The most realistic situations in quantum mechanics involve the interaction between two or more systems. In the most of reliable models, the form and structure of the interactions generate differential equations which are, in the most of cases, almost
impossible to solve exactly. In this paper, using the Schwinger Quantum Action Principle, we found the time transformation function that solves exactly the harmonic oscillator interacting with a set of other harmonic coupled oscillators. In order to do it, we have introduced a new special set of creation and annihilation operators which leads directly to the emph{dressed states} associated to the system, which are the real quantum states of the interacting emph{textquotedblleft field-particletextquotedblright} system. To obtain the closed solution, it is introduced in the same foot a set of emph{normal mode} creation and annihilation operators of the system related to the first ones by an orthogonal transformation. We find the eigenstates, amplitude transitions and the system spectra without any approximation. At last, we show that the Schwinger Variational Principle provides the solutions in a free representation way.
Here we deal in a pedagogical way with an approach to construct an algebraic structure for the Quantum Mechanical measurement processes from the concept of emph{Measurement Symbol}. Such concept was conceived by Julian S. Schwinger and constitutes a
fundamental piece in his variational formalism and its several applications.
Using Schwinger Variational Principle we solve the problem of quantum harmonic oscillator with time dependent frequency. Here, we do not take the usual approach which implicitly assumes an adiabatic behavior for the frequency. Instead, we propose a n
ew solution where the frequency only needs continuity in its first derivative or to have a finite set of removable discontinuities.
