Derivation of the Schrödinger Equation from the Laws of Classical Mechanics Taking into Account the Ether Dr.Nina Sotina

The Schrödinger equation describes many observations very well, but there are still discussions about its physical interpretation. Currently, the probabilistic interpretation is the most common one. Its proponents, however, have difficulties explaining the results of experiments with nonclassical optical effects (e.g., the two-photon interference, teleportation of polarization of the photon, etc.). It can be shown that the probabilistic approach for the case of “nonclassical” optical effects can lead to negative probabilities. It is one of the reasons to get rid of the probabilistic interpretation of the quantum formalism and to return to the idea of “hidden variables”

The probabilistic interpretation is also unable to describe the behavior of quantum systems in living matter (biomolecules). It is known that a living organism molecules act as well-tuned mechanisms. That conflicts with a concept of a molecule as a quantum system that is governed by probabilistic laws of quantum mechanics. E. Schrödinger was the first who note this conundrum. It was one of the reasons why he was an opponent of the probabilistic interpretation of - function. E. Schrödinger believed that - function is associated with some real oscillatory process in an atom.

In 1935 Einstein, Podolsky, and Rosen put forward the issue of incompleteness of the quantum mechanics description of physical reality and suggested the idea of existence of «hidden variables». Later it was proven (J.Bell and others) that «hidden variables» can be either: 1) "nonlocal" (the “nonlocality” is the existence of a connection between spatially separated measurement devices; in essence, theory of long-range forces), or 2) a field wherein disturbances can spread at speeds greater than the speed of light (hence, possibility of the existence of “superluminal” forces). The latter was inconsistent with theory of relativity therefore nobody tried to introduce such a field (or medium) to get rid of the probabilistic interpretation of the quantum formalism..

D.Bohm was an Einstein's student and the follower of theory of relativity. He supported the idea of quantum nonlocality and introduced the field of information in physics. However idea of quantum nonlocality is also not consistent with theory of relativity since there is a implicit postulate of the “locality” in it: the universe can be decompose correctly into difference and separately existing “components of reality”. To measure the speed of light it is necessary that the receiver and transmitter were not only separated in space, but were also autonomous in their behaviour. As a result more and more physicists are inclined towards neither interpretation, which was expressed by physicist David Mermin as “Shut up and calculate!”

I believe the way out of the crises in quantum mechanics is in the return to the idea of «hidden variables» as a physical field (medium, ether) wherein disturbances can spread at speeds greater than the speed of light. I will show below that the Schrödinger equation can be derived from the deterministic laws of classical mechanics. This derivation is based on the work of Russian scientist Chetaev N.G. (1936). Chetaev showed an analogy between the stable motion of a mechanical system under action of conservative forces and quantum processes described by the Schrödinger equation. His work was not translated in English, and is not known outside Russia. So it is presented here to the English speaking community for the first time.

The derivation of the time-independent Schrödinger equation Consider the motion of a mechanical system under the action of conservative forces that do not depend on time t explicitly. In this case there exists the energy integral where is potential energy, are coordinates, are momenta, and the kinetic energy of the system. The complete integral of the Jacobi equation takes the following form:, Let’s make change of variables

The following equations are correct: ( The latter equation can be represented as follows where or, using the energy integral can be written in the form It follows from the theory of stability that for a motion to be stable must be zero (necessary condition of stability). Thus for a stable motion

Let us search for the solution of the latter equation in the form (1) In this case the equation is written as the Schrödinger equation or (2) Note, that E. Schrödinger also used the Jacobi equation and the substitution (1) to derive equation (2). He found this substitution empirically as the one which yields the Rydberg- Ritz formula for the spectrum of the hydrogen atom. However, Schrödinger's approach did not clearly indicate that equation (2) contained extra solutions, because his equation is only the necessary conditions of stability, not sufficient.

