## The Schrödinger-Equation Presentation of Any Oscillatory Classical Linear System that is Homogeneous and Conservative

**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].

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### Submission history

[v1] 24 Dec 2010

[v2] 14 Mar 2011

[v3] 28 Sep 2011

[v4] 8 Nov 2011

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