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, $$i\hbar\frac d{dt}A_I=[A_I,H_0],$$ (see Sakurai 5.5.12) which I thought at first would be simple, but I keep coming up with $$i\hbar\frac d{dt}A_I=[A_I,H_0]+[A_I,V_I],$$ where ##V_I## is the interaction part of the hamiltonian in the interaction picture. The basic problem is that in the original equation ##H## contains both ##H_0## and ##V## and I don't know how to get rid of the ##V## part. Has anyone been through this calculation?(adsbygoogle = window.adsbygoogle || []).push({});

Thanks!

P.S. I know I could just start with ##A_I(t)=e^{iH_0t/\hbar}A_Se^{-iH_0t/\hbar}##, where ##A_S## is in the Schrodinger picture, and I can derive the equation this way, but I feel like it should work the other way too.

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# Interaction picture equation from Heisenberg equation

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