# Understanding the Absence of Rabi Oscillations in the Interaction Picture

• I
• Jufa
In summary, when working with the interaction picture in quantum mechanics, the Hamiltonian of a 2-level system can become uncoupled in a rotating frame. This means that there are no Rabi oscillations or other dynamic phenomena in that frame. This concept can be understood through the covariant formulation of quantum mechanics, where observable quantities are independent of the picture used. There is a discussion on this topic on Physics Forums, and a shortened derivation can be found in posting #7 on the thread. However, quantum mechanics can be completely formulated in a covariant manner from the start through time-evolution-picture transformations.
Jufa
TL;DR Summary
When working in the interaction picture one is free to choose the unitary transformation that seems helpful. How can I show that the dynamics of the problem are independent of such choice.
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 how is this understood.

Quantum mechanics can be formulated in a covariant way, i.e., such that the observable quantities (probabilities, expectation values of observables, cross sections in scattering theory etc. etc.) are by construction picture independent. There's a recent thread on this topic here at PF:

https://www.physicsforums.com/threa...ry-transformations-of-the-hamiltonian.979107/
A somewhat shortened derivation starting from the Schrödinger picture can be found on my posting #7 in this thread, though, as I said, you can formulate QT completely covariantly under "time-evolution-picture transformations" from the very beginning.

Jufa

## 1. What is the Interaction Picture in quantum mechanics?

The Interaction Picture is a mathematical representation used in quantum mechanics to describe the evolution of a quantum system over time. It is a combination of the Schrödinger Picture and the Heisenberg Picture, and it allows for easier calculations of time-dependent systems.

## 2. How does the Interaction Picture differ from the Schrödinger and Heisenberg Pictures?

In the Schrödinger Picture, the state of a quantum system evolves over time while the operators remain constant. In the Heisenberg Picture, the operators evolve while the state remains constant. In the Interaction Picture, both the state and the operators evolve, but in a simpler way compared to the other two pictures.

## 3. What are the advantages of using the Interaction Picture?

The Interaction Picture allows for easier calculations of time-dependent systems, as it separates the time evolution of a system from its intrinsic properties. It also simplifies the calculations of perturbations and interactions between different quantum systems.

## 4. Can the Interaction Picture be used for all quantum systems?

Yes, the Interaction Picture can be used for all quantum systems, as it is a mathematical representation that is applicable to all quantum mechanical problems. However, it is most commonly used for systems with time-dependent Hamiltonians or for studying the effects of interactions between different systems.

## 5. Are there any limitations to using the Interaction Picture?

While the Interaction Picture is a useful tool in quantum mechanics, it does have some limitations. It is not suitable for describing systems in which the Hamiltonian is not time-dependent, and it cannot be used to calculate expectation values of observables. It is also important to note that the results obtained in the Interaction Picture must be transformed back to the Schrödinger or Heisenberg Pictures for physical interpretation.

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