/2\)) and \(\vert\downarrow\rangle\) (energy \(-\hbar\omega_0/2\)), perturbed by

\[ \hat{V}(t)=\hbar\Omega\cos(\omega t)\,\hat{\sigma}^x. \]

(a) Rewrite \(\hat{V}(t)\) in the eigenbasis of \(\hat{H}_0\) and identify the Bohr frequency \(\omega_{\uparrow\downarrow}=(E_\uparrow-E_\downarrow)/\hbar\).

(b) Compute \(\langle\uparrow\vert\hat{V}_{\mathcal{I}}(t)\vert\downarrow\rangle\) in the interaction picture and show that it splits into two exponentials at frequencies \(\omega_0\pm\omega\).

(c) The rotating-wave approximation keeps only the slowly oscillating contribution. State the resonance condition on \(\omega\) that selects the slow term, and write down the matrix element kept by the approximation.

6. Misconception check. One might argue: In the interaction picture, the perturbation \(\hat{V}\) has been absorbed into the change of frame and no longer affects the dynamics. Identify what is correct and what is wrong in this claim. In two or three sentences, explain in what sense \(\hat{V}\) survives the transformation and what is actually removed by passing to the comoving frame.