Quantum mechanics assumes that it is fundamental because it defines all interactions within it in terms of its quantized properties while one could say that Einstein’s General Theory of Relativity defines it in terms of an emergent property of continuous space-time manifold because that’s how it defines reality.

Most would agree the best way of which to determine which one is fundamental would be to see if one can be explain in terms of the other.
For example it is impossible to explain the apparent continuous properties of space-time in terms of the discrete properties quantum mechanics associates with energy/mass because by definition something that is discrete cannot by definition be continuous.   However it is possible to explain how the continuous properties of space-time can be broken up into the discrete components of energy/mass that allows quantum mechanics to define it in those terms.

Quantum mechanics assumes that energy/mass is quantized based, in part on Schrödinger wave equation which is used to predict and define the quantized energy distribution of electrons in an atom in terms of the Principal number (n),  the Angular Momentum “â„“”   (l), Magnetic (m) and Spin Quantum Number(+1/2 and -1/2).

However as mentioned earlier it may be possible to define an emergent mechanism based on the reality of four dimensional space-time that can explain why the energy distribution in a atom is quantized.

Einstein gave us the ability to qualitatively and quantitatively convert the relativistic properties of a space-time environment to an equivalent one consisting of only four *spatial* dimensions when he defined its geometric properties in terms of the equation E=mc^2 and the constant velocity of light. This is because it allows one to redefine a unit of time he associated with energy in his space-time universe to unit of space in one consisting of 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 his space-time universe in terms of the geometry of four *spatial* dimensions.

However this would allow explain how the spatial characteristics of the energy distribution quantum mechanics associated with the four quantum numbers can emerge from reality of environment consisting of four dimensional space-time or its four *spatial* dimension equivalent.  

For example in the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown one can explain the quantum mechanical properties of energy/mass by extrapolating the “reality” of 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 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 discreet or incremental values associated a fundamental or a harmonic of the fundamental frequency of its environment.

In other words this defines the quantization or the particle properties of energy/mass in terms of an emergent property of four *spatial* dimensions. 

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 also be able to derive the quantum numbers that define the properties of the atomic orbitals in those same terms.

As mentioned earlier there are four quantum numbers.  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.

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 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?” associated with atom orbitals 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.

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 four dimensional space-time or four *spatial* dimensions 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 unit of the universe.

This defines physicality of the environment associated with the first quantum number in terms of an emergent property of four *spatial* dimensions 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.

Briefly the article “Defining potential and kinetic energy?” Nov. 26, 2007 showed 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 why only two electrons can occupy a quantum orbital and why they have slightly different energies can also be derived by extrapolating the laws of a classical three-dimensional environment to a fourth *spatial* dimension.

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 the article “Defining potential and kinetic energy?”.  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 level associated with the electron in that orbital while the other would be raise it up to that energy level .

However the energy required by each method will not be identical for the same reason that it requires slightly less energy to fill a bucket of water by pushing it down below its surface than using a cup to fill it.

However it also explains why no two quantum particles can have the same quantum number 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 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 if they had the same number.

This shows how one can derive the physicality of the four quantum numbers of an emergent property of four *spatial* dimension or its space-time equivalent.

Later Jeff

Copyright Jeffrey O’Callaghan 2016

The post Quantum mechanics as an emergent property of space-time. appeared first on Unifying Quantum and Relativistic Theories.

Later Neils Bohr sought to explain them by using the Rutherford model of the atom as a nucleus surrounded by electrons and the new ideas of quantum mechanics.  Bohr assumed that electrons orbit the nucleus at certain discrete, or quantized, radii, each with an associated energy.  He also assumed that when electrons “fall” from larger to smaller orbits, they release electromagnetic radiation obeying the Planck-Einstein relationship.  Because the energies of the orbits are quantized, so are the wavelengths. Bohr’s model explains both the Balmer series and the Rydberg constant and ushered in a new era of understanding atoms through quantum mechanics.

However Bohr felt that that no explanation of why electrons orbited in discrete, or quantized radii was needed because using that theoretical model based on that assumption was able to make very accurate prediction of energies of Balmer series.

Einstein disagreed because he felt that “If a new theory (such as that associated with Bohr’s model of the hydrogen atom) was not based on a physical image simple enough for a child to understand, it was probably worthless.”

In other words he felt that if Bohr’s explanation of the Balmer series was to have any value one should be able to form a physical image of how and why the spectral lines in the Balmer series have the energy they do.

The importance of explaining theoretical concept in physical terms was demonstrated by Einstein when addressing one of the more troubling aspect of Newton’s gravity theory.

Most, including Newton were troubled by the fact that that his gravitational theory meant ” that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact…That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.”

However Einstein realized that one can understand how gravity “may act upon another at a distance through a vacuum” by extrapolating the physical image of how objects move on a curve surface in a three-dimensional environment to a curved four dimensional space-time manifold. This allowed him to conceptually understand gravity in terms of a physical image based on our three-dimension environment.

In other words the mathematics developed by Newton was only able to quantitatively predict gravitational forces while Einstein gave us the ability to conceptually understand how and why “one body may act upon another at a distance” by physically connecting it to the reality of what we can see and touch.

However up until now no one has been able to define a physical model clear enough to explain the quantum mechanical model Bohr hypnotized was responsible for the spectral emissions associated with the Balmer series in terms of a space-time environment.

