Via the Schwinger action principle, we derive a formula for the transformation function (or propagator) of time-dependent systems that can be expressed into a quadratic form. We present also a new type of two-time rotation-translation of coordinates which makes it possible to evaluate the exact transformation function of the three-dimensional time-dependent charged harmonic oscillator in cross time-varying, arbitrary magnetic and laser fields. We demonstrate that such a transformation function can be obtained from that for a free-particle in the new two-time and space coordinate system, whereby the scalar and vector (quadratic) potentials are transformed away from the quantum-mechanical motion. We also show how the quantum-mechanical eikonal theory of this problem arises in a natural way from the Schwinger action principle.
Via the Schwinger action principle, we derive a formula for the transformation function (or propagator) of time-dependent systems that can be expressed into a quadratic form. We present also a new type of two-time rotation-translation of coordinates which makes it possible to evaluate the exact transformation function of the three-dimensional time-dependent charged harmonic oscillator in cross time-varying, arbitrary magnetic and laser fields. We demonstrate that such a transformation function can be obtained from that for a free-particle in the new two-time and space coordinate system, whereby the scalar and vector (quadratic) potentials are transformed away from the quantum-mechanical motion. We also show how the quantum-mechanical eikonal theory of this problem arises in a natural way from the Schwinger action principle.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
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-particle\textquotedblright} 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.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.
In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.