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 space in terms four *spatial* dimensions instead of four-dimensional space-time.
One is that it would allow for understanding of the physical significance of Planck’s constant in terms of the laws of classical physics.
In the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown it is possible to explain and predict the quantum mechanical properties of energy/mass by extrapolating the laws of classical resonance in a three-dimensional environment to a matter wave on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
Briefly it showed the four conditions required for resonance to occur in a classical 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 meet by a matter wave in an environment of four *spatial* dimensions.
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.
The oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established in four *spatial* dimensions.
Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with its resonant or a harmonic of its resonant frequency
This means that one can theoretically derive the quantum mechanical properties of Schrödinger’s wave function in terms of the physicality of resonant properties of four *spatial* dimensions if one assumes as is done here that its mathematical properties are representative of wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
However it also gives one the ability to understand the physical meaning of Planck’s constant or 6.626068 × 10-34 (kg*m2/s) by extrapolating the laws of classical physics in a three-dimensional environment to a fourth *spatial* dimension.
In classical physics, a point on the two-dimensional surface of paper is confined to that surface. However, that surface can oscillate up or down with respect to three-dimensional space.Â
Similarly an object occupying a volume of three-dimensional space would be confined to it however, it could, similar to the surface of the paper oscillate “up” or “down” with respect to a fourth *spatial* dimension.
The confinement of the “upward” and “downward” oscillations of a three-dimension volume with respect to a fourth *spatial* dimension is what defines the geometric boundaries or the “box” containing the wave or wave function the article “Why is energy/mass quantized?” Oct. 4, 2007 associated with a particle.Â
Planck’s constant is one of fundamental components of Quantum Physics and along with Heisenberg’s Uncertainty Principle it defines the uncertainty in the ability to measure more than one quantum variable at a time. For example attempting to measure an elementary particle’s position (â–²x) to the highest degree of accuracy leads to an increasing uncertainty in being able to measure the particle’s momentum (â–²p) to an equally high degree of accuracy. Heisenberg’s Principle is typically written mathematically as â–²xâ–²p  ³ h / 2 where h represents Planck constant
As mentioned earlier the resonant wave that corresponds to the quantum mechanical wave function defined in the article “Why is energy/mass quantized?” predicts that a particle will most likely be found in the quantum mechanical “box” whose dimensions would be defined by that resonant wave. However quantum mechanics treats particles as a one dimensional points and because it could be anywhere in it there would be an inherent uncertainty involved in determining the exact position of a particle in that “box”.
For examine the formula give above ( â–²xâ–²p  ³ h / 2 ) tells us that uncertainty of measuring the exact position of the point in that “box” defined by its wavefunction would be equal to â–²xâ–²p  ³ h / 2.  However because we are only interested in determining its exact position we can eliminate all references to its momentum.
However if we eliminate the momentum component from the uncertainty in a particle position become 6.626068 × 10-34 meters or Planck’s constant.
As mentioned earlier the uncertainty involved in determining the exact position of a particle is because it is impossible to determine were in the “box” defined earlier the quantum mechanical point representing that particle is located. However as mentioned earlier Planck’s constant tells us that one cannot determine the position of a particle to an accuracy greater that 6.626068 × 10-34. This suggest that Planck constant 6.626068 × 10-34 defines the physical parameters or dimensions of that “box” because it defines the parameters of where in a given volume of space a quantum particle can be found.
This shows how one can define and understand the physicality of Planck’s constant by extrapolating the laws of classical physics in three-dimensional environment to a fourth *spatial* dimension if one assumes as is done here that the quantum mechanical properties of the wave function are cause by a resonant structure in four *spatial* dimensions.
Later Jeff
Copyright Jeffrey O’Callaghan 2012