Interaction picture Definition and 19 Threads

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. The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations 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.

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  1. K

    I Atom-photon interactions in the interaction picture (self answered)

    EDIT: I'M SO DUMB! I can't believe I can't multiply matrices together. Of course the result is not zero, the matrix on the left will be: $$ \begin{pmatrix} 0 & e^{i\omega_at/2}\\ e^{-i\omega_at/2}&0 \end{pmatrix} $$ So i was solving problem 3 from...
  2. jorgerp24601

    Interaction picture of a phonon-electron coupling

    What I have tried to do is to separate the exponential of the unitary transformation operator to the interaction picture into three different Hilbert "subspaces" like: $$e^{i\frac{H_0}{\hbar}t}=e^{i\omega_m \hat{b}^+\hat{b}}\otimes e^{-i\hbar\nu|1><1|} \otimes e^{-i\frac{g}{\omega_m}|e><e|t}$$...
  3. M

    I Field Renormalization vs. Interaction Picture

    When introducing renormalization of fields, we define the "free Lagrangian" to be the kinetic and mass terms, using the renormalized fields. The remaining kinetic term is treated as an "interaction" counterterm. If we write down the Hamiltonian, the split between "free" and "interaction" terms...
  4. J

    I Understanding the Absence of Rabi Oscillations in the Interaction Picture

    When working on the interaction picture you can show that in a certain rotating frame the Hamiltonian of a 2-level system (for example) becomes uncoupled. This implies that in such frame there are no Rabi oscillations or other dynamical phenomena, this seems weird to me and I would like to know...
  5. binbagsss

    A Kubo Formula Deriviation - interaction picture, complete eigenbasis

    Hi I'm looking at David Tong notes on QHE http://www.damtp.cam.ac.uk/user/tong/qhe/two.pdf (page 56), I've attached the relevant screenshot below also. I understand we are working in the interaction picture whereby states evolve via the Unitary Operataor EQ 2.10 in the notes(I think this is...
  6. C

    How can I derive Eq 9.5.11 in Scully's Quantum Optics

    Firstly, I don't know in which Picture this equation holds (if I hadn't missed some words in the previous text...). I think it may be the Heisenberg Picture. But if it is, the rest target is to prove $$\frac{i}{\hbar}[H_R+H_{FR},(a^\dagger) ^ma^nO_A]=\langle\frac{d}{dt}((a^\dagger)...
  7. A

    I Interactions in QFT and consistent field energy....

    I have a not-very-well formulated question about the interaction picture of QFT. I understand that, in an interaction picture, particle numbers are not well defined (except as t goes to infinity and you're back at free fields). However, at the very least, in an interaction picture, a field...
  8. L

    I Hamiltonian after transformation to interaction picture

    Dear all, I am encoutering some difficulties while calculating the Hamiltonian after the transformation to the interaction picture. I am following the tutorial by Sasura and Buzek: https://arxiv.org/abs/quant-ph/0112041 Previous: I already know that the Hamiltonian for the j-th ion is given...
  9. M

    Hydrogen in Magnetic Field, Interaction Representation

    The hydrogen is placed in the external magnetic field: $$ \textbf{B}=\hat{i}B_1 cos(\omega t) + \hat{j} B_2 sin(\omega t) + \hat{k} B_z ,$$ Using the relation ## H = - \frac{e\hbar}{2mc} \mathbf \sigma \cdot \mathbf B ##, then I got the form $$ H = H_0 + H' , $$ where $$ H'= - \frac{e...
  10. F

    Interaction picture - time evolution operator

    Hey all, I got some question referring to the interaction picture. For example: I have the Hamiltonian ##H=sum_k w_k b_k^\dagger b_k + V(t)=H1+V(t)## When I would now have a time evolution operator: ##T exp(-i * int(H+V))##. (where T is the time ordering operator) How can I transform it...
  11. naima

    What is the equivalence to the free field in the interaction picture?

    I read in this http://arxiv.org/abs/1501.05658']paper[/PLAIN] "Unfortunately, the derivation of the Dyson series perturbation expansion relies on the use of the interaction picture, in which the interacting field is unitarily equivalent to the free field" It may be obvious but could you tell...
  12. C

    Interaction picture equation from Heisenberg equation

    The standard Heisenberg picture equation of motion is $$i\hbar\frac d{dt}A_H=[A_H,H],$$ assuming no explicit ##t##-dependence on the Heisenberg-picture operator ##A_H##. I've been trying to go directly from this equation to the corresponding interaction-picture equation, $$i\hbar\frac...
  13. X

    Time Evolution operator in Interaction Picture (Harmonic Oscillator)

    Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...
  14. G

    Interaction picture and s-matrix

    Consider the S-matrix: <f|U(t,-t)|i> When going into the interaction picture, this becomes: <f_I|U_I(t,-t)|i_I> where the propagator is the interaction picture propagator, and the states are interaction states. Can you say that: <f_I|U_I(t,-t)|i_I>= <f|U_I(t,-t)|i> ? It...
  15. C

    Interaction Picture: Explaining H = H_o + H_int

    So this concept of H = H_o + H_int has been extremely confusing to me. Wikipedia offers the best explanation, but there a couple things that still confuses me http://en.wikipedia.org/wiki/Interaction_picture Why is the state vector in the Interacting picture defined as |\psi_{I}(t)> = e^{i...
  16. K

    Some confusions about interaction picture, in peskin

    In peskin chap 4 on interaction field theory, he first introduced some basics about interaction picture(mostly in pg. 83~87) , it seems he assumed the Hamiltonians H=H_0+H_int in Schrodinger picture are all time-independent, because he used quite a lot of notations like exp(iHt), exp(iH_0t) and...
  17. D

    Equation of motion and operators in the interaction picture.

    Homework Statement I have a question that says: What is the equation of motion for a general operator in the interaction picture. I.e. how does the time derivative of the operators behaves ? Show this. And then I have to find the time development for the annihilation and creation operator...
  18. L

    Proving Interaction picture field satisfies KG eqn

    Hi, I'm just trying to convince myself that the field in the interaction picture (IP) \phi_I(x,t)=e^{iH_0t}\phi(x,0)e^{-iH_0t} satisfies the Klein Gordan equation: (\tfrac{\partial^2}{\partial t^2}-\nabla^2+m^2)\phi_I(x,t)=0 . I have so far worked out that the time derivative is...
  19. J

    Interaction picture (Is there a mistake in QFT books?)

    It is often stated that the transition amplitude between eigenstates of the free-field Hamiltonian H_0 is encoded by the S-matrix, defined by \langle \mathrm{f} | U{_\mathrm{I}} (\infty,-\infty)|\mathrm{i} \rangle. where U_{\mathrm{I}} is the time-evolution operator in the interaction...
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