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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 the curvature in space-time can be the causality of gravitational forces in terms of the physical image of a marble on a curved surface.  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 for the energy distribution within the atom because quantum mechanics defines it in terms of a non-deterministic probability function. This deeply trouble Einstein because he felt that the laws governing the entire universe must deterministic including those of the atom.  He spent the next few years attempt to define physical model of why energy levels of atoms behave the way they do. However by 1926 the problem of chance remained and Einstein became increasingly alienated from the developments in quantum theory; he insisted that “God does not play dice,” and thus there is no room for fundamental randomness in physical theory.

As mentioned earlier Einstein believed that a viable theory of nature should be base on determinism which should be describable by a physical image.

One reason for his inability to create a physical image of the quantum energy distribution in an atom may have been because he chose to define it in terms of time or space-time instead of its spatial properties.  In other words because the distribution of energy in an atom is related to its spatial not its time characteristics it may have been easier to do if he had define energy in terms of its spatial instead of time properties.

However he gave us the ability to do this when he defined the geometric properties of a space-time universe in terms of  the constant velocity of light because that allows one to redefine a unit of time he associated with energy in his space-time universe to unit of space in only four *spatial* dimensions. 

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

This would have allowed him to describe a physical image for why the energy levels of Principal Quantum number (n), the Angular Momentum “â„“” (l), Magnetic (m) and Spin Quantum Number (+1/2 and -1/2) are what they are. 

For example in the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown one can derive the quantum mechanical properties of energy/mass by extrapolating the physical image of resonance in a three-dimensional environment to a matter wave moving 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 Newtonian 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 the “surface” of a three-dimensional space manifold (the substance) the ability to oscillate spatially with respect to it 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.

Therefore, these oscillations on a “surface” of three-dimensional space, would meet the requirements mentioned above for the formation of a resonant system or “structure” in space.

Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.

Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.

These resonant systems in four *spatial* dimensions are responsible for the incremental or discreet energy associated with quantum mechanical systems.

Additionally this also allows one to derive the physical boundaries of a particle in terms of the geometric properties of four *spatial* dimensions.

For example 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 which always must occur, as was shown in the article “The observer effect in quantum mechanics a classical explanation” Sept. 1, 2015 when an observation is made is what defines the spatial boundaries of the resonant system associated with the particle component of its wave properties in the article “Why is energy/mass quantized?“

However the fact that one can derive the quantum mechanical properties of energy/mass by extrapolating the resonant properties of a wave in three-dimensional environment to a fourth *spatial* dimension means that one should be able to derive a physical image of the four quantum numbers that define the physical properties of the atomic orbitals in those same terms.

In other words one should be able to define a physical reasons in terms of the classical physics why the first the Principal Quantum number is designated by the letter “n”, the second or Angular Momentum by the letter “â„“” the third or Magnetic by the letter “m” and the last is the Spin or “s” Quantum Number are what they are.

In three-dimensional space the frequency or energy of a resonant system is defined by the vibrating medium and the boundaries of its environment.

For example the resonant energy of a standing wave generated when a violin string plucked is determined in part by the length and tension of its strings.

Similarly the energy of the resonant system the article “Why is energy/mass quantized?” Oct. 04 2007 associated with atom orbital would be defined by the “length” or circumference of the three-dimensional volume it is occupying and the tension on the space it is occupying.

Therefore the physicality of “n” or the principal quantum number would be defined by the fundamental vibrational energy of three-dimensional space that article associated with the quantum mechanical properties of energy/mass.

The circumference of its orbital would correspond to length of the individual strings on a violin while the tension on its spatial components would be created by the electrical attraction of the positive charge of the proton.

Therefore the integer representing the first quantum number would correspond to the physical length associated with the wavelength of its fundamental resonant frequency of the volume of electrons in orbit.

However, classical mechanics tells us that each environment has a unique fundamental resonant frequency which is not shared by others.

Additionally it also tells us why in terms of the physical properties of space-time an electron cannot fall into the nucleus is because, as was shown in that article all energy is contained in four dimensional resonant systems. In other words the energy released by an electron “falling” into it would have to manifest itself in terms of a resonate system. Since the fundamental or lowest frequency available for a stable resonate system in either four dimensional space-time or four spatial dimension corresponds to the energy of an electron it becomes one of the fundamental energy units of the universe.

This defines physicality of the environment associated with the first quantum number and why it is unique for each subdivision of electron orbitals. Additionally observations tell us that resonance can only occur in an environment that contains an integral or half multiples of the wavelength associated with its resonant frequency and that the energy content of its harmonics are always greater than those of its fundamental resonate energy.

This allows one to derive the physicality of the second “â„“” or azimuth quantum number in terms of how many harmonics of the fundament frequency a given orbital can support. 

In the case of a violin the number of harmonics a given string can support is in part determined by its length.   As the length increase so does the number of harmonics because its greater length can support a wider verity of frequencies and wavelengths.  However, as mentioned earlier each additional harmonic requires more energy than the one before it.  Therefore there is a limit to the number of harmonics that a violin string can support which is determined in part by its length.

Similarly each quantum orbital can only support harmonics of their fundamental frequency that will “fit” with the circumference of the volume it occupies.

For example the first harmonic of the 1s orbital would have energy that would be greater than that of the first because as mentioned earlier the energy associated with a harmonic of a resonant system is always greater than that of its fundamental frequency.  Therefore it would not “fit” into the volume of space enclosed by the 1s orbital because of its relatively high energy content.  Therefore second quantum number of the first orbital will be is 0. 

