Interaction picture

In quantum mechanics, the interaction picture (also known as the interaction representation or Dirac picture after Paul Dirac, who introduced it)^[1]^[2] 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.^[3] The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations^[4] 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.

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

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:

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

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

State vectors

Let $|\psi _{\text{S}}(t)\rangle =\mathrm {e} ^{-\mathrm {i} H_{\text{S}}t/\hbar }|\psi (0)\rangle$ be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture, $|\psi _{\text{I}}(t)\rangle$ , is defined with an additional time-dependent unitary transformation.^[5]

$|\psi _{\text{I}}(t)\rangle ={\text{e}}^{\mathrm {i} H_{0,{\text{S}}}t/\hbar }|\psi _{\text{S}}(t)\rangle .$

Operators

An operator in the interaction picture is defined as

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

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. Another instance of explicit time dependence may occur when A_S(t) is a density matrix (see below).

Hamiltonian operator

For the operator $H_{0}$ itself, the interaction picture and Schrödinger picture coincide:

H_{0,{\text{I}}}(t)=\mathrm {e} ^{\mathrm {i} H_{0,{\text{S}}}t/\hbar }H_{0,{\text{S}}}\mathrm {e} ^{-\mathrm {i} H_{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 $H_{0}$ without ambiguity.

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

H_{1,{\text{I}}}(t)=\mathrm {e} ^{\mathrm {i} H_{0,{\text{S}}}t/\hbar }H_{1,{\text{S}}}\mathrm {e} ^{-\mathrm {i} H_{0,{\text{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 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

{\begin{aligned}\rho _{\text{I}}(t)&=\sum _{n}p_{n}(t)\left|\psi _{n,{\text{I}}}(t)\right\rangle \left\langle \psi _{n,{\text{I}}}(t)\right|\\&=\sum _{n}p_{n}(t)\mathrm {e} ^{\mathrm {i} H_{0,{\text{S}}}t/\hbar }\left|\psi _{n,{\text{S}}}(t)\right\rangle \left\langle \psi _{n,{\text{S}}}(t)\right|\mathrm {e} ^{-\mathrm {i} H_{0,{\text{S}}}t/\hbar }\\&=\mathrm {e} ^{\mathrm {i} H_{0,{\text{S}}}t/\hbar }\rho _{\text{S}}(t)\mathrm {e} ^{-\mathrm {i} H_{0,{\text{S}}}t/\hbar }.\end{aligned}}

Time-evolution

Time-evolution of states

Transforming the Schrödinger equation into the interaction picture gives

\mathrm {i} \hbar {\frac {\mathrm {d} }{\mathrm {d} t}}|\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.^[6] A proof is given in Fetter and Walecka.^[7]

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

\mathrm {i} \hbar {\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 $H' = H 0$ .

Time-evolution of the density matrix

The evolution of the density matrix in the interaction picture is

\mathrm {i} \hbar {\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.

Expectation values

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

\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

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

Schwinger–Tomonaga equation

The term interaction representation was invented by Schwinger.^[8]^[9] In this new mixed representation the state vector is no longer constant in general, but it is constant if there is no coupling between fields. The change of representation leads directly to the Tomonaga–Schwinger equation:^[10]^[9]

ihc{\frac {\partial \Psi [\sigma ]}{\partial \sigma (x)}}={\hat {H}}(x)\Psi (\sigma )

{\hat {H}}(x)=-{\frac {1}{c}}j_{\mu }(x)A^{\mu }(x)

Where the Hamiltonian in this case is the QED interaction Hamiltonian, but it can also be a generic interaction, and $\sigma$ is a spacelike surface that is passing through the point $x$ . The derivative formally represents a variation over that surface given $x$ fixed. It is difficult to give a precise mathematical formal interpretation of this equation.^[11]

This approach is called the 'differential and field' approach by Schwinger, as opposed to the 'integral and particle' approach of the Feynman diagrams.

The core idea is that if the interaction has a small coupling constant (i.e. in the case of electromagnetism of the order of the fine structure constant) successive perturbative terms will be powers of the coupling constant and therefore smaller.^[12]

Use

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,^[13]^: 355ff e.g., in the derivation of Fermi's golden rule,^[13]^{: 359–363} or the Dyson series^[13]^{: 355–357} in quantum field theory: in 1947, Shin'ichirō Tomonaga and Julian 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 pictures

For a time-independent Hamiltonian H_S, where H_0,S is the free Hamiltonian,

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

References

↑ Duck, Ian; Sudarshan, E.C.G. (1998). "Chapter 6: Dirac's Invention of Quantum Field Theory". Pauli and the Spin-Statistics Theorem. World Scientific Publishing. pp. 149–167. ISBN 978-9810231149.
↑ https://courses.physics.illinois.edu/phys580/fa2013/interaction.pdf
↑ 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, ISBN 0738200522.
↑ The Interaction Picture, lecture notes from New York University.
↑ 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 space-like foliations of spacetime.
↑ Fetter, Alexander L.; Walecka, John Dirk (1971). Quantum Theory of Many-particle Systems. McGraw-Hill. p. 55. ISBN 978-0-07-020653-3.
↑ Schwinger, J. (1958), Selected papers on Quantum Electrodynamics, Dover, p. 151, ISBN 0-486-60444-6
1 2 Schwinger, J. (1948), "Quantum electrodynamics. I. A covariant formulation.", Physical Review, 74 (10): 1439–1461, Bibcode:1948PhRv...74.1439S, doi:10.1103/PhysRev.74.1439
↑ Schwinger, J. (1958), Selected papers on Quantum Electrodynamics, Dover, p. 151,163,170,276, ISBN 0-486-60444-6
↑ Wakita, Hitoshi (1976), "Integration of the Tomonaga-Schwinger Equation", Communications in Mathematical Physics, 50 (1): 61–68, Bibcode:1976CMaPh..50...61W, doi:10.1007/BF01608555, S2CID 122590381
↑ Schwinger, J. (1958), Selected papers on Quantum Electrodynamics, Dover, p. 152, ISBN 0-486-60444-6
1 2 3 Sakurai, J. J.; Napolitano, Jim (2010), Modern Quantum Mechanics (2nd ed.), Addison-Wesley, ISBN 978-0805382914

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