Using a nonlinear Schrödinger equation for the wave function of all systems, continuous transitions between quantum and classical motions are demonstrated for (i) the double-slit set up, (ii) the 2D harmonic oscillator and (iii) the hydrogen-like atom, all of which are of empirical interest.
Microscopic particles, like electrons, do not move following the classical laws of motion, given by Newtonian Mechanics. These particles, however, follow other laws that are associated to the propagation of waves. This becomes qualitatively clear when we observe an interference pattern arise in an experiment in which an electron beam passes through a double slit. In this brief study, we will deal with the quantum analysis of the dynamics of particles, by means of its postulates and its precise mathematical formulation. That is, we intend to introduce the wave function and the Schrodinger equation, as well as to interpret physically such a function. The scientific bases of this essay (1),(2),(3),(4),(5) can be found in the books listed in the bibliographic references.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
Failure of the classical theories wave packets wave functions and probability Schrodinger's Equation quantum mechanical operators Schrodinger's Equation with forces continuous spectrum Eigensolutions the harmonic oscillator three-dimensional problems matrices matrices in different representations approximate methods I applications of WKB method alpha decay approximate methods II continuous spectrum perturbation approximate methods III spin and spinors fine structure of the hydrogen atom anomalous Zeeman effect assemblages of identical particles the Dirac Equation non-relativistic limits of the Dirac Equation Dirac operator and wave function transformations negative energy solutions - positron theory relativistic theory of Coulomb scattering Compton scattering.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.