O NE OF the most basic ideas of quantum mechanics is that the condition of a system is represented mathematically by a wavefunction . This is a complex valued mathematical function of that tells us everything we can know about the system. In order to make predictions we apply mathematical operations to the wavefunction that tell us the probability of various outcomes that we can compare to experimental results. The wavefunction is a complex valued function. In other words when you evaluate it for a particular set of inputs (e.g., position, time, momentum, energy, etc.) the output is a complex number. The magnitude of the complex number gives information about the relative likelihood of a particular outcome. That seems pretty odd I'm sure. The best way to get comfortable with the notion is to play around with it. Let's consider an example.
O NE OF the most basic ideas of quantum mechanics is that the condition of a system is represented mathematically by a wavefunction . This is a complex valued mathematical function of that tells us everything we can know about the system. In order to make predictions we apply mathematical operations to the wavefunction that tell us the probability of various outcomes that we can compare to experimental results. The wavefunction is a complex valued function. In other words when you evaluate it for a particular set of inputs (e.g., position, time, momentum, energy, etc.) the output is a complex number. The magnitude of the complex number gives information about the relative likelihood of a particular outcome. That seems pretty odd I'm sure. The best way to get comfortable with the notion is to play around with it. Let's consider an example.
Quantum Mechanics is appropriately named because it is mostly about the mechanics of working out probability problems as they apply to the mysterious effect of measurement that makes energy present as quantum particle-waves. It is easy to visualize a quantum particle but there seems to be no way to visualize a “probability wave” so there has never been a clear interpretation of what quantum physics tells us – or should tell us – about the underlying essence of reality. Fortunately, a new approach was discovered and proven to work in biology that treats the mystery of life like a locking system and provides not only the key, but also the keyhole and the direction in which to turn it. In this paper the systems approach is introduced, including some background information on the history of how the systems approach has integrated elements of substance philosophy with process philosophy and has become a powerful tool for use in science. Here, it is applied to physics and used to represent the transformation of implicit energy into an explicit space-time quantum domain superimposed on a relativistic time-space background. These two “products” are then correlated with two of four blocks in a basic control-system diagram (input and output) commonly used in control-systems engineering. (The four blocks are input, transfer function, output and feedback function) Because they are explicit, they can be drawn explicitly on the space versus time plot as a map of motion. We emphasize that this is a map – an explicit projection of an implicit function (in this case, a plot of space versus time is a projection of the variance we call motion) represented as an implicit domain (a perpendicular dimension). We recognizing that a feedback function is equivalent to and therefore implies a back-projection or reflection back “up” into that “implicate order.” These two implicit functions (projection and reflection) are then correlated with the transfer function and feedback function in the control system. The motion function is related to the probability wave by the name used in statistics – the “variance” of a statistical distribution – and we point out that each axis of the space-time domain is scaled by standard “deviations,” which define the explicit scales. Because a standard deviation is the square root of a variance, the square space that maps as a relativistic scalar plot (a standard Cartesian coordinate system) of space versus time or “space-time” actually represents one of two square roots of the implicit variance we call motion. In essence, our measurements of this domain keep us rooted in physical reality. However, this coordinate system is shown to be just the root – one part of a “whole system” that includes a non-physical implicit feedback function. The feedback function is correlated with the “reflection” of the explicit domain, i.e. inverse-space and inverse-time, presented as a phase space called time-space and represented as a unit of spatial frequency versus temporal frequency. These frequency representations are shown to be the two well-known equations for quantum energy from quantum mechanics and they are used to identify energy-space as “the quantum domain” inside of, or superimposed on the background “relativistic domain,” which is a scalar space. These two domains are then shown to be coupled at the point where both scales are equal to “1”. As a whole, and in the language of quantum field theory, this model is a graphical representation of an energy-momentum tensor and as a visual model, it provides a clear conceptual interpretation of complexity theory, in which reality is expressed as the superposition of a self-organizing control system that convolves with a dissipative open system or “sea of disorder” and transforms it into physical units of order. This “holomorphic systems approach” is similar to, but much simpler and visually illustrative than the mathematically rigorous discussion On Symmetry and the Reality of Holomorphic Hartree–Fock Wavefunctions [1] or the Kohn-Sham Density Functional Theory [2]. Furthermore, it reveals that the question about the beginning of time is a question fallacy, by representing the equivalence of space and time as S=Tc2, and presents it geometrically to be the exact same relation as the mass-energy equivalence equation E=mc2. The model also reveals that quantized energy (which Huynh and Thom refer to as self-consistent fields or SCF) projects as a characteristic and allows one to visualize the solution to the particle-wave duality “problem” as being a change in perspective. It’s the change in perspective that makes the particle-like property and wave-like property appear as “emergent” – analogous to what would appear when one visualizes an object from two different perspectives – at rest with respect to one’s own body yet in motion relative to some other “moving” body. The approach also reveals how the transfer function and feedback function act to transform matter, adding the fourth component (the Controller) and making it a complete control system: a self-organizing, self-sensing system that can see itself explicitly and reflect upon itself implicitly. How that correlates with living systems becomes obvious.
