Discusses the course of quantum mechanics setting up and the application of analogy meathod in quantum mechanics, explains analogy method an important seientific research method.
This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the 2D harmonic oscillator with degenerate excited states; and a nonlinear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either with coding or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can, for example, use our derived results and the code (supplemental material (https://stacks.iop.org/EJP/41/065406/mmedia)) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.
Abstract This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the 2D harmonic oscillator with degenerate excited states; and a nonlinear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either with coding or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can, for example, use our derived results and the code (supplemental material ( https://stacks.iop.org/EJP/41/065406/mmedia )) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.
Abstract In this article, we work with an optical-mechanical analogy model that exploits the analogy between wave mechanics in quantum theory and the scalar wave treatment of optics. We discuss the application of this model to the evanescent waves and the frustrated total internal reflection of light. Further, we demonstrate that this model can also be used to derive the Goos–Hänchen shift for s -polarised light. This work provides the advanced undergraduate students and instructors of optics and quantum mechanics courses with a platform, where they can apply the methods of the quantum theory to optics, which is not discussed in depth in the standard texts.