The Schroedinger-equation presentation of any oscillatory classical linear system that is homogeneous and conservative

TL;DR

This paper demonstrates that any homogeneous, conservative, linear oscillatory classical system can be linearly transformed into a Schrödinger equation, providing a new perspective for quantization and analysis of such systems.

Contribution

It introduces a general method to map classical oscillatory systems into Schrödinger equations, including relativistic cases like Klein-Gordon and Maxwell equations.

Findings

Mapping classical systems to Schrödinger equations simplifies quantization.

The Hamiltonian matrix relates directly to the system's coupling matrix.

The approach is compatible with the correspondence principle.

Abstract

The time-dependent Schroedinger 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 Schroedinger 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 Schroedinger 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 Schroedinger equation it maps into is h 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 Schroedinger 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 Schroedinger equation, it is automatically in canonical form, making second quantization of that Schroedinger equation a technically simple as well as a physically very interpretable way to quantize the original classical system.