Philosophical Reflections in Physics and Math

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Philosophical Reflections in Physics and Math

Quantum Quandries - Part 1

Black-body radiation: a crisis In turn-of-the-century physics

When black-bodies are heated, they glow with different colors at different temperatures. However, using, as a basis for calculations, the relationship between light and matter discovered by Maxwell, classical physicists had predicted that the color of light given off by heated black-bodies should be blue at all temperatures. The difference between observed values and predicted values could not be accounted for within the framework of classical physics.

Max Planck addressed himself to this anomaly in 1900. Part of his search for a way of resolving the issue, involved making an assumption which helped to render the problem more mathematically tractable. Essentially, this move consisted in placing constraints on the vibrational character of a particle. Instead of permitting particles to have unlimited degrees of freedom with respect to how they vibrate, he required particle vibrations to conform to the following principle: E = nhf.

In this equation, E represents the energy of the particle under consideration, and 'f stands for that particle's vibrational frequency. 'h' is a constant which was specified by Planck, and "n' gave expression to an integer value. In effect, Planck's mathematical equation limited the energy of a given particle to integer multiples of the product of the particle's frequency and the constant 'h'.

Planck's intention had been to use his simplifying assumption during the process of calculating energy values for different particles and, then, to, subsequently, cancel the effect of that assumption by permitting 'h' to fall to zero. However, when he gave 'h' the value of zero, he arrived at precisely the same result as everyone else in classical physics - namely, that the color of light radiation given off by a heated black-body should be blue, irrespective of the temperature to which the body had been heated. On the other hand, when he assigned 'h' a specific value, now known as Planck's constant, he was able to calculate answers which were accurately reflective of the observed results from experiments with black-body radiation.

Over the next five years, classical physicists, for the most part, dismissed Planck's results. They did this despite the fact that Planck's results removed the anomaly of differences between predicted and observed values for black-body radiation.

Their unwillingness to accept Planck's approach to the problem was largely due to the arbitrary character of Planck's simplifying assumption. There was no compelling physical reason for making such an assumption. The assumption appeared to be merely a mathematical device for removing some of the complications surrounding the process of calculating energies.

Waves and/or particles?

In 1905, however, Einstein published his paper on the photoelectric effect. This paper gave a central role to Planck's constant.

Because Einstein's explanation of the photoelectric effect was highly successful, the prominence he had given to Planck's constant suggested, rather strongly, that this constant was something more than an arbitrary, mathematical convenience. Apparently, something had been stumbled upon which was of considerable significance.

Einstein's exploration of the photoelectric effect focused on what happens when light particles or photons are absorbed by the electrons of the surface atoms of a given metal. Arthur Compton, on the other hand, investigated what happens when light, in the form of x-rays, is scattered by the atoms of a gas. Compton discovered that if one gave the scattered photons a momentum of p , where p is equal to hk and k is the spatial frequency (i.e., the number of wavelengths per centimeter) of the photon, then, the results of the scattering experiment seemed to indicate that light behaves as if it is particle-like.

Planck's study of black-body radiation in which he introduced the constant 'h', together with Einstein's account of the photoelectric effect as well as Compton's x-ray scattering experiments, all pointed in the direction of light having particle-like properties under certain conditions. These findings were in direct contradiction to earlier optical studies (e.g., Thomas Young's two-slit interference experiment) which indicated that light has wave-like properties. Moreover, these findings were totally at odds with the prevailing interpretation of light which was provided by Maxwell's electromagnetic theory. Maxwell had said that light is a wave phenomenon which is propagated by the electromagnetic field.

The plot of the story of physics in the early 20th century thickened, so to speak, when Louis de Broglie introduced a new wrinkle into the discussion by means of his doctoral thesis. De Broglie maintained that not only could one show, as Einstein had done, that light was capable of behaving as a particle, one could also show that particles had wave properties.

More specifically, de Broglie attempted to put forth arguments which indicated there were spatial and temporal wave frequencies associated with a particle. The spatial frequency, k, was given in the relation advanced by Compton- namely, p = hk, whereas the temporal frequency, f, was given in Einstein and Planck's equation: E = hf.

