When is the Interaction representation used and why?

In summary, the interaction picture is a convenient representation that falls between the Schrödinger and Heisenberg pictures. It can be easily changed between using unitary transformations and is useful for solving problems such as a driven Jaynes-Cummings Hamiltonian on resonance. In this representation, the time dependence disappears and the problem is simplified. The choice of which representation to use is based on convenience and can be likened to choosing different coordinate systems in classical mechanics.
  • #1
DonnerJack
7
0
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!
 
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  • #2
the interaction picture is useful because it's decomposition of the hamiltonian allows for time-dependent perturbation methods
 
  • #3
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 [tex]$\omega_l=\omega_0=\omega_r$, where $\omega_0, \omega_r$ [/tex] 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.
 

What is "Interaction representation"?

Interaction representation is a mathematical technique used to simplify complex systems by separating the interactions between different components. It allows for a more manageable analysis of the system by breaking it down into smaller, more easily understandable parts.

How is "Interaction representation" used in scientific research?

Interaction representation is used in a variety of scientific fields, including physics, chemistry, and biology. It is often used to study the behavior of particles in quantum mechanics, the dynamics of molecules in chemistry, and the interactions between cells in biology.

What are the benefits of using "Interaction representation" in scientific research?

There are several benefits to using interaction representation in scientific research. It allows for a more comprehensive understanding of complex systems, it simplifies the analysis process, and it can reveal hidden relationships and patterns within the system. Additionally, interaction representation can help to identify key components or interactions that are crucial for the overall behavior of the system.

Are there any limitations to using "Interaction representation" in scientific research?

While interaction representation is a useful tool in scientific research, it does have some limitations. It may not be suitable for all types of systems, and it can sometimes oversimplify the interactions between components. Additionally, it may be challenging to accurately represent all interactions within a system, which can affect the accuracy of the results.

How has "Interaction representation" evolved over time?

The concept of interaction representation has been around for decades and has evolved significantly over time. In the early days, it was primarily used in quantum mechanics, but it has since been applied to various other fields of science. With advancements in technology and computing power, interaction representation techniques have become more sophisticated and can now handle more complex systems.

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