The material of the previous chapter has set out the foundations of quantum mechanics. It has been shown that a particle has wave character. Once this is accepted, it becomes possible to use Fourier synthesis to construct a hypothetica particle. Since matter has been shown to have wave-like properties and since it is possible to define it by means of the superposition of component waves, it follows that it should be possible to use the mathematical tools which are available for treating wave motion. This forms the basis of wave mechanics.
The material of the previous chapter has set out the foundations of quantum mechanics. It has been shown that a particle has wave character. Once this is accepted, it becomes possible to use Fourier synthesis to construct a hypothetica particle. Since matter has been shown to have wave-like properties and since it is possible to define it by means of the superposition of component waves, it follows that it should be possible to use the mathematical tools which are available for treating wave motion. This forms the basis of wave mechanics.
Nearly all of quantum physics is concerned with the properties of waves. The mathematical description of waves is covered in this second chapter, which makes use of the material covered in Chapter 1 to explain the wave equation, superposition and beats. The complex representation of waves is particularly important. The notion of a wave function, which is a solution of the wave equation, forms the basis of all quantum mechanics. Superposition and beat frequency will later be shown to account for the way in which light is emitted from atoms.
This introductory text emphasizes Feynman's development of path integrals and its application to wave theory for particles. Suitable for undergraduate and graduate students of physics, the well-written, clear, and rigorous text was written by two of the nation's leading authorities on quantum physics. A solid foundation in quantum mechanics and atomic physics is assumed. Early chapters provide background in the mathematical treatment and particular properties of ordinary wave motion that also apply to particle motion. The close relation of quantum theory to physical optics is stressed. Subsequent sections emphasize the physical consequences of a wave theory of material properties, and they offer extensive applications in atomic physics, nuclear physics, solid state physics, and diatomic molecules. Four helpful Appendixes supplement the text.
More than a hundred years ago, a series of discoveries, especially in physics, began to subtly (or maybe not) reveal some mysterious properties of matter, properties that in some fashion or other were not acceptable to our established way of perceiving Reality. Strangely enough, these properties were not taken seriously by science because they pointed to a reality that seemed too fantastical to be true, a reality that seemed to have no substance or permanence. These so called mysterious properties matter, for lack of a better word, turn out to be fundamentally related to the following discoveries: The ambiguous wave-particle duality. As in de Broglie matter-waves and the Compton properties of particles. The ubiquitous appearance of the cycle constant (τ=2π=C⁄r) in physical mathematical relations. The general applicability of the Fourier Transform to represent/model physical motion and other natural processes. The Heisenberg Uncertainty Principle and its mysterious constraint between conjugate variables, momentum and position for example. Quantum Mechanics and its collapse of the wavefunction. The surprising mass-energy equivalence. As in Einstein’s E=mc^2. The perplexing isotropy of the speed of light. The mind boggling time dilation and length contraction of Einstein’s Special Relativity (SR). The contradictory fabric of SR/GR’s spacetime and its assignment of properties to the void. The surprisingly discrete nature of atomic properties. As in Quantum Mechanics (QM). Needless to say, no matter what science has tried in order to fit the above discoveries into our physical world, their implications continue to elude an intuitively acceptable explanation in material terms. This monograph, although not necessarily in chronological order, is a recapitulation of previous monographs written by the author in an attempt at a single-source theoretical explanation/clarification for the above mentioned list of “mysterious” properties of matter, in terms of one fundamental set of properties, the motional geometry of matter.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
In the domain of light emissions, quantum mechanics has been an immensely successful guiding tool for us. In the propagation of light and optical instrument design, Huygens-Fresnel diffraction integral (HFDI) (or its advanced versions) and Maxwell’s wave equation are continuing to be the essential guiding tools for optical scientists and engineers. In fact, most branches of optical science and engineering, like optical instrument design, image processing, Fourier optics, Holography, etc., cannot exist without using the foundational postulates behind the Huygens-Fresnel diffraction integral. Further, the field of structured light is also growing where phases and the state of polarizations are manipulated usually with suitable classical macro-devices to create wave fronts that restructured through light-matter interactions through these devices. Mathematical modeling of generating such complex wave fronts generally follows classical concepts and classical macro tools of physical optics. Some of these complex light beams can impart mechanical angular momentum and spin-like properties to material particles inserted inside these structured beams because of their electromagnetic dipolar properties and/or structural anisotropy. Does that mean these newly structured beams have acquired new quantum properties without being generated through quantum devices and quantum transitions? In this chapter, we bridge the classical and quantum formalism by defining a hybrid photon (HP). HP is a quantum of energy, hν, at the initial moment of emission. It then immediately evolves into a classical time-finite wave packet, still transporting the original energy, hν, with a classical carrier frequency ν (oscillation of the E-vector). This chapter will raise enquiring questions whether all these observed “quantum-like” behaviors are manifestations of the joint properties of interacting material particles with classical EM waves or are causal implications of the existence of propagation of “indivisible light quanta” with exotic properties like spin, angular momentum, etc.