Many of the most intriguing, yet perplexing, characteristics of quantum theory arise in relation to the strange way this theory amalgamates or juxtaposes particle and wave properties in one and the same entities. There are a number of fundamental differences involved in construing quantum entities as waves and construing such entities as particles:

(a) waves are capable of superpositional - that is, waves can interpenetrate one another without this process altering their characteristics as individual waves. Particles do not exhibit such superpositional ;

(b) waves are dispersed across large regions of space, whereas particles are localized in space;

(c) waves are able to travel in a variety of directions simultaneously, but a particle is restricted to a single direction of movement at any one time.

In addition to the blurring of the differences between particle and wave, there was a further, though related, difficulty. Classical physicists made a very clear demarcation between two fundamental concepts: (a) matter and (b) fields. For the classical physicist, matter generated fields, and fields were the source of the forces which affected the dynamics of matter.⁷ With the advent of quantum theory and the influence of the work of Planck, Einstein, Compton and de Broglie, the classical distinction between matter and field was becoming obscured and, therefore, no longer clearly demarcated.

Four basic approaches to quantum calculations

There are four basic approaches to quantum theory, three of which came, independently, at various times in 1925, and one of which came more than 20 years later. These approaches are: Heisenberg's matrix mechanics; Schrodinger's wave mechanics; Dirac's transformation mechanics, and, finally, Feynman's "sum over histories" mechanics.

Heisenberg devised a system of matrix mechanics in order to be able to describe the various characteristics and properties of various kinds of quanta. Associated with a given quantum, are a number of matrices or arrays of values.

Each individual matrix is a summary statement, so to speak, of one of the physical aspects or quantities of a given quantum. Quantities such as: position, velocity, mass, charge, angular momentum, energy, and so on, were assigned values in separate matrices.

Moreover, each matrix was assigned values expressing the probability that a given quantum had the property being represented by the matrix. Each matrix also was assigned values that indicated the strength of linkage for various values of the property being represented by the matrix.

The rules describing how the various matricies underwent transformations, transitions and so on, were contained in what came to be known as matrix mechanics. One of these rules turned out to have considerable importance. This rule stipulated that the multiplication of matrices is not commutative, and, therefore, the sequence in which the multiplication is carried out makes a difference in the outcome of the operation.

Erwin Schrodinger represented quanta in terms of various kinds of waveforms. Quanta possessing different properties would be represented by waveforms which reflected those properties.

In order to describe the dynamics of the transitions and transformations of various kinds of waveforms, Schrodinger introduced a wave equation which specified how waveforms behaved when different variables were substituted into the equation. Thus, his quantum theory is referred to as wave mechanics.

Quantum theory uses different waveform families to represent various properties of the entities inhabiting the quantum realm. For example, the property of spin (including both magnitude as well as orientation) is expressed in terms of the spherical waveform family. The impulse waveform family, on the other hand, is used in connection with a quantum entity's positional character. Additional examples include: energy properties construed in terms of the temporal sine waveform family, as well as, the spatial sine waveform family used in conjunction with the property of momentum.

The aspect of quantization which becomes associated with many of these waveform families is apparent in the formulae which are used to calculate the values for some of the variables involved in quantum theory. Thus, the spin magnitude of a particle is determined by means of the formula S = hn, where n is the number of nodal circles, and h is Planck's constant. The energy, moreover, which is to be assigned to a given particle is provided by: E = hf, where f is the temporal frequency of the waveform associated with the particle. Furthermore, the momentum of a particle is calculated on the basis of P = hk, where, as previously indicated, k is the spatial frequency of the waveform assigned to the particle.

In each of these cases, Planck's constant is introduced as the basic unit of quantum of action which is juxtaposed next to various wave properties. Therefore, the determination of values for both dynamic and static properties are a function of the interaction, at least mathematically, of discrete quanta together with properties rooted in continuous wave functions.

