English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The time-dependent Schroedinger equation with time-independent Hamiltonianmatrix is a homogeneous linear oscillatory system in canonical form. Weinvestigate whether any classical system that itself is linear, homogeneous,oscillatory and conservative is guaranteed to linearly map into a Schroedingerequation. Such oscillatory classical systems can be analyzed into their normalmodes, which are mutually independent, uncoupled simple harmonic oscillators,and the equation of motion of such a system linearly maps into a Schroedingerequation whose Hamiltonian matrix is diagonal, with h times the individualsimple harmonic oscillator frequencies as its diagonal entries. Therefore ifthe coupling-strength matrix of such an oscillatory system is presented insymmetric, positive-definite form, the Hamiltonian matrix of the Schroedingerequation it maps into is h times the square root of that coupling-strengthmatrix. We obtain a general expression for mapping this type of oscillatoryclassical equation of motion into a Schroedinger equation, and apply it to thereal-valued classical Klein-Gordon equation and the source-free Maxwellequations, which results in relativistic Hamiltonian operators that arestrictly compatible with the correspondence principle. Once such an oscillatoryclassical system has been mapped into a Schroedinger equation, it isautomatically in canonical form, making second quantization of thatSchroedinger equation a technically simple as well as a physically veryinterpretable way to quantize the original classical system.

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