Concepts in CLASSIC

Therefore, this is a statement of our fundamental hypothesis, specifically, that the topology of the vacuum defines the field equations through group and gauge field theory. Prior to the inference and empirical verification of the Aharonov-Bohm effect, there was no such concept in classical electrodynamics, the ether having been denied by Lorentz, Poincare, Einstein, and others. Our development of 0(3) electrodynamics in this chapter, therefore, has a well-defined basis in fundamental topology and empirical data. In the course of the development of... 

Without the additional 3-symmetry condition, the resulting Whittaker 4-symmetry EM energy flow mechanism resolves the nagging problem of the source charge concept in classical electrodynamics theory. Quoting Sen [10] The connection between the field and its source has always been and still is the most difficult problem in classical and quantum mechanics. We give the solution to the problem of the source charge in classical electrodynamics. 

Elementary Concepts in Classical Machanics 197 2.2.2 Lagrangian Mechanics... 

Elementary Concepts in Classical Machanics 205 Since the Hamiltonian equations satisfy the incompressibility condition... 

Two additional conmiciits apply at tliis point. First, there is a conceptual difference between the probabilistic element in quantum and classic statistical physics. For instance, in quantum mechanics, the outcome of a measurement of properties even of a single particle can be known in principle only w ith a certain probability. In classic mechanics, on the other hand, a probabilistic element is usually introduced for many-particle systems where we would in principle bo able to specify the state of the system with absolute certainty however, in practice, this is not possible because we are dealing wdth too many degrees of freedom. Recourse to a probabilistic description within the framework of classic mechanics must therefore be regarded a matter of mere convenience. The reader should appreciate this less fundamental meaning of probabilistic concepts in classic as opposed to quantiun mechanics. 

Finally, the units of h, joule-second (or J-s), is a combination of energy and time. Energy multiplied by time yields a quantity known as action. Earlier in history, scientists developed something called the principle of least action, which is an important concept in classical mechanics. In quantum mechanics, we will find that any quantity that has units of action is intimately related to Planck s constant. 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

Basically, Newtonian mechanics worked well for problems involving terrestrial and even celestial bodies, providing rational and quantifiable relationships between mass, velocity, acceleration, and force. However, in the realm of optics and electricity, numerous observations seemed to defy Newtonian laws. Phenomena such as diffraction and interference could only be explained if light had both particle and wave properties. Indeed, particles such as electrons and x-rays appeared to have both discrete energy states and momentum, properties similar to those of light. None of the classical, or Newtonian, laws could account for such behavior, and such inadequacies led scientists to search for new concepts in the consideration of the nature of reahty. 

The third group is the continuum, models, and these are based on simple concepts from classical electromagnetism. It is convenient to divide materials into two classes, electrical conductors and dielectrics. In a conductor such as metallic copper, the conduction electrons are free to move under the influence of an applied electric field. In a dielectric material such as glass, paraffin wax or paper, all the electrons are bound to the molecules as shown schematically in Figure 15.2. The black circles represent nuclei, and the electron clouds are represented as open circles. 

The first law is closely related to the conservation of energy (Section A) but goes beyond it the concept of heat does not apply to the single particles treated in classical mechanics. 

This theory was advanced by G. N. Lewis (1916, 1923, 1938) as a more general concept. In his classic monograph of 1923 he considered and rejected both the protonic and solvent system theories as too restrictive. An acid-base reaction in the Lewis sense means the completion of the stable electronic configuration of the acceptor atom of the acid by an electron pair from the base. Thus ... 

Herring C. 1952. The use of classical macroscopic concepts in surface-energy problems. In Gomer R, Smith CS. editors. Structure and Properties of Sohd Surfaces. Chicago Chicago University Press. 

Another experiment that relates to the physical interpretation of the wave function was performed by O. Stem and W. Gerlach (1922). Their experiment is a dramatic illustration of a quantum-mechanical effect which is in direct conflict with the concepts of classical theory. It was the first experiment of a non-optical nature to show quantum behavior directly. 

Atomic partial charges are a difficult concept in quantum chemistry. On the one hand, assigning charges to individual atoms in a molecule is very close to the classical interpreta-... 

In the remainder of this chapter, we review the fundamentals that underlie the theoretical developments in this book. We outline, in sequence, the concept of density of states and partition function, the most basic approaches to calculating free energies and the essential strategies for improving the efficiency of these calculations. The ideas discussed here are, most likely, known to the reader. They can also be found in classical books on statistical mechanics [132-134] and molecular simulations [135, 136]. Thus, we do not attempt to be exhaustive. On the contrary, we present the material in a way that is most directly relevant to the topics covered in the book. 

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