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Journal of Physics Conference Series

The optical-mechanical analogy for wave mechanics: a new hope

D. H. Delphenich

Published Mar 1, 2022• 1 citation
Publisher

Abstract

Abstract The continuum-mechanical formulation of wave mechanics suggests that there is an intermediate stage of theoretical generality between wave mechanics and point mechanics, namely, continuum mechanics. When that argument is applied to the corresponding transition from wave optics to geometrical optics, the corresponding intermediate stage is essentially the geometrical theory of diffraction, i.e., the theory of diffracted geodesics.

DOI
10.1088/1742-6596/2197/1/012005

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Abstract The continuum-mechanical formulation of wave mechanics suggests that there is an intermediate stage of theoretical generality between wave mechanics and point mechanics, namely, continuum mechanics. When that argument is applied to the corresponding transition from wave optics to geometrical optics, the corresponding intermediate stage is essentially the geometrical theory of diffraction, i.e., the theory of diffracted geodesics.
Wave mechanics In the classical dynamics the state of the system whose coordinates are q 1 … q s and momenta p 1 … p s is denoted by a point in 2s-dimensional space (the q-space); changes in the system are indicated by the passage of the point along some curve in this space, a ‘ray,’ it may be called. Schrödinger's new concept is that changes in the system should be represented in the q-space not by ‘rays’ but by ‘waves.’ He regards the classical mechanics as only an approximation to actual mechanical events, which he says can only be described by wave mechanics; the former in his view bears to the latter just the same relation as geometrical (or ‘ray’) optics does to physical (or ‘wave’) optics. A large scale (macroscopic) mechanical process corresponds to a wave signal in the g-space, which can be regarded as a point in comparison with the geometrical structure of its path. But for small scale (microscopic) processes, such as atomic phenomena are, the rigorous wave theory must be used. The wave equation must replace the usual dynamical equations. The latter are just as unfruitful for the purposes of atom mechanics as geometrical optics alone would be to clear up the fine points of diffraction phenomena.
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.
Light quanta are treated by wave mechanics by analogy with electrons. It is shown that their wave equation is the ordinary optical wave equation. Heisenberg's principle of indeterminateness becomes a description of diffraction. In problems in which localization of quanta is found experimentally, wave packets are to be set up; these are applied to the experiments of Bothe and Geiger, and of Compton. The paths of quanta coincide with the rays of geometrical optics, with a deviation of the error in geometrical optics; by the principle of indeterminateness, more accurate laws for the paths are neither necessary nor possible.
The paper gives an account of the author's work on a new form of quantum theory. \textsection{}1. The Hamiltonian analogy between mechanics and optics. \textsection{}2. The analogy is to be extended to include real "physical" or "undulatory" mechanics instead of mere geometrical mechanics. \textsection{}3. The significance of wave-length; macro-mechanical and micro-mechanical problems. \textsection{}4. The wave-equation and its application to the hydrogen atom. \textsection{}5. The intrinsic reason for the appearance of discrete characteristic frequencies. \textsection{}6. Other problems; intensity of emitted light. \textsection{}7. The wave-equation derived from a Hamiltonian variation-principle; generalization to an arbitrary conservative system. \textsection{}8. The wave-function physically means and determines a continuous distribution of electricity in space, the fluctuations of which determine the radiation by the laws of ordinary electrodynamics. \textsection{}9. Non-conservative systems. Theory of dispersion and scattering and of the "transitions" between the "stationary states." \textsection{}10. The question of relativity and the action of a magnetic field. Incompleteness of that part of the theory.
According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.