However it also defines why in terms of classical wave mechanics the number of suborbital associated with the second quantum number increases as one move outward from the nucleus because a larger number of harmonics will be able to “fit” with the circumference of the orbitals as they increase is size.

This also shows that the reason the orbitals are filled in the order 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s is because the energy of the 3d or second harmonic of the third orbital is higher in energy than the energy of the fundamental resonant frequency of the 4th orbital.  In other words classical wave mechanics tells us the energy of the harmonics of the higher quantum orbitals may be less than that of the energy of the fundamental frequency of preceding one so their harmonics would “fit” into circumference of the lower orbitals

The third or Magnetic (m) quantum number physical defines how the energy associated with each harmonic in each quantum orbital is physically oriented with respect to axis of three-dimensional space.

For example it tells us that the individual energies of 3 “p” orbitals are physically distributed along each of the three axis of three-dimensional space.

The physicality of the fourth quantum or spin number has nothing to do with the resonant properties of space however as was shown in the article “Pauli’s Exclusion Principal: a classical interpretation” Feb. 15, 2012 one can derive its physicality by extrapolating the laws of a three-dimensional environment to a fourth *spatial* dimension.

That article it was shown all forms of energy including the angular momentum of particles can be defined in terms of a displacement in a “surface* of three-dimensional space manifold with respect to a fourth *spatial* dimension.

In three-dimensional space one can use the right hand rule to define the direction of the angular momentum of charged particles.  Similarly the direction of that displacement with respect to a fourth *spatial* dimension can be understood in term of the right hand rule.  In other words the angular momentum or energy of an electron with a positive spin would be directed “upward” with respect to a fourth *spatial* dimension while one with a negative spin would be associated with a “downwardly” directed one.
Therefore one can define the physically of the fourth or spin quantum number in terms of the direction a “surface” of three-dimensional space is displaced with respect to a fourth *spatial* dimension.  For example if one defines energy of an electron with a spin of -1/2 in terms of a downward directed displacement one would define a +1/2 spin as an upwardly directed one.

The physical reason for Pauli’s exclusion principal or why only two electrons can occupy a quantum orbital and why they must have slightly different energies can also be derived by extrapolating the observations of a classical three-dimensional environment to a fourth *spatial* dimension.

For example there a two ways to fill a bucket.  One is by pushing it down and allowing the water to flow over its edge or by using a cup to raise it to the level of the buckets rim.

Similarly there would be two ways fill an atomic orbital according to the concepts presented in that article.  One would be by creating a downward displacement on the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* to the energy level associated with the electron while the other would create an upward displacement in that surface.

However the energy required by each method will not be identical because it requires slightly less energy to fill a bucket by pushing it down below the surface than it would be to fill one that was above it in part because the one above the surface would be at a higher gravitational potential.

Additionally it takes considerable more energy to push two buckets on on top of the other below the surface than it does just one.

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 quantum particles with similar quantum numbers would greater than that caused by a single one.  Therefore, they will repel each other and seek the lower energy state associated with a different quantum number because the magnitude of the force resisting the displacement will be less for them than if they had the same number.

This shows how one can define a physical model for the energy distribution with an atom by extrapolating the deterministic laws of a classical three-dimensional environment to a fourth *spatial* dimension.

However it also allows one to understand why in terms of a physical image the energy distribution within the atom MUST be defined in terms of a non-deterministic probability function.

As mentioned earlier the article “The observer effect in quantum mechanics: a classical explanation” Oct. 4, 2007 showed the particle component of a quantum system is the result of the restricting its wave motion through observation.

Briefly it showed that because of the continuous properties of waves, the energy the article “Why is energy/mass quantized?” Oct. 04 2007 associated with a quantum system it is free to move and therefore be distributed over the entire “surface” of three-dimensional space with respect to a fourth *spatial* dimension similar to how a wave generated by a vibrating ball on a surface of a rubber diaphragm would be disturbed over its entire surface.  However to observe it one would have to touch its surface with a probe thereby restricting the wave motion of that surface.

In other words there is a probability that a probe could observe the vibrations of the ball anywhere on that surface with a decreasing probably as one move away from the ball or center of the diaphragm.

Similarly an electron energy which is not being observed would be distributed throughout its entire orbit.

In other words similar to the rubber diaphragm the wave properties of an electron would be distributed throughout the entire volume of its atomic orbital.

However if we decide to restrict or redirect some of its energy by probing or observing it it appears to be at a specific place in space and time because as was shown in the article “Why is energy/mass quantized? the act of observation confines its wave component to specific volume thereby allowing the resonant system that article showed defines a particle’s position.

In other words in an atom an electron’s wave energy is allow freely move or exist within a specific volume however the act of observing where it is in its orbit restricts its movement thereby allowing the resonant system the article “Why is energy/mass quantized?“ associated with a particle to form and appear or be observed in a specific position within that orbital.

However similar to the vibrations in the rubber diaphragm there is a probability that a probe could observe them anywhere in their orbital with a decreasing probably as one move away from the center or focal point of its wave component.

In other words assuming space is composed of four spatial dimensions instead of four dimensional space-time in allows one to form a physical image of probabilistic interactions individual electrons in atoms have with observers and with electrons in other orbitals in terms of the classical laws of probabilities.

Later Jeff

Copyright Jeffrey O’Callaghan 2015

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