endobj JavaScript is disabled. Before the interaction phase is acquired as \(e ^ { - i E _ { \ell } \left( \tau - t _ { 0 } \right) / \hbar }\), whereas after the interaction phase is acquired as \(e ^ { - i E _ { \ell } ( t - \tau ) / \hbar }\). ⟩ ± /SMask /None>> {\displaystyle H_{0}} /PCSp 5 0 R 0, we have the differential equation . I The usual Schrödinger picture has the states evolving and the operators constant. << , /ca 1.0 /Type /Page i 0000091546 00000 n Just plug it into Equation 1. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. S I >> 0000061565 00000 n H 0000076906 00000 n 1 0000143386 00000 n 0000103164 00000 n where \(k\) and \(l\) are eigenstates of \(H_0\). /XObject << 0000063592 00000 n Suppose that A is an Hermitean operator and [A,H] = 0. 0000006486 00000 n 0000006953 00000 n Setting \(V\) to zero, we can see that the time evolution of the exact part of the Hamiltonian \(H_0\) is described by, \[\frac { \partial } { \partial t } U _ { 0 } \left( t , t _ { 0 } \right) = - \frac { i } { \hbar } H _ { 0 } ( t ) U _ { 0 } \left( t , t _ { 0 } \right) \label{2.94}\], \[U _ { 0 } \left( t , t _ { 0 } \right) = \exp _ { + } \left[ - \frac { i } { \hbar } \int _ { t _ { 0 } } ^ { t } d \tau H _ { 0 } ( t ) \right] \label{2.95}\], \[U _ { 0 } \left( t , t _ { 0 } \right) = e ^ { - i H _ { 0 } \left( t - t _ { 0 } \right) / \hbar } \label{2.96}\]. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. Hey all, I got some question referring to the interaction picture. So let's attempt to do the part that is easy. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. >> 0000035960 00000 n Oxford University Press: New York, 2006; Ch. So what changes about the time-propagation in the interaction representation? {\displaystyle |\psi _{\text{I}}(t)\rangle } ⟩ 0000062557 00000 n /Parent 2 0 R If there is probability pn to be in the physical state |ψn〉, then, Transforming the Schrödinger equation into the interaction picture gives, 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. << 0000118554 00000 n 0000028837 00000 n >> You see, there's going to be time evolution as you go from t0 to tf. 0000145311 00000 n 0000148352 00000 n 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. [4], 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. endobj A This is called the Heisenberg Picture. ℏ For the last two expressions, the order of these operators certainly matters. >> 0000026444 00000 n , + 0000148840 00000 n 0000004750 00000 n H 0000154338 00000 n = 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. stream [3], | 0000146776 00000 n The interaction picture is often used to describe the time evolution of a quantum system whose Hamiltonian has the form H= H 0 +H I, where H 0 is the Hamiltonian for freely moving particles and H I describes the in-teractions between them. 0000145609 00000 n 0 0000018017 00000 n endobj 0000109213 00000 n | ⟩ 0 0000018885 00000 n We can now define a time-evolution operator in the interaction picture: \[| \psi _ { I } ( t ) \rangle = U _ { I } \left( t , t _ { 0 } \right) | \psi _ { I } \left( t _ { 0 } \right) \rangle \label{2.103}\], \[U _ { I } \left( t , t _ { 0 } \right) = \exp _ { + } \left[ \frac { - i } { \hbar } \int _ { t _ { 0 } } ^ { t } d \tau V _ { I } ( \tau ) \right] \label{2.104}\], \[ \begin{array} { r l } { | \psi _ { S } ( t ) \rangle } & { = U _ { 0 } \left( t , t _ { 0 } \right) | \psi _ { I } ( t ) \rangle } \\ { } & { = U _ { 0 } \left( t , t _ { 0 } \right) U _ { I } \left( t , t _ { 0 } \right) | \psi _ { I } \left( t _ { 0 } \right) \rangle } \\ { } & { = U _ { 0 } \left( t , t _ { 0 } \right) U _ { I } \left( t , t _ { 0 } \right) | \psi _ { S } \left( t _ { 0 } \right) \rangle } \label{2.105} \\ { \therefore U \left( t , t _ { 0 } \right) = U _ { 0 } \left( t , t _ { 0 } \right) U _ { I } \left( t , t _ { 0 } \right) } \end{array} \], Also, the time evolution of conjugate wavefunction in the interaction picture can be written, \[U ^ { \dagger } \left( t , t _ { 0 } \right) = U _ { I } ^ { \dagger } \left( t , t _ { 0 } \right) U _ { 0 } ^ { \dagger } \left( t , t _ { 0 } \right) = \exp _ { - } \left[ \frac { i } { \hbar } \int _ { t _ { 0 } } ^ { t } d \tau V _ { I } ( \tau ) \right] \exp _ { - } \left[ \frac { i } { \hbar } \int _ { t _ { 0 } } ^ { t } d \tau H _ { 0 } ( \tau ) \right] \label{2.107}\]. [1] The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. For a general operator H evolution operator associated with a (interaction picture) Hamiltonian depending period-ically on time. 0000078039 00000 n It is also useful to know that the time-evolution operator in the interaction picture is related to the full time-evolution operator U(t) as U(t) = e−iH 0t/~U I(t), (22) t H Wavefunctions evolve under VI , while operators evolve under, \[\text { For } H _ { 0 } = 0 , V ( t ) = H \quad \Rightarrow \quad \frac { \partial \hat { A } } { \partial t } = 0 ; \quad \frac { \partial } { \partial t } | \psi _ { S } \rangle = \frac { - i } { \hbar } H | \psi _ { S } \rangle\], \[\text { For } H _ { 0 } = H , V ( t ) = 0 \Rightarrow \frac { \partial \hat { A } } { \partial t } = \frac { i } { \hbar } [ H , \hat { A } ] ; \quad \frac { \partial \psi } { \partial t } = 0 \label{2.113}\], Earlier we described how time-dependent problems with Hamiltonians of the form \(H = H _ { 0 } + V ( t )\) could be solved in terms of the time-evolving amplitudes in the eigenstates of \(H_0\). I adjusted your post using the double # around your latex expressions and they look a lot better. What about the operators? 0000133019 00000 n 0000000016 00000 n I t We define a wavefunction in the interaction picture \(| \psi _ { I } \rangle\) in terms of the Schrödinger wavefunction through: \[| \psi _ { S } ( t ) \rangle \equiv U _ { 0 } \left( t , t _ { 0 } \right) | \psi _ { I } ( t ) \rangle \label{2.97}\], \[| \psi _ { I } \rangle = U _ { 0 } ^ { \dagger } | \psi _ { S } \rangle \label{2.98}\]. ⟩ 0000005241 00000 n t 0000151136 00000 n This is the solution to the Liouville equation in the interaction picture. 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. 0000147279 00000 n 11 0 obj 0000105938 00000 n / << %%EOF {\displaystyle H_{1,{\text{I}}}} Now we need an equation of motion that describes the time evolution of the interaction picture wavefunctions.
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