Interaction picture

Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.

The interaction picture is a special case of unitary transformation applied to the Hamiltonian and state vectors.

Haag's theorem says that the interaction picture doesn't exist in the case of interacting quantum fields.

Definition

Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation) to those same operators and state vectors in the Schrödinger picture.

To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts:

${\displaystyle H_{\text{S}}=H_{0,\text{S}}+H_{1,\text{S}}.}$

Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H_0,S is well understood and exactly solvable, while H_1,S contains some harder-to-analyze perturbation to this system.

If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H_1,S, leaving H_0,S time-independent:

${\displaystyle H_{\text{S}}(t)=H_{0,\text{S}}+H_{1,\text{S}}(t).}$

We proceed assuming that this is the case. If there is a context in which it makes sense to have H_0,S be time-dependent, then one can proceed by replacing ${\displaystyle {\mathrm{e}}^{\pm \mathrm{i}{H}_{0,\text{S}}t/\hslash }}$ by the corresponding time-evolution operator in the definitions below.

State vectors

Let ${\displaystyle |{\psi }_{\text{S}}(t)⟩={\mathrm{e}}^{-\mathrm{i}{H}_{\text{S}}t/\hslash }|\psi (0)⟩}$ be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture, ${\displaystyle |{\psi }_{\text{I}}(t)⟩}$, is defined with an additional time-dependent unitary transformation.

${\displaystyle |{\psi }_{\text{I}}(t)⟩={\text{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }|{\psi }_{\text{S}}(t)⟩.}$

Operators

An operator in the interaction picture is defined as

${\displaystyle A_{\text{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{A}_{\text{S}}(t){\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash }.}$

Note that A_S(t) will typically not depend on t and can be rewritten as just A_S. It only depends on t if the operator has "explicit time dependence", for example, due to its dependence on an applied external time-varying electric field. Another instance of explicit time dependence may occur when A_S(t) is a density matrix (see below).

Hamiltonian operator

For the operator ${\displaystyle H_0}$ itself, the interaction picture and Schrödinger picture coincide:

${\displaystyle H_{0,\text{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{H}_{0,\text{S}}{\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash }=H_{0,\text{S}}.}$

This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called ${\displaystyle H_0}$ without ambiguity.

For the perturbation Hamiltonian ${\displaystyle H_{1,\text{I}}}$, however,

${\displaystyle H_{1,\text{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{H}_{1,\text{S}}{\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash },}$

where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [H_1,S, H_0,S] = 0.

It is possible to obtain the interaction picture for a time-dependent Hamiltonian H_0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H_0,S(t), or more explicitly with a time-ordered exponential integral.

Density matrix

The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let ρ_I and ρ_S be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability p_n to be in the physical state |ψ_n⟩, then

${\displaystyle \begin{array}{cc}{\rho }_{\text{I}}(t)& =\sum _n{p}_n(t)|{\psi }_{n,\text{I}}(t)⟩⟨{\psi }_{n,\text{I}}(t)|\\ & =\sum _n{p}_n(t){\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }|{\psi }_{n,\text{S}}(t)⟩⟨{\psi }_{n,\text{S}}(t)|{\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash }\\ & ={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{\rho }_{\text{S}}(t){\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash }.\end{array}}$

Time-evolution

Time-evolution of states

Transforming the Schrödinger equation into the interaction picture gives

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}|{\psi }_{\text{I}}(t)⟩=H_{1,\text{I}}(t)|{\psi }_{\text{I}}(t)⟩,}$

which states that in the interaction picture, a quantum state is evolved by the interaction part of the Hamiltonian as expressed in the interaction picture. A proof is given in Fetter and Walecka.

Time-evolution of operators

If the operator A_S is time-independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for A_I(t) is given by

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}{A}_{\text{I}}(t)=[A_{\text{I}}(t),H_{0,\text{S}}].}$

In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian H' = H₀.

Time-evolution of the density matrix

The evolution of the density matrix in the interaction picture is

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}{\rho }_{\text{I}}(t)=[H_{1,\text{I}}(t),{\rho }_{\text{I}}(t)],}$

in consistency with the Schrödinger equation in the interaction picture.