Quantum tunneling: a classical interpretation

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

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

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