Bohr summarized his complementary perspective on reality as follows:…"however far the [quantum physical] phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word "experiment" we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.

This crucial point…implies the impossibility of any sharp separation between the behavior of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear…. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects."

The "Invisible Reality" of Quantum Theory with Alan Alda, Brian Greene

Albert Einstein and Leopold Infeld also addressed this issue in their book The Evolution of Physics, when they asked "what is light really? Is it a wave or a shower of photons? There seems no likelihood for forming a consistent description of the phenomena of light by a choice of only one of the two languages. It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do."

In other words the quantum world has duel non overlapping realities: one consisting of waves; the other particles and which one we observe depends on how we observe it.

This is in stark contrast to the classical one we all live in which defines only one reality based on cause and effect.

However one of the reasons it has been it is so difficult to understand why these dual realties exist may be because too much attention has been focused on the mathematics that describe it and not enough on their physical meaning in our classical environment. 

Another reason may be because most have tried to integrate them into a space-time environment even though they are primarily defined in terms of the spatial properties of probabilities.

This suggest we may be able to better understand why the quantum world possess two distinct realties based on the single reality of a classical world of cause and effect if we view them in terms of spatial properties instead of their space-time ones.

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 he associated with energy to unit of space associate with position.  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.

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 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 defining the dimensional properties of quantum system in terms of its spatial instead of its time components would allow one to understand why a quantum environment possess two distinct realities by extrapolating the laws governing cause and effect in the classical world to them.

For example the article “Why is energy/mass quantized?” Oct. 4, 2007 showed one can derive quantum properties of 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 its 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.

Yet it also allowed 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 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?” Oct. 4, 2007.

In other words one can use classical wave mechanics to explain why a quantum system can possess both wave and particle properties.

However it does not help us to understand why they define two non-overlapping realties. 

In other words it does not help us to understand why if one devises and experiment to observe its particle properties one cannot at the same time observe its wave reality while if one observes its particle reality one cannot observe it wave properties.

However one can use the properties of classical reality to understand the causality of the complementary or dual real of quantum mechanics.

For example Classical Mechanics tell us that because of the continuous properties of waves, the energy the article "Why is energy/mass quantized?” Oct. 4, 2007 associated with a quantum system would be free to move over the entire "surface" of three-dimensional space with respect to a fourth *spatial* dimension similar to how the wave generated by a vibrating ball on a surface of a rubber diaphragm would be free to move over its entire surface.  However to observe the movement of the rubber diaphragm one must physically touch it with a probe thereby restricting its movement

Similarly to observe the movement of a quantum system one must use a probe which would restrict its movement.

However this also allows one to understand why observing a quantum system effects its reality as is demonstrate in the double slit experiment.

The double slit experiment

In this experiment the wave reality of a quantum system is demonstrated by the bright and dark interference bands produced on the screen after being allowed to freely pass through between two slits on a screen, However, it is always found to be absorbed at the screen at discrete points, as individual particles (not waves), while the interference pattern appears as varying density of these particle hits on the screen. Furthermore, its particle reality is demonstrated when someone puts detectors at the slits he finds that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).

These results demonstrate the principle of wave–particle duality.

In the first part of this experiment the wave properties of a quantum system defines its reality because it is allowed to move freely through space. This would be analogous to classical reality of sound waves created by a random source in that they show no properties of quantization. 

However classical wave mechanics tells us that if one if one restrict the movement of a wave as is done in a pipe organ it will form a quantized resonant system.

For example if one restricts sound waves as is done in an organ pipe its output becomes quantized because it amplifies one of its wave components while diminishing all others.

As the article “Why is energy/mass quantized?” Oct. 4, 2007 showed the particle "reality" of a quantum system is the result of the restricting its wave "reality" which must always happen when an observation is made.

In other words as the reason why the particle reality of a quantum environment only occurs when an observation is made is because that act restricts the movement of its wave reality thereby creating the resonant structure associated with its particle properties defined in the article “Why is energy/mass quantized?” Oct. 4, 2007.

In other words the reason the interference pattern appears as varying density of these particle hits on the screen in the double slit experiment is because as was mentioned earlier observing a quantum system requires one to restrict its wave prosperities thereby causing the resonant system the article Why is energy/mass quantized?” Oct. 4, 2007 associated with its particle really.

Additionally their varying density occurs because these resonant systems will most likely occur where the magnitude of the wave component is maximum and drop off as it decreases.  Therefore their position on the screen will form a wave interference pattern .

The reason why the two realties are complimentary or cannot be simultaneous observed is similar to why one cannot observe ice and water at the same time.  For example in an environment consisting of water that is well below freezing the reality is frozen water while its reality when it is above freezing is water.  Additionally our classical experiences tell us that these two states are complementary because they cannot be both at the same time but have to one or the other.

Similarly in an unobserved quantum environment the wave reality exists because is it allows to move freely through space while as was shown above the restrictions caused by observation makes its particle properties or reality become predominate.

This shows even though a quantum environment consists of two non-overlapping contradictory pictures of reality one can fully understand their existence by applying the cause and effect relationship of a classical reality to it.

Later Jeff

Copyright Jeffrey O’Callaghan 2015

 

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The Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different spectral emission lines of the hydrogen atom whose energy levels can calculated using an empirical equation discovered by Johann Balmer in 1885.

Bohr’s Atomic Model for Hydrogen Atom

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 "ℓ" (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 "ℓ"  (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 "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.

Using your right-hand:
Curl your fingers into a half-circle around the wire, they point in the direction of the magnetic field, B

Point your thumb in the direction of the conventional current

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 Einstein’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

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