On quantization, the generalised Schroedinger equation and classical mechanics

V. D. Rusov, D.S. Vlasenko, S.Cht. Mavrodiev

M. Fusco Girard

The numerical version of the Hamilton-Jacobi quantization method, recently proposed, is applied to the one dimensional quartic oscillator. A suitable quantization condition is formulated and various energy levels and wave functions are computed. The results very well agree with those obtained by means of the Schroedinger equation, and confirm that the Quantum Hamilton Jacobi approach, which is the exact version of the semiclassical WKB scheme, is a self-contained quantization procedure, equivalent and independent from the Schoedinger's one but more general. Indeed, with respect to this latter, the Quantum Hamilton-Jacobi equation can be used to investigate the limit h->0, where the Schroedinger equation loses its significance, and explains how the fundamental quantities of the classical mechanics, the Hamilton's characteristic function and the classical momentum, are generated from the corresponding quantum ones.

Steven Kenneth Kauffmann

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.

Herbert C. Corben

Jinwuk Seok, Changsik Cho

This work presents a quantum mechanical framework for analyzing quantization-based optimization algorithms. The sampling process of the quantization-based search is modeled as a gradient-flow dissipative system, leading to a Hamilton-Jacobi-Bellman (HJB) representation. Through a suitable transformation of the objective function, this formulation yields the Schroedinger equation, which reveals that quantum tunneling enables escape from local minima and guarantees access to the global optimum. By establishing the connection to the Fokker-Planck equation, the framework provides a thermodynamic interpretation of global convergence. Such an analysis between the thermodynamic and the quantum dynamic methodology unifies combinatorial and continuous optimization, and extends naturally to machine learning tasks such as image classification. Numerical experiments demonstrate that quantization-based optimization consistently outperforms conventional algorithms across both combinatorial problems and nonconvex continuous functions.