There can be no doubt the probabilistic interpretation of Schrödinger’s wave equation predicts with amazing precision the results of every experiment involving the quantum world that has ever been devised to test it.

However this interpretation is at odds with the reality of the classical or deterministic world most of us appear to live in because it assumes that for a given set of initial conditions there can only be one outcome while the probabilistic interpretations of quantum mechanics assumes there can be an infinite number.

Leonard Susskind’s
Quantum probabilities

However many of the standard interpretations of quantum mechanics assume that probability is the fundamental property of the universe, while alternative interpretations explain it as an emergent or a second-order consequence of various limitations of the observer or the environment he or she is occupying when making an observation.

Determining which is the correct way of interpreting it is difficult because due to the limitation imposed on observers by uncertainty principle we can never be sure what is happening on the quantum scale when an observation is made.

Yet that does not mean that we cannot extrapolate what we can learn from observing our four dimensional space-time environment to the quantum world to help us understand what happens when we make an observation.

However we will find it beneficial to redefine Einstein space-time model of the universe into its equivalent in four spatial dimensions.

(The reason for this will become obvious later on)

Einstein gave us the ability to do this when he use the equation E=mc^2 and the constant velocity of light to define the geometric properties of space-time because it provided a method of converting a unit of it he associated with energy to unit of space we feel he would have associated with mass. 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.

As mentioned earlier Quantum mechanics assumes one can only determine the future evolution of a particle in terms of the probabilistic values associated with its wave function which is in stark contrast to the Classical "Newtonian" assumption that one can assign precise values of future events based on the knowledge of their past. 

In other words in a quantum system Schrödinger’s wave equation plays the role of Newtonian laws in that it predicts the future position or momentum of a particle in terms of a probability distribution.

This accentuates the fundamental difference between quantum and classical mechanics because the latter defines the reality of future events in terms of pervious events whereas quantum mechanics defines them based on the "non-classical" reality of the sum total of all possible events that can occur.  

However as mentioned earlier one may be able to understand the physical reason why these two theories define the reality of events differently if, as was done earlier one redefine Einstein’s space-time concepts in terms of four spatial dimensions.

In the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown one can understand the physicality of quantum properties energy/mass by extrapolating the laws of classical wave mechanics 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 occur in one consisting 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 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 space.

Therefore, these oscillations in a "surface" of a three-dimensional space manifold would meet the requirements mentioned above for the formation of a resonant system or "structure" in four-dimensional space if one extrapolated them to that environment. 

Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with it fundamental or a harmonic of its fundamental frequency.

Hence, these resonant systems in four *spatial* dimensions would be responsible for the discrete quantized energy associated with the quantum mechanical systems.

(In the article "The geometry of quarks" Mar. 15, 2009  the internal structure of quarks, a fundament component of particles was derived in terms of a resonant interaction between a continuous energy/mass component of space and the geometry of four *spatial* dimensions.)

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 it’s 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?“ showed energy must be packaged in terms of its discrete resonant properties.  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 that 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 defines the physical reason in terms of four *spatial* dimensions for why we are unable to measure the exact position and moment of a quantum system.

However it also defines the reason why the probably functions of quantum mechanics are an emergent or a second-order consequence of various limitations of the observer or the environment and not a fundamental property of our universe because as was just shown the physicality of four *spatial* dimension places limitations on our ability to define the initial conditions or momentum and position of a quantum system we are measuring. 

In other words the reason quantum mechanics can only predict the probability of an event occurring is because of the limitations the physical properties of four *spatial* dimension places on an observer.

This shows why we should view the probabilistic properties of quantum mechanics as an emergent or a second-order consequence of the limitations of the four *spatial dimension or space-time environment he or she is occupying when making an observation and not a fundamental property of the universe.

Later Jeff

Copyright Jeffrey O’Callaghan 2014


 

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Einstein was often quoted as saying "If a new theory was not based on a physical image simple enough for a child to understand, it was probably worthless."

