One is that it would allow for the integration of the quantum mechanical and wave properties of energy/mass by extrapolating the classical laws of a three-dimensional environment to a fourth *spatial* dimension.
In 1924 Louis de Broglie was the first to theorize that all particles had a transverse matter wave component.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their wave 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 theories presented by Born, Bohr, and Heisenberg regarding the uncertainties and probabilistic interpretation of quantum particles.

One of the difficulties he may have faced in this endeavor is that he assume 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 time or a space-time dimension can only move in one direction, forward.  Therefore, it could not support bidirectional movement required for the propagation of a transverse wave. 

However Einstein provided a solution to this problem when he use the equation E=mc^2 and the constant velocity of light to define the geometric properties of space-time because that gave a method of converting a unit of time associated with energy to its equivalent unit of space in four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one quantitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

This may have allowed Louis de Broglie to define the quantum or particle properties, as was done in the article “Why is mass and energy quantized?” Oct. 4, 2007 of the transverse wave he theorized they were made up of in terms of a spatial displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly that article showed that the four conditions required for resonance to occur in a classical 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 be satisfied by a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

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* dimensions 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 in space.

As was shown in that article these resonant systems in a continuous form of mass/energy are responsible for its quantum mechanical properties.

He then many have been able defined a causal interpretation of the Quantum Mechanical equation for a particles energy or E=hv (where “h” is Planck’s constant “v” is a particles frequency and “E” is the magnitude or its energy) based on the existence of these resonant systems

Classical mechanics tells us that the energy of a resonant system is quantized in terms of multiples of the harmonics of the fundamental frequency of its environment. 

Therefore, he could have interpreted the equation E=hv as defining the quantization of a particle’s energy in terms of the incremental energies “h” associated with the fundamental or harmonic of the resonant frequency of an environment consisting of four *spatial* dimensions..

However, this would have also allowed him to define a casual mechanism responsible for the uncertainty principal and the probability functions of Quantum Mechanics, again by extrapolating the three-dimensional laws of classical resonance to four *spatial* dimensions.

Because he may have realized the causality of the uncertainty in oneâ€s ability to define the exact position or momentum of a particle was due to the fact that the resonant system that the article “Why is mass and energy quantized?” derived was responsible for the quantum mechanical properties of energy/mass is distributed over the finite volume associated with the wavelength of its resonant system.  Therefore, one could only define its specific position or momentum in terms of an uncertainty related to where relative to its finite extended volume a measurement is made.

This indicates may have been able to derive a causal interpretation of the quantum mechanical properties of energy/mass in terms of classical properties of a matter wave if he had assumed there were a result of the geometric property of four *spatial* dimensions.

However, as mentioned earlier this cannot be done if one assumes space it made up of four-dimensional space-time because its geometry cannot support the transverse wave properties Louis de Broglie associated with particles.

Later Jeff

Copyright Jeffrey O’Callaghan 2010

The post The geometry of Quantum Mechanics appeared first on Unifying Quantum and Relativistic Theories.

One is that it would allow one understand quantum tunneling or how particles can tunnel or move through potential barriers that are higher than their energy in terms of the laws of classical physics.

Quantum tunneling is defined “as a microscopic phenomenon in which a particle violates the principles of classical mechanics by penetrating or passing through a potential energy barrier or impedance higher than its potential energy.  A barrier, in terms of quantum tunneling, may be a form of energy state analogous to a “hill” or incline in classical mechanics, which classically suggests that passage through or over such a barrier would be impossible without sufficient energy.  However, on the quantum scale, objects exhibit wave-like behavior.  Therefore, in quantum theory, quanta moving against a potential energy “hill” can be described by their wave function, which represents the probability amplitude of finding that particle in a certain location at either side of the “hill”.  If this function describes the particle as being on the other side of the “hill”, then there is the probability that it has moved through, rather than over it, and has thus “tunneled“.
However, does this behavior really violate the laws of classical mechanics because there are numerous examples of how energy can be relocated or tunneled through what appears to be an impenetrable barrier for the potent energy of the mediums associated with the transmission of that energy?

For example, classical wave mechanics tells us that the energy of wave on the surface of water can be transmitted or “tunnel” through a flexible steel plate separating two volumes of water by the flexing of that plate.  This plate acts as a potential energy barrier between the water in each tank because it is “higher than the potential energy” of the two volumes however due to steel plate’s ability to flex its energy can be transmitted beyond it.

This demonstrates that due to the spatial properties of a wave, its energy can penetrate or be transmitted to other side of a flexible steel plate without the water having to go over it.

However, as was showed in the article “Why is energy/mass quantized?” Oct 4, 2007 on can define the energy of a quantum mechanical system in terms of a resonant system or “structure” generated by a matter wave on a “surface” of a three-dimensional space manifold with respect to fourth *spatial* dimension.

In other words one can derive its physical structure in terms of the classical properties of a wave

(Louis de Broglie was the first to realize this when theorized that all particles have a wave component.  His theories were confirmed by the discovery of electron diffraction by crystals in 1927 by Davisson and Germer. )

This means the energy associated with a particle could tunnel through a potential energy barrier that had an energy “higher” than it and still not violate the laws of classical mechanics for the same reason as the energy associated with the water, in the earlier example could tunnel though a potential energy barrier that was higher than its potential energy and not violate those laws. 

In other words, the physical wave component that Davission and Germer observed a particle to have could “penetrate or pass through a potential energy barrier or impedance higher than its potential energy” for the same reason as the energy of a wave on water can penetrate one

The reason why quantum tunneling appears to contradict the laws of Classical Mechanics is because Quantum Mechanics does not define a particle in terms of its wave prosperities but only in terms of a rigid structure of a particle. 

However, as was shown in the article “Why is energy/mass quantized?” a particle can be defined in terms of the dynamics of a resonant wave on a “surface” of a three-dimensional space manifold with respect a fourth *spatial* dimension. 

This means, on a quantum level a particle does not have the rigid point like structure quantum mechanics associates with it but a dynamic spatial one associated with a classical wave.  Therefore, its energy can “tunnel” through a potential barrier or impedance higher than the energy associated with the point in space quantum mechanics associated with a particle for the same reason the energy of a water wave can tunnel through the flexible steel plate in the earlier example.

The probability function quantum mechanics uses to predict whether or not a particle will tunnel through a potential energy barrier can be thought of as a measure of the energy distribution in the space surrounding a particle because as was shown in the article “Why is energy/mass quantized?” a particles volume would be defined by wave properties. 

Therefore, according to classical mechanics its position could not be determined with an accuracy smaller than the wavelength of the resonant system associated with a particle.  This means classical mechanics tells us that the probability a particle can be observed on either side of an energy barrier will depend on the “energy width” of that barrier.

Additionally classical mechanics tells us that due to the time variant properties of waves one could only determine its position in a probabilistic manner defined by were one observed it in that time varying environment.

This shows because particles have wave-like properties on a quantum scale they can quantum tunnel or penetrate through a potential barrier or impedance higher than the energy of the particle without violating the laws of classical mechanics.

Later Jeff

Copyright 2008 Jeffrey O’Callaghan

The post Quantum tunneling: a classical interpretation appeared first on Unifying Quantum and Relativistic Theories.