Editing Interaction picture (section)

==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 (quantum mechanics)|Hamiltonian]] into two parts:
{{Equation box 1
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|equation = <math>H_\text{S} = H_{0,\text{S}} + H_{1,\text{S}}.</math>
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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''<sub>0,S</sub> is well understood and exactly solvable, while ''H''<sub>1,S</sub> 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''<sub>1,S</sub>, leaving ''H''<sub>0,S</sub> time-independent:{{Equation box 1
|indent =:
|equation = <math>H_\text{S}(t) = H_{0,\text{S}} + H_{1,\text{S}}(t).</math>
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|bgcolor=#F9FFF7}}We proceed assuming that this is the case. If there ''is'' a context in which it makes sense to have ''H''<sub>0,S</sub> be time-dependent, then one can proceed by replacing <math>\mathrm{e}^{\pm \mathrm{i} H_{0,\text{S}} t/\hbar}</math> by the corresponding [[Schrödinger picture|time-evolution operator]] in the definitions below.

===State vectors===

Let <math>|\psi_\text{S}(t)\rangle = \mathrm{e}^{-\mathrm{i}H_\text{S}t/\hbar}|\psi(0)\rangle</math> be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture, <math>|\psi_\text{I}(t)\rangle</math>, is defined with an additional time-dependent unitary transformation.<ref>{{cite web|url=http://www.nyu.edu/classes/tuckerman/stat.mechII/lectures/lecture_21/node2.html|title=The Interaction Picture, lecture notes from New York University.|archive-url=https://web.archive.org/web/20130904235610/http://www.nyu.edu/classes/tuckerman/stat.mechII/lectures/lecture_21/node2.html|archive-date=2013-09-04|url-status=dead}}</ref>
{{Equation box 1
|indent =:
|equation =  <math> | \psi_\text{I}(t) \rangle = \text{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} | \psi_\text{S}(t) \rangle.</math>
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===Operators===

An operator in the interaction picture is defined as
{{Equation box 1
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|equation =  <math>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}.</math>
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Note that ''A''<sub>S</sub>(''t'') will typically not depend on {{mvar|t}} and can be rewritten as just ''A''<sub>S</sub>. It only depends on  {{mvar|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''<sub>S</sub>(''t'') is a density matrix (see below).

====Hamiltonian operator====

For the operator <math>H_0</math> itself, the interaction picture and Schrödinger picture coincide:
:<math>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}}.</math>
This is easily seen through the fact that operators [[commutativity|commute]] with differentiable functions of themselves. This particular operator then can be called <math>H_0</math> without ambiguity.

For the perturbation Hamiltonian <math>H_{1,\text{I}}</math>, however,
:<math>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},</math>
where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [''H''<sub>1,S</sub>, ''H''<sub>0,S</sub>] = 0.

It is possible to obtain the interaction picture for a time-dependent Hamiltonian ''H''<sub>0,S</sub>(''t'') as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by ''H''<sub>0,S</sub>(''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 {{math|''ρ''<sub>I</sub>}} and {{math|''ρ''<sub>S</sub>}} be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability {{math|''p<sub>n</sub>''}} to be in the physical state  |''ψ''<sub>''n''</sub>⟩, then
:<math>\begin{align}
\rho_\text{I}(t)
&= \sum_n p_n(t) \left|\psi_{n,\text{I}}(t)\right\rang \left\lang \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\rang \left\lang \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{align}</math>

===Time-evolution===

====Time-evolution of states<!--'Schwinger–Tomonaga equation' redirects here-->====

Transforming the [[Schrödinger equation]] into the interaction picture gives

:<math> \mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d}t} |\psi_\text{I}(t)\rang = H_{1,\text{I}}(t) |\psi_\text{I}(t)\rang, </math>

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.<ref>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.</ref> A proof is given in Fetter and Walecka.<ref>{{cite book |last1=Fetter |first1=Alexander L. |last2=Walecka |first2=John Dirk |title=Quantum Theory of Many-particle Systems |date=1971 |publisher=McGraw-Hill |isbn=978-0-07-020653-3 |page=55 |url=https://books.google.com/books?id=Y1HwAAAAMAAJ |language=en}}</ref>

====Time-evolution of operators====

If the operator ''A''<sub>S</sub> is time-independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for ''A''<sub>I</sub>(''t'') is given by
:<math> \mathrm{i}\hbar\frac{\mathrm{d}}{\mathrm{d}t}A_\text{I}(t) = [A_\text{I}(t),H_{0,\text{S}}].</math>

In the interaction picture the operators evolve in time like the operators in the [[Heisenberg picture]] with the Hamiltonian {{math|''H{{'}}'' {{=}} ''H''<sub>0</sub>}}.

====Time-evolution of the density matrix====

The evolution of the [[density matrix]] in the interaction picture is

:<math> \mathrm{i}\hbar \frac{\mathrm{d}}{\mathrm{d}t} \rho_\text{I}(t) = [H_{1,\text{I}}(t), \rho_\text{I}(t)],</math>

in consistency with the Schrödinger equation in the interaction picture.