One reason for both Einstein’s and modern scientist’s inability to define one can be traced to the fact that they chose to define their energies in terms of four dimensional space-time instead four *spatial* dimensions because most view reality in terms of the physicality of the spatial dimensions instead of a time or space-time dimension.

This is true even though Einstein’s space-time theories give us a detailed physical image how a curvature in a space-time manifold can be responsible for gravity by extrapolating the image of an object moving on a curved two dimensional “surface” in a three dimensional environment to four dimensional space-time.  However this image only contains reference to the physicality of the spatial dimensions and not a time or space-time dimension.

This suggests that one may be able to develop a physical image how and why the energy levels in a hydrogen atom are what they are by converting or transposing Einstein’s space-time universe which defines energy in terms of geometry of space-time to one that defines it in terms of  the physicality of the spatial dimensions..

Einstein gave us the ability to do this when he used the constant velocity of light and the equation E=mc^2 to define the dynamic balance between mass and energy because that provided a method of converting the time displacement he associated with energy in a space-time universe to one to a spatial one in a universe consisting of only four *spatial* dimensions.  Additionally because the velocity of light is constant he also allows us to defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

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

This fact is the bases for assuming as was done in the article “Defining energy” Nov 27, 2007 that all forms of energy 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.

However if true one should be able to form a physical image of why the energy of each of the Blamer lines are what they are by extrapolating the physicality of the spatial dimensions to a fourth *spatial* dimension.

In other words one would should be able to define why the elections associated with the Principal Quantum number (n), the Angular Momentum “â„“” (l) Magnetic (m) and Spin Quantum Number (+1/2 and -1/2) have the energy they do by extrapolating the laws of a classical environment to a fourth *spatial* dimension while at the same time excluding all other energies.

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 laws governing 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 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.

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 as mentioned earlier be able to define why Principal Quantum number (n),  the Angular Momentum “â„“”  (l) Magnetic (m) and Spin Quantum Number(+1/2 and -1/2) have the energy they do by extrapolating the laws of a classical environment to a fourth *spatial* dimension while at the same time excluding all other energies.

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?” associated with atom orbitals 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 fundamental vibrational energy of three-dimensional space which in terms is dependent on the tension created by the electrical attraction of the proton and electron.

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

The reason an electron does not fall into the nucleus is because as was shown in the article “Why is energy/mass quantized?” all energy is contained in four dimensional resonant systems.  Therefore the fundamental frequency or wavelength of four dimensional space would define the minimum energy and therefore the physical size of the first quantum orbital.

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 “Pauliâ€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.

Briefly the article “Defining potential and kinetic energy?” Nov. 26, 2007 showed 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 why only two electrons can occupy a quantum orbital and why they have slightly different energies can also be derived by extrapolating the laws of a classical three-dimensional environment to a fourth *spatial* dimension.

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 the article “Defining potential and kinetic energy?“.  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 level associated with the electron in that orbital while the other would be raise it up to that energy level .

However the energy required by each method will not be identical for the same reason that it requires slightly less energy to fill a bucket of water by pushing it down below its surface than using a cup to fill it.

However it also explains why no two quantum particles can have the same quantum number 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 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 if they had a different number.

This shows how one can explain why spectral emissions specifically those of the Balmer series have the energy they do and the four quantum numbers in terms of emergent property of four *spatial* dimensions or four dimensional space-time by extrapolating the laws of a classical three-dimensional environment to them.

It should be remember that Einsteinâ€s genius allows us to choose whether to define the physicality of the atomic orbitals 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.

Later Jeff

Copyright Jeffrey O’Callaghan 2015

The post Spectral emissions as an emergent property of space-time appeared first on Unifying Quantum and Relativistic Theories.

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

However it may be possible to develop a classical understanding of why the four quantum numbers define the arrangement of electron in atoms by converting or transposing Einstein’s space-time universe to one made up of fourth *spatial* dimension.

The reason this is necessary is because the quantum numbers deal more with the spatial than the time properties of three-dimensional space therefore eliminating time will allow for a more direct application of classical laws to the solution.

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 time in a space time environment  to 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 should allow one to define the physicality of the Principal Quantum number (n),  the Angular Momentum “â„“”  (l), Magnetic (m) and Spin Quantum Number(+1/2 and -1/2) by extrapolating the laws of a classical Newtonian environment to a fourth *spatial* dimension.
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 laws governing 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.

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 the quantum numbers that define the properties of the atomic orbitals in those same terms.

As mentioned earlier there are four quantum numbers.  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.

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?” associated with atom orbitals 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.

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

The reason an electron does not fall into the nucleus is because as was shown in the article “Why is energy/mass quantized?” all energy is contained in four dimensional resonant systems.  Therefore the fundamental frequency or wavelength of four dimensional space would define the minimum energy and therefore the physical size of the first quantum orbital.

This defines physicality of the environment associated with the first quantum number.  (The reason why it is unique for each subdivision of electron orbitals will be developed later) . 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 “Pauliâ€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 why only two electrons can occupy a quantum orbital and why they have slightly different energies can also be derived by extrapolating the laws of a classical three-dimensional environment to a fourth *spatial* dimension.

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 for the same reason that 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 because the one above the surface would be at a higher gravitational potential.

However it also explains why no two quantum particles can have the same quantum number 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 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.

Later Jeff

Copyright Jeffrey O’Callaghan 2012

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