Interaction picture

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In quantum mechanics, the interaction picture (also known as the Dirac picture) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.^[1] The interaction picture is useful in dealing with the changes to the wave functions and observable due to the interactions. Most field theoretical calculations^[2]^[3] use the interaction representation because they construct the solution to the many body Schrödinger equation as the solution to the free particle problem plus some unknown interaction part.

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.

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,

$H_{S}=H_{{0,S}}+H_{{1,S}}~.$

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. 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 $e^{{\pm iH_{{0,S}}t/\hbar }}$ by the corresponding time-evolution operator in the definitions below.

State vectors

A state vector in the interaction picture is defined as^[4]

$|\psi _{{I}}(t)\rangle =e^{{iH_{{0,S}}t/\hbar }}|\psi _{{S}}(t)\rangle ~,$

where |ψ_S(t)〉is the state vector in the Schrödinger picture.

Operators

An operator in the interaction picture is defined as

$A_{{I}}(t)=e^{{iH_{{0,S}}t/\hbar }}A_{{S}}(t)e^{{-iH_{{0,S}}t/\hbar }}.$

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.

Hamiltonian operator

For the operator H₀ itself, the interaction picture and Schrödinger picture coincide,

$H_{{0,I}}(t)=e^{{iH_{{0,S}}t/\hbar }}H_{{0,S}}e^{{-iH_{{0,S}}t/\hbar }}=H_{{0,S}}.$

This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called H₀ without ambiguity.

For the perturbation Hamiltonian H_1,I , however,

$H_{{1,I}}(t)=e^{{iH_{{0,S}}t/\hbar }}H_{{1,S}}e^{{-iH_{{0,S}}t/\hbar }},$

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 matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability $p n$ to be in the physical state |ψ_n〉, then

$\rho _{I}(t)=\sum _{n}p_{n}(t)|\psi _{{n,I}}(t)\rangle \langle \psi _{{n,I}}(t)|=\sum _{n}p_{n}(t)e^{{iH_{{0,S}}t/\hbar }}|\psi _{{n,S}}(t)\rangle \langle \psi _{{n,S}}(t)|e^{{-iH_{{0,S}}t/\hbar }}=e^{{iH_{{0,S}}t/\hbar }}\rho _{S}(t)e^{{-iH_{{0,S}}t/\hbar }}.$

Evolution	Picture
of:	Heisenberg	Interaction	Schrödinger
Ket state	constant	$\|\psi _{{I}}(t)\rangle =e^{{iH_{{0,S}}~t/\hbar }}\|\psi _{{S}}(t)\rangle$	$\|\psi _{{S}}(t)\rangle =e^{{-iH_{{S}}~t/\hbar }}\|\psi _{{S}}(0)\rangle$
Observable	$A_{H}(t)=e^{{iH_{{S}}~t/\hbar }}A_{S}e^{{-iH_{{S}}~t/\hbar }}$	$A_{I}(t)=e^{{iH_{{0,S}}~t/\hbar }}A_{S}e^{{-iH_{{0,S}}~t/\hbar }}$	constant
Density matrix	constant	$\rho _{I}(t)=e^{{iH_{{0,S}}~t/\hbar }}\rho _{S}(t)e^{{-iH_{{0,S}}~t/\hbar }}$	$\rho _{S}(t)=e^{{-iH_{{S}}~t/\hbar }}\rho _{S}(0)e^{{iH_{{S}}~t/\hbar }}$

Time-evolution equations in the interaction picture

Time-evolution of states

Transforming the Schrödinger equation into the interaction picture gives:

$i\hbar {\frac {d}{dt}}\mid \psi _{{I}}(t)\rangle =H_{{1,I}}(t)\mid \psi _{{I}}(t)\rangle .$

This equation is referred to as the Schwinger–Tomonaga equation.

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

$i\hbar {\frac {d}{dt}}A_{I}(t)=\left[A_{I}(t),H_{0}\right].\;$

In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian $H' = H 0$ .

Time-evolution of the density matrix

Transforming the Schwinger–Tomonaga equation into the language of the density matrix (or equivalently, transforming the von Neumann equation into the interaction picture) gives:

$i\hbar {\frac {d}{dt}}\rho _{I}(t)=\left[H_{{1,I}}(t),\rho _{I}(t)\right].$

Use of interaction picture

The purpose of the interaction picture is to shunt all the time dependence due to H₀ onto the operators, thus allowing them to evolve freely, and leaving only H_1,I to control the time-evolution of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term, H_1,S, being added to the Hamiltonian of a solved system, H_0,S. By utilizing the interaction picture, one can use time-dependent perturbation theory to find the effect of H_1,I, e.g., in the derivation of Fermi's golden rule, or the Dyson series, in quantum field theory: In 1947, Tomonaga and Schwinger appreciated that covariant perturbation theory could be formulated elegantly in the interaction picture, since field operators can evolve in time as free fields, even in the presence of interactions, now treated perturbatively in such a Dyson series.

References

↑ Albert Messiah (1966). Quantum Mechanics, North Holland, John Wiley & Sons. ISBN 0486409244 ; J. J. Sakurai (1994). Modern Quantum Mechanics (Addison-Wesley) ISBN 9780201539295 .
↑ J. W. Negele, H. Orland (1988), Quantum Many-particle Systems,
↑ Piers Coleman, The evolving monogram on Many Body Physics
↑ The Interaction Picture, lecture notes from New York University

Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed. Sausalito, CA: University Science Books. ISBN 1-891389-13-0.