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International Journal of Fundamental Physical Sciences

Determination of Escape Speed from de Broglie-Bohm Interpretation

Ch’ng Han Siong, Shahidan Radiman

Published Mar 31, 2013• 0 citations
Publisher

Abstract

In this study, we apply the standard quantization procedure to the Newtonian equation to obtain the Schrödinger equation. The wave function is obtained and subsequently the de Broglie-Bohm interpretation is applied to the wave function to yield the formulas for escape speed. It is shown that the usual Newtonian formula for escape speed is purely resulted from taking the asymptotic form of Bessel functions. We then extend our work to hydrogen atom and show that the work done to eject the electron away from proton is in discrete form. The ionization energy for ground state of hydrogen atom from escape kinetic energy method is obtained.

DOI
10.14331/ijfps.2013.330043

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In this study, we apply the standard quantization procedure to the Newtonian equation to obtain the Schrödinger equation. The wave function is obtained and subsequently the de Broglie-Bohm interpretation is applied to the wave function to yield the formulas for escape speed. It is shown that the usual Newtonian formula for escape speed is purely resulted from taking the asymptotic form of Bessel functions. We then extend our work to hydrogen atom and show that the work done to eject the electron away from proton is in discrete form. The ionization energy for ground state of hydrogen atom from escape kinetic energy method is obtained.
We proposed a new de Broglie-Bohm interpretation, applied it to various cases, and derived the particle dynamics of de Broglie-Bohm mechanics from the wave dynamics of quantum mechanics.De Broglie-Bohm dynamics, constructed from the ground state of quantum mechanics, is often consistent with classical dynamics. Even when the two differ, the discrepancy remains within the range of minute quantum effects. Consequently, for many quantum mechanical ground states, quantum mechanics and de Broglie-Bohm mechanics, linked by the de Broglie-Bohm interpretation, are consistent and free of contradictions.
The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.
A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.
There are three possible interpretations of the wave function in the de Broglie-Bohm theory: taking the wave function as corresponding to a physical entity or a property of the Bohmian particles or a law. In this paper, we argue that the first interpretation is favored by an analysis of protective measurements. The de Broglie-Bohm theory is an alternative to standard quantum mechan-ics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. The wave function follows the linear Schrödinger equation and never collapses. The motion of the particles, which are usually called Bohmian particles, follows the so-called guiding equation. Although the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics, there is no clear consensus with regard to its physical interpretation. In particular, the interpre-
The validity of the de Broglie formula λ=h⁄p for matter waves has been confirmed by numerous experiments such as Davisson-Germer and G. P. Thomson experiments. But there is a big problem here. If λ=h⁄p is valid for matter waves, we must be able to obtain it for these waves using the mathematical calculations. But as you will see here, according to de Broglie's hypothesis, which considers matter wave energy is E=hf, this work is impossible; and so, we must change the matter wave energy equation to extract λ=h⁄p. The change in the equation of energy causes two important changes in the Schrodinger’s wave equation. These two changes bring us to a new wave equation. In the fourth part of the article, you will see that we need a new atomic model for the hydrogen atom due to some problems. The new model leads us to new results and findings. In this Article, we show what the cause of the quantization of the orbits in the Bohr atomic model is.