A vector or arrow which rotated in a multi-dimensional abstract space was used by Dirac to represent the properties of a given quantum. According to Dirac, one could describe the dynamics of a quantum's transformations and transitions by keeping track of the way the vector rotated over time.

Since there were various ways of establishing a coordinate system through which to assign values to the rotating vector, Dirac developed a method for translating between different modes of devising such coordinate systems. His method of translation is known as transformation theory.

Dirac maintained that the quantum theories of both Heisenberg and Schrodinger were, in fact, limiting cases of his more generalized theory. Consequently, the three theories are really a matter of using different mathematical means to provide a method of description for quanta and the transformations.

Schrodinger's wave equations are usually used to solve for certain variables involving relatively slowly moving quantum entities. When, however, one needs to solve for variables involving quantum entities moving near the speed of light, physicists often use Dirac's equations which are capable of taking corrective, relativistic effects into account.

According to Feynman, quanta could be represented by means of a technique referred to as the 'sum over histories' method. In effect, one takes into account all the states possible for a given particle and, then, adds together the amplitudes of these possibilities to arrive at the wave function for the particle.

In the case of any given actual particle, many of the possibilities will cancel out one another. As a result, whatever particle-histories remain at the end of the summing-over process will express a range of probability values that will encompass the way the particle actually behaves in a given set of circumstances.

In order to keep tabs on which histories had already been summed-over, Feynman invented a method of diagraming which provided a summary account of such histories. These are known as Feynman diagrams.

A question of ontological status

From the perspective of classical physics, properties such as mass, charge, momentum, velocity and position were intrinsic aspects of a particle. Consequently, the classical physicist believed a particle had determinate properties at any given point in time.

The vast majority of quantum physicists, however, do not accept the classical viewpoint concerning the ontological status of the various properties of any given quantum entity. In other words, neither the belief that all of the properties of a particle are necessarily intrinsic to the character of the particle, nor the idea that particles must always have, in every respect, definite, determinate properties at each point in time, coincide with the perspective of the vast majority of quantum theorists.

The predominant view of quantum physicists (namely, the so-called Copenhagen interpretation) does acknowledge that any given species of particle can be described, i part, by a fixed set of properties. These sort of property values will not vary from one member of a particle-species to the next. Such properties include spin magnitude (i.e., larger, heavier particles have a larger spin magnitude than do smaller, lighter particles), as well as mass and charge.

Nonetheless, quantum theory maintains there are other properties associated with a particle (such as momentum, position, and spin orientation) that are not intrinsic to the particle per se. These variable properties are a function of the way the process of measurement and observation engage, or interact with, a given particle.

Questions about the nature of quantum reality focus on the so-called non-intrinsic or dynamic properties of a particle, as opposed to its fixed or static properties. For example, quantum physicists ask: How does a particle acquire such non-intrinsic or dynamic properties? Or, once acquired, what is the precise character of the relationship between a particle and its non-intrinsic or dynamic properties?

In addition to the static versus dynamic theme, there is another related theme which highlights an important difference between the classical and quantum perspectives. This second theme concerns the extent to which different dynamic variables can be said to be independent of one another.

On the classical view, the various properties of, say, an electron can be measured independently from one another. This means, for example, that one can focus on establishing the value of a given electron's momentum in a particular context while remaining indifferent to the character of the values for the other properties of the electron.

From the perspective of quantum theory, the various properties of, for instance, an electron are not always independent from one another. Some of these properties involve conjugate variables.

Such variables are so inextricably intertwined that measuring one conjugate variable will affect, and be affected by, what is happening with the counterpart to the conjugate variable. In other words, one can measure one of the two conjugate pairs with some degree of precision, but in doing this, one loses the capacity to measure the second of the conjugate pair with any degree of accuracy.

The character of the relationship of conjugate variables was first expressed by Werner Heisenberg in the form of an uncertainty principle concerning measurements involving momentum and position. Heisenberg claimed that (delta-p)(delta-x) > h; where: (delta-p) represents measured variability involving a particle's momentum; (delta-x) refers to measured variability in a particle's position, and h is Planck's constant. Essentially, the uncertainty principle states that the combined precision of simultaneously measuring a particular conjugate pair (in this case, momentum and position) can never be less than the value of Planck's constant.

