Wave Equation

The time-independent Schroedinger and Klein-Gordon equations-as well as any other Helmholtz-like equation-were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory. RÉSUMÉ-On a démontré récemment que les équations indépendantes du temps de Schroedinger et de Klein-Gordon, ainsi que toutes les autres équations d'Helmholtz, sont associées à des systèmes de trajectoires hamiltoniennes couplées par une function (le "potentiel d'onde") codée dans leur structure même, qui permettent de décrire tous le phénomènes ondulatoires, comme le diffraction et l'interférence. Cette propriété suggère d'envisager la Mécanique Ondulatoire comme une extension directe, causale et réaliste de la Mécanique Classique (basée sur les trajectoires exactes de particules ponctiformes pilotées par des ondes materielles monochromatiques) évitant à la fois le probabilisme et les paquets d'ondes de l'interpretation de Copenhagen et de la théorie de Bohm.

This note is presented to the undergraduate students who are interested in the oscillations and waves. One of the simplest models in the classical mechanics is a simple harmonics. A more general oscillation is described by a superposition of the so-called modes. This mode is quantized into elementary excitation in quantum mechanics. In this sense, the concept of the oscillations and waves is fundamental but is essential to understanding the physics from the classical mechanics to the quantum mechanics. The duality of waves and particles plays a central role in quantum mechanics. This note is written on the basis of a book (Oscillations and waves) [in Japanese] 1 written by Prof. M. Ogata of the University of Tokyo. This summer (July, 2009), we visited Japan. We stopped by Kanda Book Stores near the University of Tokyo, in order to buy used books on physics (mainly written in Japanese), which are usually much cheaper than the new books. Fortunately we found a very interesting book on...

In the paper the Schrodinger equation (SE) with gravity term is developed and discussed. It is shown that the modified SE is valid for particles with mass m<M_P, M_P is the Planck mass, and contains the part which, we argue describes the pilot wave. For m \to M_P the modified SE has the solution with oscillatory term, i.e. strings. Key words: Schrodinger-Newton equation; Planck time; pilot wave.

Journal of Applied Mathematics and Physics

Both classical and wave-mechanical treatments of monochromatic wave-like features may be faced in terms of exact trajectories, mutually coupled by a "Wave Potential" function (encoded in the structure itself of all Helmholtz-like equations) which is the cause of any diffraction and/or interference process. In the case of Wave Mechanics, an energy-dependent Wave Potential establishes a bridge between de Broglie's matter waves and the relevant particles by piloting them along their path by means of an energy-preserving "gentle drive", allowing an insight into the dynamical mechanism of waveparticle duality. Thanks to a numerical treatment which is not substantially less manageable than its classical counterpart each particle is seen to "dance", under the action of the Wave Potential, "a wavemechanical dance around the classical motion".

The objective of this work is to investigate the Schrödinger equation, analyzing the mathematical concepts employed and relating them to other areas of knowledge. In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system evolves over time. It was formulated in late 1925 and published in 1926 by the Austrian physicist Erwin Schrödinger. In quantum mechanics, the analogue of Newton's law is the Schrödinger equation for the quantum system (usually atoms, molecules, and subatomic particles are free, bound, or located). It is not a simple algebraic equation but, in general, a linear partial differential equation. The solutions to the Schrödinger equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems.

2006

We look at the mathematical theory of partial differential equations as applied to the wave equation. In particular, we examine questions about existence and uniqueness of solutions, and various solution techniques.

Encyclopedia of Thermal Stresses, 2014

In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic equations, there also exists a general theory for finding wave solutions. In addition, nonlinear parabolic partial differential equations are sometimes said to posses wave solutions, though they lack hyperbolicity, because it may be possible to find solutions that translate in space with time. Unfortunately, an all-encompassing methodology for solution of partial differential equations with any

Physical Review E, 1998

Dissipative quantum systems are sometimes phenomenologically described in terms of a non-Hermitian Hamiltonian H, with different left and right eigenvectors forming a biorthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second quantization.