Furthermore, it turned out that the equation (2) has more solutions than the original problem. Probably that was the reason why Schrödinger approach that led him to his famous equation was not accepted by a scientific community and the equation (2) was taken as a postulate. Chetaev, however, noticed that the Schrödinger equation contains other solutions besides those that are determined by the potential energy. He gave a method that allows to find all the possible motions of a mechanical system under the action of conservative forces which as a necessary condition of stability have the Schrödinger equation (2). Let me present now my derivation of the Schrödinger equation based on Chetaev’s method, with the further analysis of the solutions.

Assume that in addition to the primary forces described by, there are also forces described by potential. Represent the time-independent component of the psi- function as where A is some real function of coordinates. In this case the following equations are correct: As follows from the theory of stability for a motion to be stable the expression L[S] must be zero.

The energy integral can be represented as Conditions (3) and (4) take the form of the stationary Schrödinger equation: if the additional forces have structure defined by equations Note that the forces are different corresponding to different solutions of the Schrödinger equation. Forces of such type appear as an object is moving in medium (field) as a result of object-medium interaction.

Chetaev consider forces to be small. However, I showed that they can be of the same order of magnitude as the primary forces. Let me demonstrate this for a hydrogen atom. For a case of one particle the Schrödinger equation takes the form where the speed of the particle, i.e. the Schrödinger equation is the law of conservation of energy on a stable orbit.

In the hydrogen atom the electron moves along a circular orbit with the speed - the radius The net force that provides centripetal acceleration equals The question arises whether this deterministic approach, in which the characteristics such as trajectory and velocity of an elementary particle were introduced, contradicts the second postulate of quantum mechanics : “collapse of the wave function”? The answer to this question was given in the article by D. Bohm and Hiley “Measurement Understood Through the quantum potential approach”. (1983)

Bohm and Hiley reviewed the quantum potential approach to quantum theory, and showed that it yields a completely consistent account of the measurement process. Let me remind you, that D.Bohm supported the idea of quantum nonlocality. He has noticed, that the Schrödinger equation can be represented as the law of conservation of energy of stable orbits. He called the potential “the quantum potential”. Bohm explained his quantum potential not as a physical quantity associated with real forces in medium, but by introducing the field of information in physics. I would like to make a point here that inclusion of the “information" as a term in an equation that contains such physical quantities as time, force and displacement looks like a logical inconsistency.

Bohm and his followers often use an example of a ship controlled from a shore to explain their ideas. Here the ship controlled by a radio signal is an analogy of a quantum particle controlled by the information field. In the case of the ship, however, the EM signal turn on some real forces that act on the ship (such as interaction between the ship mechanisms and fluid). I think that Bohm‘s information approach lacks the explanation of what is the nature of the forces that change the particle's motion, even if the control is done at a distance. Moreover, D. Bohm derived his formulas directly from the Schrödinger equation. As was mentioned above this equation contains extra solutions, i.e. such trajectories that do not realize in nature. This should be taken in to consideration when one analyzes concrete solutions.

Another question is: how well does my deterministic interpretation of the Schrödinger equation agree with the probabilistic interpretation? The answer to this question was in fact given by another Russian physicist V.A. Kotel'nikov (2009). As a starting point in his study Kotel’nikov uses the Shrödinger equation and probabilistic interpretation of the psi- function. He raised a question: how can an elementary particle move according to the laws of classical mechanics if its probabilistic behavior is determined by the Schrödinger equation. As result of his mathematical derivations, he concluded that the particle should move under the action of two forces: a classical force, defined by the potential U and a quantum force defined by “quantum potential”.

The role of physics is to establish correspondence between mathematical properties of psi-function and physical quantities. Psi-function must change continuously along the trajectory and into the space surrounding the particle. From the latter formula we can see that an oscillations with exactly the same frequency are at each point of space surrounding the particle. Unfortunately the theory does not address the nature of this process yet. We can make some speculations only. In my opinion the ether is similar in its properties to superfluid. In this case the following interpretation of the mathematical formalism of quantum mechanics for a hydrogen atom can be suggested.

The centre of mass of an electron in a hydrogen atom moves along a circle. Due to the Barnette’s effect, a uniformly processing domain can appear in superfluid, which was observed experimentally. The frequency of precession of the domain. Simultaneously the electrical polarization can appear in such a medium.