A classical interpretation of the Schrödinger’s wave equation

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We have shown throughout this blog and its companion book “The Reality of the Fourth *Spatial* Dimension” there would be many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

One is that it would allow for a classical interoperation of Schrödinger’s wave equation and its associated probabilities.

In 1924 Louis de Broglie was the first to realize all particles are, in part composed of a matter wave.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their quantum mechanical properties in the classical terms of space and time.  He later abandoned it in the face of the almost universal adherence of physicists to the probabilistic interpretation of Schrödinger’s wave equation.
However one of the difficulties he may have faced is that he assumed along with most other scientists of his day the universe was composed of four-dimensional space-time.

This presented a problem because observations of a space-time environment indicate that a time or a space-time dimension things can only move in one direction, forward.  Therefore, it could not support bidirectional movement required for the propagation of the observed wave properties of particles.

However when Einstein defined the geometric properties of a space-time in terms of the constant velocity of light and a dynamic balance between mass and energy he provided a method of converting a unit of time in a space-time environment with unit of space in four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

As mentioned earlier Louis de Broglie may have been able to derive the probabilities associated Schrödinger equation in terms of the wave properties he had associated with particles by extrapolating the laws of classical wave mechanics in a three-dimensional environment as was done in the article  “Why is energy/mass quantized?” Oct. 4, 2007 to a resonant system created by matter wave on a “surface” of a three-dimensional space manifold with respect to fourth “spatial” dimension.

Briefly it showed the four conditions required for resonance to occur in a three-dimensional environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in one made up of four.

The existence of four *spatial* dimensions would give a matter wave the ability to oscillate spatially on a “surface” between a third and fourth *spatial* dimension thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

However, the oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established on a surface of a three-dimensional space manifold. 

Yet the classical laws of three-dimensional space tell us the energy of resonant systems can only take on the discontinuous or discreet energies associated with their fundamental or harmonic of their fundamental frequency.

However, these are the similar to the quantum mechanical properties of energy/mass in that they can only take on the discontinuous or discreet energies associated with the formula E=hv where “E” equals the energy of a particle “h” equal Planck’s constant “v” equals the frequency of its wave component.

Therefore, he may have been able to defined the quantum mechanical properties of his matter wave in terms of the discrete incremental energies associated with a resonant system in four *spatial* dimensions if he had assume space was composed of it instead of four dimensional space-time.

However, if a quantum mechanical properties of particle is a result of a matter wave on a “surface” of three-dimensional space with respect to a fourth *spatial* dimension, as this suggests one should be able to show that it is responsible for the uncertainties and probabilistic predictions made by Schrödinger and his wave equation regarding the position and momentum of particles.

Classical wave mechanics tells us a wave’s energy is instantaneously constant at its peaks and valleys or the 90 and 270-degree points as its slope changes from positive to negative while it changes most rapidly at the 180 and 360-degree points.

Therefore, the precise position of a particle could be only be defined at the “peaks” and “valleys” of the matter wave responsible for its resonant structure because those points are the only place where its energy or “position” is stationary with respect to a fourth *spatial* dimension.  Whereas its precise momentum would only be definable with respect to where the energy change or velocity is maximum at the 180 and 360-degree points of that wave.  All points in between would only be definable in terms of a combination of its momentum and position.

However, to measure the exact position of a particle one would have to divert or “drain” all of the energy at the 90 or 270-degree points to the observing instrument leaving no energy associated with its momentum left to be observed by another instrument.  Therefore, if one was able to precisely determine position of a particle he could not determine anything about its momentum.  Similarly, to measure its precise momentum one would have to divert all of the energy at the 180 or 360 point of the wave to the observing instrument leaving none of its position energy left to for an instrument which was attempting to measure its position.  Therefore, if one was able to determine a particles exact momentum one could not say anything about its position.

The reason we observe a particle as a point mass instead of an extended wave is because, as mentioned earlier the article  “Why is energy/mass quantized?” Oct. 4, 2007 showed its energy must be packaged in terms of its resonant frequency.  Therefore, when we observe or “drain” the energy continued in its wave function, whether it be related to its position or momentum it will appear to come from a specific point in space similar how the energy of water flowing down a sink drain appears to be coming from a “point” source with respect the extended volume of water in the sink.

As mentioned earlier, all points in-between are a dynamic combination of both position and momentum.  Therefore, the degree of accuracy one chooses to measure one will affect the other. 

For example, if one wants to measure the position of a particle to within a certain predefined distance “m” its wave energy or momentum will have to pass through that opening.  However, Classical Wave Mechanics tells us that as we reduce the error in our measurement by decreasing the predefine distance interference will cause its energy or momentum to be smeared our over a wider area thereby making its momentum harder to determine.  Summarily, to measure its momentum “m”kg / s one must observe a portion the wavelength associated with its momentum.  However, Classical wave mechanics tell us we must observe a larger portion of its wavelength to increase the accuracy of the measurement of its energy or momentum.  But this means that the accuracy of its position will be reduced because the boundaries determining its position within the measurement field are greater.

However, this dynamic interaction between the position and momentum component of the matter wave would be responsible for the uncertainty Heisenberg associated with their measurement because it shows the measurement of one would affect the other by the product of those factors or m^2 kg / s.

Yet because of the time varying nature of a matter wave one could only define its specific position or momentum of a particle based on the amplitude or more precisely the square of the amplitude of its matter wave component.

This shows how Louis de Broglie may have been able to derive uncertainties and probabilistic properties associated with Schrödinger’s equation in terms of his matter wave theory of particles if he had assumed that they were a result of one moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Later Jeff

Copyright Jeffrey O’Callaghan 2010

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