Abstract In stark contrast to classical Newtonian mechanics, the standard interpretation of quantum mechanics associates the Schrödinger wavefunction, not in a determ- inistic way with definite predictions about a specific, individual experimental result, but only in a probabilistic manner, with |ψ(x, t )|2 acting as a probability density, in this case, for position measurements. Before analyzing this approach in more detail, we will find it profitable to briefly review some of the basic concepts of discrete and
Quantum Mechanics is appropriately named because it is mostly about the mechanics of working out probability problems as they apply to the mysterious effect of measurement that makes energy present as quantum particle-waves. It is easy to visualize a quantum particle but there seems to be no way to visualize a “probability wave” so there has never been a clear interpretation of what quantum physics tells us – or should tell us – about the underlying essence of reality. Fortunately, a new approach was discovered and proven to work in biology that treats the mystery of life like a locking system and provides not only the key, but also the keyhole and the direction in which to turn it. In this paper the systems approach is introduced, including some background information on the history of how the systems approach has integrated elements of substance philosophy with process philosophy and has become a powerful tool for use in science. Here, it is applied to physics and used to represent the transformation of implicit energy into an explicit space-time quantum domain superimposed on a relativistic time-space background. These two “products” are then correlated with two of four blocks in a basic control-system diagram (input and output) commonly used in control-systems engineering. (The four blocks are input, transfer function, output and feedback function) Because they are explicit, they can be drawn explicitly on the space versus time plot as a map of motion. We emphasize that this is a map – an explicit projection of an implicit function (in this case, a plot of space versus time is a projection of the variance we call motion) represented as an implicit domain (a perpendicular dimension). We recognizing that a feedback function is equivalent to and therefore implies a back-projection or reflection back “up” into that “implicate order.” These two implicit functions (projection and reflection) are then correlated with the transfer function and feedback function in the control system. The motion function is related to the probability wave by the name used in statistics – the “variance” of a statistical distribution – and we point out that each axis of the space-time domain is scaled by standard “deviations,” which define the explicit scales. Because a standard deviation is the square root of a variance, the square space that maps as a relativistic scalar plot (a standard Cartesian coordinate system) of space versus time or “space-time” actually represents one of two square roots of the implicit variance we call motion. In essence, our measurements of this domain keep us rooted in physical reality. However, this coordinate system is shown to be just the root – one part of a “whole system” that includes a non-physical implicit feedback function. The feedback function is correlated with the “reflection” of the explicit domain, i.e. inverse-space and inverse-time, presented as a phase space called time-space and represented as a unit of spatial frequency versus temporal frequency. These frequency representations are shown to be the two well-known equations for quantum energy from quantum mechanics and they are used to identify energy-space as “the quantum domain” inside of, or superimposed on the background “relativistic domain,” which is a scalar space. These two domains are then shown to be coupled at the point where both scales are equal to “1”. As a whole, and in the language of quantum field theory, this model is a graphical representation of an energy-momentum tensor and as a visual model, it provides a clear conceptual interpretation of complexity theory, in which reality is expressed as the superposition of a self-organizing control system that convolves with a dissipative open system or “sea of disorder” and transforms it into physical units of order. This “holomorphic systems approach” is similar to, but much simpler and visually illustrative than the mathematically rigorous discussion On Symmetry and the Reality of Holomorphic Hartree–Fock Wavefunctions [1] or the Kohn-Sham Density Functional Theory [2]. Furthermore, it reveals that the question about the beginning of time is a question fallacy, by representing the equivalence of space and time as S=Tc2, and presents it geometrically to be the exact same relation as the mass-energy equivalence equation E=mc2. The model also reveals that quantized energy (which Huynh and Thom refer to as self-consistent fields or SCF) projects as a characteristic and allows one to visualize the solution to the particle-wave duality “problem” as being a change in perspective. It’s the change in perspective that makes the particle-like property and wave-like property appear as “emergent” – analogous to what would appear when one visualizes an object from two different perspectives – at rest with respect to one’s own body yet in motion relative to some other “moving” body. The approach also reveals how the transfer function and feedback function act to transform matter, adding the fourth component (the Controller) and making it a complete control system: a self-organizing, self-sensing system that can see itself explicitly and reflect upon itself implicitly. How that correlates with living systems becomes obvious.
In the current work, we extend and incorporate the five-axioms probability system of Andrey Nikolaevich Kolmogorov, set up in 1933 the imaginary set of numbers, and this by adding three supplementary axioms. Consequently, any stochastic experiment can thus be achieved in the extended complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. The purpose here is to evaluate the complex probabilities by considering additional novel imaginary dimensions to the experiment occurring in the “real” laboratory. Therefore, the random phenomenon outcome and result in C=R+M can be predicted absolutely and perfectly no matter what the random distribution of the input variable in R is since the associated probability in the entire set C is constantly and permanently equal to one. Thus, the following consequence indicates that chance and randomness in R are replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge of the chaotic factor in the probabilistic experiment. Moreover, I will apply to the established theory of quantum mechanics my original complex probability paradigm (CPP) in order to express the quantum mechanics problem considered here completely deterministically in the universe of probabilities C=R+M.
The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.