Authors: Steven Kenneth Kauffmann
The time-dependent Schrödinger equation with time-independent Hamiltonian matrix is a homogeneous linear oscillatory system in canonical form. We investigate whether any classical system that itself is linear, homogeneous, oscillatory and conservative is guaranteed to linearly map into a Schrödinger equation. Such oscillatory classical systems can be analyzed into their normal modes, which are mutually independent, uncoupled simple harmonic oscillators, and the equation of motion of such a system linearly maps into a Schrödinger equation whose Hamiltonian matrix is diagonal, with h times the individual simple harmonic oscillator frequencies as its diagonal entries. Therefore if the coupling-strength matrix of such an oscillatory system is presented in symmetric, positive-definite form, the Hamiltonian matrix of the Schrödinger equation it maps into is h-bar times the square root of that coupling-strength matrix. We obtain a general expression for mapping this type of oscillatory classical equation of motion into a Schrödinger equation, and apply it to the real-valued classical Klein-Gordon equation and the source-free Maxwell equations, which results in relativistic Hamiltonian operators that are strictly compatible with the correspondence principle. Once such an oscillatory classical system has been mapped into a Schrödinger equation, it is automatically in canonical form, making second quantization of that Schrödinger equation a technically simple as well as a physically very interpretable way to quantize the original classical system.
Comments: 12 pages, The physically most appropriate linear mapping into a Schroedinger equation of such an oscillatory classical system that has a symmetric, positive-definite coupling-strength matrix is given in general closed form. Also archived as arXiv:1101.0168 [physics.gen-ph].
Unique-IP document downloads: 253 times
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.