## 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].

**Download:** **PDF**

### Submission history

[v1] 24 Dec 2010

[v2] 14 Mar 2011

[v3] 28 Sep 2011

[v4] 8 Nov 2011

**Unique-IP document downloads:** 330 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.*

*
*