When is the Interaction representation used and why?

[DonnerJack]

Hi,

Can someone explain to me why do we actually need the Interaction/Intermediate representation?
In my past, each course in QM touched it only for a few minutes and then it got... forgotten.

Can someone please give me an example as to how (and when) it is used (and a good reason why)?

Thanks!

 

[quetzalcoatl9]

the interaction picture is useful because it's decomposition of the hamiltonian allows for time-dependent perturbation methods

 

[f95toli]

We don't really "need" the interaction picture, but it is very convenient sometimes.
The interaction picture is a representation which is somewhere in-between the Schrödinger and the Heisenberg picture. Note, however, that you can easily move between all of these representations using unitary transformations.

I think the word "picture" is somewhat missleading. Today there is no "philosophical" reason why you choose one over the other, you use whichever one s the most convenient for the problem you are trying to do. Moving between pictures is therefore somewhat akin to e.g. moving between coordinate-systems in classical mechanics.

A good example would be a driven Jaynes-Cummings Hamiltonian on resonance (being driven at some frequency $$\omega_l=\omega_0=\omega_r, where \omega_0, \omega_r$$ are the splitting of the 2-level system and the resonance frequency of the resonator, respectively).
Moving to the interaction picture here essentially means that you are solving your problem in a 'rotating coordinate system' which simplifies the problem A LOT since all but two terms become zero and the time dependence dissapears.