The Relationship between Classical Mechanics and Wave Mechanics

George Adolphus Schott

Enrico G. Beltrametti, Slawomir Bugajski

R. Faure

Runsheng Tu

In reality, micro objects and macro objects coexist harmoniously, and each macro object depends on micro particles to exist (macro is a collection of micro objects, indicating that they have a very extensive relationship and are symbiotic). However, classical mechanics and quantum mechanics have long been regarded as binary oppositions. This is a serious disconnect between theory and phenomena, requiring the establishment of new theories and methods that are compatible between quantum and classical. Prior to this, many reasons and methods for combining quantum theory with classical theory have been found (providing numerous computational examples). The wave mechanics equations of classical mechanics laws have been written. The technical method used is still the conversion rule of the Hermitian operator in the fundamental postulates of quantum mechanics. There are no new assumptions. The key is to use the relationship between force and energy in the planetary model to obtain wave mechanics equations that include force, Newton's second law, and the law of conservation of momentum. Thus, quantum mechanics, which is compatible with classical mechanics, truly becomes a form of 'mechanics'. This kind of mechanics is generalized wave mechanics. Quantum and classical are coordinated and compatible, resulting in no clear boundary between quantum and classical. It will seriously affect the credibility of quantum decoherence and wave function collapse.

Francesco R. Ruggeri

In quantum mechanics, it is often stressed that if one knows position (x) with complete certainty, then momentum (p) is completely uncertain. A similar result is applied to energy and time. It is also stressed that quantum mechanics is mathematically a wave type theory describing probability waves. In a previous note (1), we argued that one may obtain the Schrodinger equation from exp(i Action), where Action is the classical action, by varying x and t in v=x/t independently. A main idea of classical mechanics is that x and t are not independent, but are directly linked i.e. x(t). In this note, we argue that wave behaviour in quantum mechanics and in the wave equation for a photon decouples time and space which is the opposite notion of classical mechanics. One has partial derivatives (d/dx, d/dt) in an equation describing a problem, but there is a kind of decoupling of changes in x and t. There is still a relationship related to frequencies between the decoupled time and space functions as the wave-equation contains both d/dt d/dt and d/dx d/dx, but one does not have a scenario of x depending directly on t as in classical mechanics. This, we argue, leads to wave or a periodic type behaviour for both x and t. In the case of a potential V(x), there can be multiple x waves exp(ipx) stirred up. In the time-independent Schrodinger equation one does not even consider time (one has EW(x)) even though it is clearly present through exp(iEt), but x and t are not linked As a result, we argue that p and x are not linked because p=mv and if one had v(x), one could integrate dt = dx/v(x) to find an explicit relationship between x and t, as in classical mechanics, which negates the idea of wave mechanics. Thus, we argue that it is the “decoupling” of space and time that is important in quantum mechanics

Elmar Sauter, A. K. Ghatak

For pt.I see ibid., vol.10, p.136-43 (1989). In the preceding paper the authors used the Lagrangian formalism to show the close relationship between the classical mechanics of the harmonic oscillator and the ray optics of the parabolic index optical waveguide. The authors have now studied the corresponding quantum mechanical and wave optical problems and, in particular, have studied the time evolution of a coherent state and have shown its close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide. The corresponding transitions to classical mechanics and ray optics have been shown. The authors have also presented an analysis of the recently developed squeezed states and have shown its correspondence to the waveguide problem. They have demonstrated that the relationship between classical and quantum mechanics is the same as between ray optics and wave optics. All such analyses should be of considerable pedagogical value.