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International Journal of Applied Science and Engineering Review

A COMPLETE RE-DERIVATION OF THE HIDDEN VARIABLE EQUATION IN DAVID BOHM’S 1952 PAPERS

Yeuncheol Jeong

Published Jan 1, 2022• 0 citations
Publisher

Abstract

Schrodinger’s equation is a basic mathematical expression for a single particle in quantum mechanics. However, its physical meaning has never been assigned in any standard quantum mechanics text book. Here, a detailed complete re-derivation of hidden variable equation in David Bohm’s 1952 papers is given to assign some physical meanings to Schrodinger’s equation. This work shows some required detailed transformation of Schrodinger’s Equation into a form of Newtonian second law of motion with definite trajectory with some initial conditions given. All the quantum effects come from the single localized term called the quantum potential.

DOI
10.52267/ijaser.2022.3304

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Schrodinger’s equation is a basic mathematical expression for a single particle in quantum mechanics. However, its physical meaning has never been assigned in any standard quantum mechanics text book. Here, a detailed complete re-derivation of hidden variable equation in David Bohm’s 1952 papers is given to assign some physical meanings to Schrodinger’s equation. This work shows some required detailed transformation of Schrodinger’s Equation into a form of Newtonian second law of motion with definite trajectory with some initial conditions given. All the quantum effects come from the single localized term called the quantum potential.
In our derivation of Planck's Law (showing that this Formula is an exact mathematical identity that describes the interaction of measurement) the 'accumulation of energy' locally at time t was a key variable. By starting with this as the primary variable and defining energy, momentum and force in terms of it we are able to mathematically derive such basic laws of Physics as Conservation of Energy and Momentum, and Newton's Second Law of Motion. Following the same approach we demonstrate in this short note how Schrodinger's Equation relates to this formulation. This leads to a physical meaning of the wave-function ψ and a plausible explanation of the double-slit experiment.
In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.
David Bohm transformed Schrödinger equation into a form of Newton’s second law of motion, revoking causality, determinism and realism in quantum mechanics. However, the newly created term of quantum potential in Bohm’s transformation is taken to be non-physical to many physicists. Besides, Bohm’s quantum mechanics came one generation later after Copenhagen counterpart. At the same time, Bohm could not advocate his quantum mechanics with other physicists, due to his involuntary exile from the US, leading to the marginality of his theory in the general physics community today.
Abstract This work presents a further development of causal interpretation of quantum mechanics. Existence of ‘non-local hidden variables’ like a physical field is taken as an assumption. Under this assumption, on the basis of the causal approach it is proven that Schrödinger equation is a necessary condition for stability of the motion of a particle. For the case of a hydrogen atom this approach gives a mathematical base with which to suggest that (1) the electron’s spin in an atom is precessing; (2) the energy of the precessional motion on Bohr orbits satisfies the Rydberg formula, and (3) structures are formed in the physical vacuum which stabilize the motion of an electron on Bohr orbits.
A brief account of the world view of classical physics is given first. We then recapitulate as to why the Copenhagen interpretation of the quantum mechanics had to renounce most of the attractive features of the clasical world view such as a causal description, locality, scientific realism and introduce a fundamental distinction between system and apparatus. The crucial role is played in this by the Bohr's insistence on the wavefunction providing the most complete description possible for an even individual system. The alternative of introducing extra dynamical variables, called hidden variables, in addition to the wavefunction of the system so as to be able to retain at least some of the desirable features of classical physics, is then explored. The first such successful attempt was that of Bohm in 1952 who showed that a realistic interpretation of the quantum mechanics can be given which maintains a causal description as well as does not treat systems and measuring appeartus differently. We begin with the construction of the Bohm's theory. He introduces particle positions as the hidden variables. The particle positions play a special role in Bohm theory. The particle trajectories are guided by the wavefunction. The Bohm theory is deterministic. The probability enters through a special assumption, ``quantum equilibrium'' hypothesis, for the initial conditions on the ensemble of particle trajectories. The ``wave or particle'' dilemma is resolved by a ``wave and particle'' resolution. The measurements in Bohm theory can be described without mysticism. Bohm's theory is however nonlocal.
The relationship between the modified potential and Bohm's quantum-mechanical potential is developed. For a given bound-state energy eigenvalue of the Schr\"odinger equation, the modified potential, which is not unique, describes various possible continuous motions. It is shown that all of the various continuous motions of the same energy eigenvalue do generate the same corresponding Schr\"odinger wave function. The parameters which specify these different continuous motions are identified as hidden variables. The equations of motion for a particle with the modified potential differ from those of Bohm.