For example one can easily understand how a curvature in space-time can cause gravity in terms of the physical image of a marble on a curved surface of a rubber diaphragm.  The marble follows a circular pattern around the deformity in the surface of the diaphragm. Similarly planets revolve around the sun because they follow a curved path in the deformed "surface" of space-time.

However the same cannot be said about black body radiation.

This is because classical physics suggests that all harmonics in a black body have an equal chance of being produced even when their number goes up in proportion to the square of the frequency.  However this classical concept works reasonably well at low frequency yet it begins to diverge at higher frequencies so much so that its energy content at those frequencies should approach infinity.  This discrepancy between the classical description of a black body and its reality has come to be called the Ultraviolet Catastrophe.

The Ultraviolet catastrophe

However Planck realized it could be explained by assuming that energy is not continuous but comes in discreet packaged define by the equation E=hv.  Their observation that the energy in a black box is quantized was the basis for the development of Quantum theory.

Yet no one, up until now has been able to provide a physical image of how and why this should be so.

Up until now because in the article "The Photon: a matter wave?" Oct. 1, 2007 it was shown that one can use the observed wave properties of electromagnetic radiation and Einstein Theory of Relativity to form a physical image of how energy is disturbed in a black body.

However it is easier if one converts or transposes Einstein space-time universe to one consisting of only four *spatial* dimensions.

(The reason will become obvious later.)

Einstein gave us the ability to do this when used the constant velocity of light to define the geometric properties of space-time because it provided a method of converting the space-time displacement he associated with energy in a space-time universe to a spatial one in a universe consisting of only four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words by mathematically defining the geometric properties of time in his space-time universe in terms of the constant velocity of light he provided a qualitative and quantitative means of redefining it in terms of the geometry of four *spatial* dimensions.

The fact that one can use Einstein’s equations to qualitatively and quantitatively redefine the curvature in space-time he associated with energy in terms of four *spatial* dimensions is one bases for assuming, as was done in the article “Defining energy?” Nov 27, 2007 that all forms of energy including electromagnetic can be derived in terms of a spatial displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

As mentioned earlier the article "The Photon: a matter wave?" Oct. 1, 2007 shows how one can understand properties of electromagnetic energy and how it is disturbed in space and a black body in terms of a physical image based on the classical properties of wave motion if one assume that space is composed of four *spatial* dimensions instead of four dimensional space-time.

For example a wave on the two-dimensional surface of water causes a point on that surface to be become displaced or rise above or below the equilibrium point that existed before the wave was present.  A force will be developed by this displacement, which will result in the elevated and depressed portions of the water moving towards or become "attracted" to each other and the undisturbed surface of the water.

Similarly a matter wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that "surface" to become displaced or rise above and below the equilibrium point that existed before the wave was present.

However, classical wave mechanics, if extrapolated to four *spatial* dimensions tells us the force developed by the differential displacements caused by a matter wave moving on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension will result in its elevated and depressed portions moving towards or become "attracted" to each other.

This would define in terms, of a physical image the causality of the attractive forces of unlike charges associated with the electromagnetic wave component of a photon in terms of a force developed by a differential displacement of a point on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitudes of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly the magnitude of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one.  Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two similar charges than it would be for a single charge.

One can define the causality of electrical component of electromagnetic radiation in terms of the energy associated with its "peaks" and "troughs" that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.  This is because classical mechanics tells us a horizontal force will be developed by that perpendicular or vertical displacement which will always be 90 degrees out of phase with it.  This force is called magnetism.

This is analogous to how the vertical force pushing up of on mountain also generates a horizontal force, which pulls matter horizontally towards from the apex of that displacement.

This provides a physical image that would allow on to understand the electromagnetic wave component of black body radiation 

However its quantum mechanical or particle properties can also be derived in terms of a physical image by extrapolating the laws of classical resonance to that same wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

For example as the article "Why is energy/mass quantized?" Oct. 4 2007 showed one could derive a physical image of the particle or photonic properties electromagnetic energy by extrapolating the laws of classical wave 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 four *spatial* dimensions.