Quantum theory deals only with predicting the character of the results one will get during the process of measurement. There is universal agreement among physicists about how quantum theory is to be applied and about what sorts of results are to be expected in any given measurement process involving the quantum realm. However, quantum theory does not provide an account of what occurs outside the context of measurement or between measurements.

Each sub-atomic entity has a wave function, psi , assigned to it. This wave function is not necessarily considered by physicists to be an actual wave in the sense that an ocean wave is a concrete, observable wave out there in the real world. Usually, physicists tend to treat the wave function as a means of obtaining certain values that are useful in making predictions about the likely character of a given measurement process.

The wave function associated with a given quantum entity exhibits most of the features of 'real' waves such as phase, interference, amplitude, and the principle of superposition. However, unlike a normal wave, the quantum wave possesses no energy.

The amplitude of a wave is an indication of the maximum displacement which occurs during the course of a given wave's cycle in relation to some arbitrarily chosen 'rest point of that wave. Normally speaking, the intensity of a wave provides an index of the amount of energy which exists at any given point in the wave and is proportional to the square of the amplitude.

According to quantum theory, however, a quantum wave has no energy associated with it. The intensity of a quantum wave is a reflection of a set of probability values which is to be associated with the quantum waveform. These probability values are indications of the likelihood that a given quantum will have certain properties under specified circumstances.

Because the quantum wave has no energy, it is not directly observable or detectable. Nevertheless, scientists are able to infer the general properties of the probability or proxy wave by keeping track of the character of the pattern which is formed over time by, for example, a series of electrons striking a phosphor coated screen.

The principle of superposition and random phase

One also can approach some of the foregoing issues by considering the principle of superposition. Essentially, this principle stipulates that when two waveforms engage one another, they combine to form one waveform in which the amplitude is the sum of the amplitudes of the two individual waves.

Moreover, when the two waves disengage from one another they retain their original amplitude identities, as it were. In other words, they emerge with their pre-engagement amplitude intact.

Sometimes, however, when waves engage one another under certain circumstances, the waves do not conform to the superposition principle. For example, if the amplitudes of regular, everyday sorts of waves are too large, such waves will not exhibit the superposition principle when they combine with one another. These sorts of waves and conditions are referred to as non-linear.

When one considers the principle of superposition in the context of oscillatory systems, one must take into account the phase relations of the waves when adding together the amplitudes of the waves which are interacting with one another. The character of the combined amplitude of the interacting waves will be shaped by the patterns of constructive or destructive interference which arises as a result of the way in which the interacting waves are, respectively, in phase or out of phase with one another.

Thus, although the principle of superposition still applies and, therefore, the amplitudes of the interacting waves are added together, the manifestation of the combined amplitudes can range all the way from: (a) the amplitudes canceling out one another when the waves are totally out of phase, to: (b) a doubling of the individual amplitudes when the waves are totally in phase and provided that the amplitudes of the two waves are the same. Between the extremes of being totally out of phase or being completely in phase, are an indefinite number of combinatorial possibilities which depend on the extent to which two waves are in or out of phase.

In oscillatory systems, therefore, the principle of superposition manifests itself as function of the interference patterns which arise according to how the waves interact with one another. More specifically, interference patterns are an expression of the way the interacting oscillatory systems are responsive to the property of phase in one another.

When the phase of a wave exhibits variability during the process of measurement, one observes what is referred to as random phase. Random phase includes a mixture of phase values.

If one adds together the amplitudes waves which are identical in every way except that they exhibit random phase characteristics, one gets a result which is, roughly, intermediate (but tending toward the 'in phase' side of things) in range between what one would get if one were adding together the amplitudes of these waves under conditions of: (a) being totally out of phase; (b) being completely in phase. This means that when random phase occurs, peaks tend to be flattened out somewhat, and troughs tend to be lessened somewhat. In short, the interference properties become somewhat blurred, with peaks and troughs becoming less distinct.

