Abstract In this paper we explore the analogies between optical physics and quantum mechanics. The formal similarity between the Helmholtz wave equation and the time-independent Schrödinger equation is presented. Details of the analogy between plane interface and potential step and between thin film and potential barrier are developed and a detailed discussion is provided to reinforce the analogy. We hope that the present work can be useful for undergraduate students.
Abstract In this paper we explore the analogies between optical physics and quantum mechanics. The formal similarity between the Helmholtz wave equation and the time-independent Schrödinger equation is presented. Details of the analogy between plane interface and potential step and between thin film and potential barrier are developed and a detailed discussion is provided to reinforce the analogy. We hope that the present work can be useful for undergraduate students.
Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light . Erwin Schrödinger, Nobel lecture , 1933 In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics. Isomorphism of the Schrödinger and Helmholtz equations In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.
In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics.The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials.The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions -like behaviour at infinity and boundary conditions -are imposed, they generate similar eigenvalues problems, with similar solutions.The benefit of such analogies is twofold.First, it could help a researcher, specialized in a specific field, to better understand a new one.For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics.Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1].Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different."Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.",considers Francis Bacon, quoted by [1].Some two centuries later, Goethe was looking for the "ultimate fact" -the Urphänomenon -specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role.However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.Let us describe now shortly the structure of this chapter.The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3).The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts
Abstract A new derivation of Schrödinger’s equation is presented, based on Schrödinger’s original discussions on refraction and the optical-mechanical analogy, but adopting a much simpler formalism: Newtonian mechanics and some basic elements of classical wave theory (such as Snell’s law). We compare how particles and waves refract and show that the ‘law of particle refraction’ and the ‘law of wave refraction’ may become consistent if one assumes that a particle can be represented by a wave group. In this case, the differential equation whose solutions represent the waves forming such wave group is the Schrödinger equation. Due to the simplicity of the adopted mathematical formalism, we argue that this derivation can be used in quantum mechanics courses at introductory level to give students an idea of Schrödinger’s original path to his wave equation.
Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.
It is shown that the optics-mechanics analogy which originally led Schrödinger to his famous equation stops at the time-independent level. For potentials involving time explicitly, the time-dependent Schrödinger equation cannot be deduced in a similar manner. Instead, a variational principle for the time-dependent Schrödinger equation is established. It minimizes the total quantum fluctuations of a newly defined Hamilton-Jacobi function about its limiting classical value over space-time, thereby demonstrating the existence of a deeper relationship between classical and quantum mechanics beyond the simple optics-mechanics analogy. This principle is similar in spirit to Feynman's space-time approach to quantum mechanics.
th , 2012; revised: Jan. 8 th , 2013; accepted: Jan. 16 th , 2013 Abstract: In order that students have a whole realization of the wave mechanics foundation, we review the two impor- tant enlightened steps in details, i.e. the analogues between mechanics and optics, de Broglie matter wave. Based on the duality of wave and particle, we report the whole process that Schrodinger founded his wave mechanics. We introduce Schrodinger's interesting and deep method of deriving wave equation to quantum mechanics teaching.