Is the quantization of energy/mass a fundamental or an emergent characteristic of reality.
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 "ℓ" (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.Yet because quantum mechanics defines its operational environment in terms of the spatial properties of position or momentum and not in terms of temporal properties of time or a space-time environment it would be easier to understand how by redefining that environment in terms of its spatial equivalent
Einstein gave us the ability to qualitatively and quantitatively convert the geometric properties of his space-time environment to an equivalent one consisting of only four *spatial* dimensions when he defined the geometric properties of a space-time universe and the dynamic balance between mass and energy 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 we believe he would have associated with mass in a universe consisting of only four *spatial* dimensions.
In other words by defining the geometric properties of a space-time universe in terms of the equation E=mc^2 and 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 " ℓ" 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 "ℓ" 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.
Copyright Jeffrey O’Callaghan 2016
Vol. 5 — 2014
In physics, the conservation laws state the measurable property of an isolated physical system does not change as the system evolves over time. They include the laws of conservation of energy, linear momentum, angular momentum, and electric charge.
However these laws suggest the existence of another more fundamental one that physically defines their causality.
For example Einstein told us that time dilates and space contracts as the energy and momentum of reference frames increase.
In other words there appears to a one to one correspondence between the effects momentum and energy has on the dimensional properties of space-time.
However the fact that the energy and momentum have a common effect on those properties suggests there may be a physical connection between them and their conservation laws.
For example Einstein told us the mass of a particle created in accelerators increases the curvature in space-time causing the physical distance between two points external to it to decrease by a measurable amount. If that particle decays that curvature returns to where it was before that mass was created. In other words physical properties of space are conserved in the creation, destruction or redistribution of mass. Additionally he also told us that concentrating it in the form of a particle causes time to dilate by a measurable amount with respect to its external space-time environment and when that particle decays time is returned to normal rate of change.
In other words in all reactions involving mass the physical properties of space-time are conserved because they always return to their original value before it was either created or destroyed.
One can also connect the causality of the law of conservation of all forms of energy to the physical properties of a space-time environment.
For example it can be shown the causality of charge conservation is also directly related to the symmetries of the space-time environment defined by Einstein.
However it will be easier to explain if one coverts it to its equivalent in four *spatial* dimensions.
(The reason will become obvious later on in this discussion.)
Einstein gave us the ability to do this when defined the geometric properties of space-time in terms of the constant velocity of light because that provided a method of converting a unit of time in a space-time environment to a unit of space in four *spatial* dimensions. Additionally because the velocity of light is constant he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.
The fact that one can use Einstein’s theories to qualitatively and quantitatively derive the displacement he associated with energy in a space-time universe in terms of four *spatial* dimensions is the bases for assuming as was done in the article “Defining energy” Nov 27, 2007 that all forms of energy including those associated with charge 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.
This allows one to derive the physical properties of charge in terms a displacement in that "surface" with respect to a fourth *spatial* dimension.
For example if one raises a cup of water above its surface it will be given a measurable amount of potential energy with respect to that surface while at the same time a force will be developed that will be directed downward towards it. Additionally the level of the water will be lowered by the exact amount that was removed by the lifting of the cup above its surface. If one pours the water back the levels will return it original depth. In other words the level of the water is conserved due to the symmetry of its surface levels.
However as was shown in the article “Defining energy” Nov 27, 2007 if one raises, with respect to a fourth *spatial* dimension the volume of three-dimension space associated with a charge it will be given a measurable amount of potential energy with respect to that "surface" while at the same time a force will be developed that will be directed downward towards it. Additionally the energy level of three-dimensional space not associate with that charge will be lowered by the exact same amount. If one calls the volume space that was raised up a negative charge one would call the lowering of the "surface" of three dimension space caused by that a positive charge. If one neutralizes the negative charge by bring it in contact with that "surface" it will return to its original level and the charge will be neutralized. This shows how one can derive the causality of charge conservation in term of the symmetry imposed by Einstein theories.
In other words symmetry imposed by Einstein’s space-time environment means that charge must be conserved because the creation of one must always be offset by the other.
This is true in environments consisting of either four *spatial* dimensions or four dimensional space-time because as was shown earlier they are quantitative and qualitative interchangeable.
However it also allows one to understand how the conservation laws of nature are physically connected to each other in terms of the physical geometry of our universe.
It should be remember Einstein’s genius allows us to choose to derive the conservation laws either a space-time environment or one consisting of four *spatial* dimension when he defined their environments in terms energy and the constant velocity of light. This interchangeability broadens the environment encompassed by his theories thereby giving us a new perspective on the origins of the conservation laws of physics.
Copyright Jeffrey O’Callaghan 2016
Vol. 5 — 2014