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College Physics

Planck’s theory about"Geometrical optics of non-homogeneous bodies relationships to quantum mechanics"

Fan Hong-Yi

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The theory about how Planck presenting the stationary Schrdinger equation by the analogy between geometrical optics equations in non-homogeneous bodies with fundamental equations of classical mechanics for a free particle,which moves in a given static conservative field of force,is introduced.

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According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear 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. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
According to Schrodinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear 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. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrodinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
According to De Broglie's idea of analogy, the relation between quantum mechanics and classical mechanics is similar to that between wave optics and geometric optics, we have given the quantum equation of the gravitational field intensity Eg(r⃗,t)E_g(\vec{r},t)Eg​(r,t) for matter particles, since the gravitational field intensity Eg(r⃗,t)E_g(\vec{r},t)Eg​(r,t) relatives to the particle position distribution function ψ(r⃗,t)ψ(\vec{r},t)ψ(r,t), the quantum equation convert into the Schrodinger equation. In addition, we have further studied the essence and origin of quantum theory, and obtained some new results.
Abstract：Geometrical optics has high similarity with classical particle mechanics. After first quantization program classical particle mechanics can obtain non-relativistic quantum mechanics that suitable for micro, but quantum mechanics does not describe the photon that its static mass is zero. So whether there is a kind of "quantization " approach for geometrical optics, by which non-relativistic quantum mechanics that can be used to describe the photon can be obtained. It’s quantum mechanics of photon similar to the Schrödinger wave mechanics under the neglection of the formation and annihilation of photon. In this paper, several methods that can be used to find