In the case of ordinary, everyday oscillatory wave systems characterized by non-random, or determinate phases, adding the amplitudes of waves conforming to the principle of superposition means the energies of the interacting waves will not add everywhere, but only at those points involving constructive interference. However, if these interacting waves exhibited random phase, then, the energies of the interacting waves will add everywhere.

The foregoing also applies, with certain qualifications, to quantum waveforms. In the case of quantum waves, one is not talking about energies, since, according to quantum theory, quantum waves have no energy. One is, instead, talking about the intensity of the quantum wave as expressed in terms of probabilities.

When quantum waves with determinate phase interact with one another, the probabilities do not add everywhere. The probabilities add only where their phase relations permit it. On the other hand, when the quantum waves being added together involve random phase, then, the probabilities add together everywhere in the wave.

The set of ideas involving: the principle of superposition, constructive and destructive interference, as well as random phase, may have considerable implications for the educational context involving the relationship between teacher and students, or curriculum and students, or system and students, or system and teachers. More specifically, suppose one treats a hermeneutical system as a latticework of linked oscillators. How the complex, multi-faceted amplitude of a given individual's hermeneutical system (which is the sum of the amplitudes of the various lattice-oscillators which shape that individual's hermeneutical system) interacts with the hermeneutical systems of other individuals (whether students, teachers, textbooks or the educational system in general) will depend, to a large extent, on the phase characteristics of the hermeneutical latticework waveforms which are interacting with one another.

Where the aforementioned latticeworks are out of phase, destructive interference occurs. Where they are in phase, constructive interference occurs, and if they exhibit random phase characteristics (in which there are a mixture of phase elements), there will be summing process which leads to no particular hermeneutical focus, direction, orientation, significance, purpose or value. In other words, in the latter case there is a blurring of the structural character of the hermeneutical waveforms which are engaging one another, both with respect to the student as well as the teacher and the system in which they are rooted.

The task of a teacher, of an educational system, of a curriculum, and of the student is to work toward creating an interaction of waveforms with a determinate phase and in which the amplitudes are in phase. The determinate phase comes with a clear-cut, hermeneutical orientation, purpose, or significance. The in-phase aspect comes from people who share that orientation and who cooperate to make sure that all of the different parameters of the waveforms complement one another.

However, in addition to the need for waveforms with determinate phases as well as wave forms which are in phase with one another, there also is a further need. There is a need to ensure that the phase character or orientation character of the hermeneutical latticework waveforms which emerge in the educational context are capable of accurately reflecting the structural character of certain themes of ontology.

Stated slightly differently, there is a need to ensure that the phase character or orientation character of the hermeneutical latticework waveforms which guide a teacher, student, system or society are capable of being accurately reflective of, or give expression to, a methodology which is capable of leading the individual, the teacher, the system, and the greater society toward a system of unobtrusive measures. These sorts of measures permit individuals increasingly less distorted access to various aspects of experience and reality.

Unfortunately, too frequently, modern day education is beset with tendencies toward either destructive interference or waveforms with random phase. In the former case, educational problems are due to the way in which the hermeneutical latticework waveforms that are brought to the educational setting by teachers, by students, by institutions, or by society, work antagonistically or destructively against one another, to varying degrees.

On the other hand, sometimes, educational problems are due to the components of randomization which are prevalent in the sort of diversified environment which often exists in democratic societies. When education is randomized, student, teacher, the educational system, as well as the surrounding society are pulled in a variety of different phase orientation directions.

For example, when a society or an educational system attempts to please everyone, one ends up with a randomization of the phase relations of the various hermeneutical latticework waveforms which shape that society or educational system. In effect, the randomization of phase relations gives expression to anarchy in which energy is distributed or dispersed everywhere and, as a result, there is no concerted, in-phase, dialectical interaction of the waveforms making up that society or educational system.

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| Quantum - Part 4 | Quantum - Part 5 |

| Quantum - Part 6 |

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