Interaction picture

Paul Dirac, together with Werner Heisenberg and Erwin Schrödinger. The Dirac picture may be thought of as in between the Schrödinger picture and the Heisenberg picture.

In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) 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 changes to the wave functions and observables due to interactions. Most field-theoretical calculations[2] 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 parts.

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.

DefinitionEdit

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 H0,S is well understood and exactly solvable, while H1,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 H1,S, leaving H0,S time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have H0,S be time-dependent, then one can proceed by replacing ${\displaystyle e^{\pm iH_{0,{\text{S}}}t/\hbar }}$  by the corresponding time-evolution operator in the definitions below.

State vectorsEdit

A state vector in the interaction picture is defined as[3]

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

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

OperatorsEdit

An operator in the interaction picture is defined as

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

Note that AS(t) will typically not depend on t and can be rewritten as just AS. 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 operatorEdit

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

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

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

For the perturbation Hamiltonian H1,I, however,

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

where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [H1,S, H0,S] = 0.

It is possible to obtain the interaction picture for a time-dependent Hamiltonian H0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H0,S(t), or more explicitly with a time-ordered exponential integral.

Density matrixEdit

The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let ρI and ρS be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability pn to be in the physical state |ψn〉, then

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

Time-evolution equations in the interaction pictureEdit

Time-evolution of statesEdit

Transforming the Schrödinger equation into the interaction picture gives

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

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[4].

Time-evolution of operatorsEdit

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

${\displaystyle i\hbar {\frac {d}{dt}}A_{\text{I}}(t)=[A_{\text{I}}(t),H_{0,{\text{S}}}].}$

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 matrixEdit

The evolution of the density matrix in the interaction picture is

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

in consistency with the Schrödinger equation in the interaction picture.

Expectation valuesEdit

For a general operator ${\displaystyle A}$ , the expectation value in the interaction picture is given by

${\displaystyle \langle A_{\text{I}}(t)\rangle =\langle \psi _{\text{I}}(t)|A_{\text{I}}(t)|\psi _{\text{I}}(t)\rangle =\langle \psi _{\text{S}}(t)|e^{-iH_{0,{\text{S}}}t}e^{iH_{0,{\text{S}}}t}\,A_{\text{S}}\,e^{-iH_{0,{\text{S}}}t}e^{iH_{0,{\text{S}}}t}|\psi _{\text{S}}(t)\rangle =\langle A_{\text{S}}(t)\rangle .}$

Using the density-matrix expression for expectation value, we will get

${\displaystyle \langle A_{\text{I}}(t)\rangle =\operatorname {Tr} {\big (}\rho _{\text{I}}(t)\,A_{\text{I}}(t){\big )}.}$

Use of interaction pictureEdit

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 H1,I to control the time-evolution of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term, H1,S, being added to the Hamiltonian of a solved system, H0,S. By utilizing the interaction picture, one can use time-dependent perturbation theory to find the effect of H1,I,[5]:355ff e.g., in the derivation of Fermi's golden rule,[5]:359–363 or the Dyson series[5]:355–357 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.

Summary comparison of evolution in all picturesEdit

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

ReferencesEdit

1. ^ Albert Messiah (1966). Quantum Mechanics, North Holland, John Wiley & Sons. ISBN 0486409244; J. J. Sakurai (1994). Modern Quantum Mechanics (Addison-Wesley) ISBN 9780201539295.
2. ^ J. W. Negele, H. Orland (1988), Quantum Many-particle Systems, ISBN 0738200522.
3. ^ The Interaction Picture, lecture notes from New York University.
4. ^ Quantum Field Theory for the Gifted Amateur, Chapter 18 - for those who saw this being called the Schwinger-Tomonaga equation, this is not the Schwinger-Tomonaga equation. That is a generalization of the Schrödinger equation to arbitrary time-like foliations of spacetime.
5. ^ a b c Sakurai, J. J.; Napolitano, Jim (2010), Modern Quantum Mechanics (2nd ed.), Addison-Wesley, ISBN 978-0805382914
• Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed. Sausalito, California: University Science Books. ISBN 1-891389-13-0.