The existence of four *spatial* dimensions would give space (the substance) 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 to oscillate with the frequency associated with the energy of that event.

However, these oscillations in a continuous non-quantized field of energy/mass caused by such an event would cause a resonant system or "structure" to be established in it. 

Classical mechanics tells us the energy of a resonant system can only take on the discreet quantized values associated with its resonant or a harmonic of its resonant frequency.

However, one can also use the above concepts of four *spatial* dimensions to develop a physical image of the particle or photonic properties

Max Planck, as was mentioned earlier associated with black body radiation.

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. 

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 spatial boundaries associated with a particle in the article “Why is energy/mass quantized?

These resonant systems in a continuous non-quantized field of energy/mass are responsible for the discrete or incremental energies associated with the quantum component of black body radiation.

The ultraviolet catastrophe is the error at short wavelengths in the Rayleigh–Jeans law (depicted as "classical theory" in the graph) for the energy emitted by an ideal black-body. The error, much more pronounced for short wavelengths, is the difference between the black curve (the wrong curve predicted by the Rayleigh–Jeans law) and the blue curve (the correct curve predicted by Planck’s law).

However, these two articles also provide a physical image of why the energy distribution in a black body is what it is in terms of the concepts of classical physics.

A black body is an idealized object that absorbs all electromagnetic radiation that falls on it.  Because no light is reflected or transmitted, the object appears black when it is cold.  However, a black body emits a temperature-dependent spectrum of light.  This thermal radiation from a black body is termed black body radiation.

At room temperature, black bodies emit mostly infrared wavelengths, but as the temperature increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths, appearing red, orange, yellow, white, and blue with increasing temperature.  By the time an object is white, it is emitting substantial ultraviolet radiation.

The problem is, as was mentioned earlier the laws of classical mechanics, specifically the equipartition theorem, states that black-bodies which have achieved thermodynamic equilibrium are mathematically obligated (by classical, pre-quantum, laws) to radiate energy in the form of ultraviolet light, gamma rays and x-rays at a certain level, depending on the frequency of emitted light.

However, as mentioned earlier observations of black body radiation indicate that there was less and less energy given off at high end of the spectrum.

Einstein pointed out this difficulty could be avoided by making use of a hypothesis put forward five years earlier by Max Planck.  He had hypothesized that electromagnetic energy did not follow classical laws, but could only oscillate or be emitted only in discrete packets of energy proportional to the frequency, as given by Planck’s law.  In other words, the light waves of each frequency in a black body could not have any energy but are limited to a few discrete values.

However, as mentioned earlier the article "Why is energy/mass quantized?" showed the quantity of energy of a photon at each frequency could be understood by extrapolating a physical image of a resonant system in three-dimensional space to a fourth *spatial* dimension similar to how Einstein was able to from a physical image of gravity.

For example as the above theoretical model showed using only the concepts of classical physics and Einstein’s theory of  Relativity a photon could only have the discrete energies or frequencies that are a fundamental or harmonic of the energy of an environment which would be determined by the temperature of the one it was occupying. Therefore, according to the above theoretical model any frequency other than that would be irregular and non-repeating and would be absorbed into the fundamental or harmonic frequency of that environment.

In other words it explains in terms of a physical image based on our classical reality why black-bodies which have achieved thermodynamic equilibrium are mathematically obligated by (classical, pre-quantum, laws) to radiate energy in the form of ultraviolet light, gamma rays and x-rays at a certain discrete levels, depending on the frequency of emitted light.

It should be remember Einstein’s genius allows us to choose if we want to view the physical properties of electromagnetic energy and black body radiation in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of the constant velocity of light. This interchangeability broadens the environment encompassed by his theories by making them applicable to both the sensory as well as the non-sensory time properties of our universe thereby giving us a new perspective on the physical relationship between particles and waves

Later Jeff

Copyright 2014 Jeffrey O’Callaghan


 

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