Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom

Abstract

This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schrö dinger equation. Using complex canonical variables, a formal proof of the quantization axiom p fi p = Ài h$, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov-Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.

Classical universe emerging from quantum cosmology without horizon and flatness problems

“…From the microwave background radiation it is possible to trace it up to a red-shift z ∼ 10 3 , while nucleosynthesis probes it up to z ∼ 10 11 . We do not have observational evidence regarding the correctness of this scenario at larger red-shifts, for example the standard GUT area, z ∼ 10 27 .…”

“…Indeed, it not only provides an alternative interpretation of quantum mechanics but may also serve as a powerful tool to solve quantum mechanical problems [25][26][27][28]. The starting point of the CQHJ formalism of quantum mechanics, instead of (11), is to use the following ansatz [9][10][11]:…”

Flux continuity and probability conservation in complexified Bohmian mechanics

“…The resultant "complex-valued Bohmian trajectories" offer certain advantages; for instance, they are known not to be fixed-points, in general, for nondegenerate stationary states, so that it is possible to achieve nontrivial trajectory dynamics in this context. Although complexvalued Bohmian mechanics may still be in its infancy, interest has grown tremendously in the last few years [1,9,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The field appears to have started in the 1980's with a paper by Leacock and Padgett [15] and a less well known (and very brief) article by Tourenne [16].…”

“…The complexified flux is naturally defined from Eqs. (18) and (20), and the analytic continuation of Eq. (6), as…”

“…The field appears to have started in the 1980's with a paper by Leacock and Padgett [15] and a less well known (and very brief) article by Tourenne [16]. More recent authors have explored the complex Bohmian approach both for time-independent (stationary) and time-dependent (wavepacket propagation) problems, in both the analytical and synthetic contexts [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].…”