de Broglie’s concept of particles as sources of phase is extended to the analysis of systems in which the potential is not constant. The treatment assumes that the phase information propagates as waves according to classical electrodynamics including reflection with phase changes from potential energy barriers. The energies and weighting functions in such cases are obtained by comparison of the phase of the accelerated particle relative to the phases of reflections from the change in potential. Application to the chemically interesting systems of the harmonic oscillator and hydrogen atom are considered using the time evolution of the relative phases of the reflected de Broglie phase waves, whereby the weighting functions are obtained from the distance dependence of the phase differences between the reflections and the source. This is accomplished conveniently by taking time derivatives of the relative de Broglie phases and considering the periodicity of the motion. Optimum trajectories are determined by minimization of the action 2i(mv2-r∂V/∂r)/ħγ which depends on the potential V and the velocity v. The treatment leads to the Newtonian force law for circular orbits and affords the same energies as derived using the Schrödinger equation.
de Broglie’s concept of particles as sources of phase is extended to the analysis of systems in which the potential is not constant. The treatment assumes that the phase information propagates as waves according to classical electrodynamics including reflection with phase changes from potential energy barriers. The energies and weighting functions in such cases are obtained by comparison of the phase of the accelerated particle relative to the phases of reflections from the change in potential. Application to the chemically interesting systems of the harmonic oscillator and hydrogen atom are considered using the time evolution of the relative phases of the reflected de Broglie phase waves, whereby the weighting functions are obtained from the distance dependence of the phase differences between the reflections and the source. This is accomplished conveniently by taking time derivatives of the relative de Broglie phases and considering the periodicity of the motion. Optimum trajectories are determined by minimization of the action 2i(mv2-r∂V/∂r)/ħγ which depends on the potential V and the velocity v. The treatment leads to the Newtonian force law for circular orbits and affords the same energies as derived using the Schrödinger equation.
A formulation is proposed for the behavior of a particle in a box based on two assumptions: (1) particles have an associated phase which changes periodically with time as defined by de Broglie, and (2) the transmission of phase information follows optical principles, including reflection at potential boundaries and relativistic Doppler effects. This approach defines optimal trajectories by matching the phase of a source and its reflections. The momentum of the particle is guided by a de Broglie wave which follows the equation R exp {i[w0 (1-v^2/c^2)^-1/2][(t - vz)/c^2]} cos {[w0(1-v^2/c^2)^-1/2/c](vt-z)} This wave is generated by reflections occurring an even number of times, the phase and amplitude of which correspond to Lorentz transforms relative to the time and position, respectively. Nodes are produced by reflections occurring an odd number of times, which can be conceived to generate a wave traveling in the opposite direction and shifted by a phase constant π. Application to an individual particle in an infinite potential well affords the same results for energies and nodes as derived by the Schrödinger equation.
Bohmian mechanics considering particles as de Broglie sources of phase which propagates according to classical electrodynamics affords a methodology for evaluating the diffraction of individual particles. Treating the phasing as optical waves allows for reflections at potential energy barriers by which the phase information of an individual particle can return to direct the motion of the particle. Using this approach, crystallographic diffraction is shown to exemplify the behavior of a particle in a box, and the predictions for single and double slit diffraction also are consistent with optical and particle experiments.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367
In this work we are presenting a modified Coulomb potential function to describe the interaction between two micro-scopic electric charges. In particular, concerning the interaction between the proton and the electron in the hydrogen atom. The modified potential function is the product of the classical Coulomb potential and an oscillatory function dependent on a quantized phase factor. The oscillatory function picks up only selected points along the Coulomb potential, creating potential wells and barriers around the nucleus of the atom. The new potential reveals us new features of the hydrogen atom. Searching for a manner to determine the phase factor, we are using the concept of the de Broglie particle wavelike behavior and the quantum analogue of the virial theorem for describing the bound motion of a particle in a central force field. This procedure is a kind of feedback action, where we are making use of well established concepts of the quantum mechanics aiming to determine the phase factor of the new interaction potential.
The atom model of Niels Bohr is commonly considered as antiquated even if it describes the atomic spectrum of hydrogen quite accurately. The later published relation of Louis De Broglie could arithmetically be implemented into Bohr’s formulation, leading to the concept of standing waves as the existence cause of excited electron states. However, at that time it was not possible to find well defined electron trajectories being classically describable. As a consequence, the »quantum mechanics« were developed by several authors, particularly by Schrodinger and by Heisenberg, but delivering a hardly under¬standable formalism where the classical physical laws appear being abrogated while abstract terms replace concrete and imaginable ones, abandoning the particle perception of mass points. In particular, Heisenberg’s »uncertainty principle« and the assumption of state probabilities seem to be in a striking variance to the idea of standing waves. In contrast, a formulation is given here which exactly describes the electron trajectories in the exited states solely by applying classical physical laws. Firstly, the original Bohr-model - in combination with De Broglie’s relation - is rolled up. From this formula system a vibration frequency - corresponding to the De-Broglie frequency – is deduced which is n-times larger than the rotation frequency of the Bohr-model. Furthermore, a direct coherence between that vibration-frequency of the electron and the frequency of the involved light is evident being explainable as a resonance effect. Then, a three-dimensional model is proposed where the electron oscillates and pulses perpendicularly to a virtual rotation plane i.e. rotating around a vertical axis, accompanied by a perpetual energy exchange between potential and kinetic energy. This leads at the excited, metastable energy states to well-defined, three-dimensional and wavy trajectories winding up on a surface similar to the one of a hyperboloid, whereas at the ground state the trajectory is planar and stable.