Special relativity combined with the stochastic vacuum flux impact model lead to an explicit interpretation of many of the phenomena of elementary quantum mechanics. We examine characteristics of a repetitively impacted submicroscopic particle in conjunction with examination of the ways in which effects associated with the particle's behavior appear in moving frames of reference. As seen from relatively moving frames of reference, the time and location of impacts and recoils automatically exhibit wave behavior. This model leads to free particle waves with frequencies proportional to the energy and wavelengths inversely proportional to the momentum. As seen from relatively moving frames of reference, impacts and their associated recoils can appear to an observer to take place simultaneously at multiple locations in space. For superposed free particle waves corresponding to bidirectional motion, an amplitude that varies sinusoidally with distance results. A governing equation identical in form to the Schroedinger equation is developed that describes the behavior of the impacts and their associated recoils. This approach permits many features of quantum mechanics to be examined within an intuitively visualizable framework.
Special relativity combined with the stochastic vacuum flux impact model lead to an explicit interpretation of many of the phenomena of elementary quantum mechanics. We examine characteristics of a repetitively impacted submicroscopic particle in conjunction with examination of the ways in which effects associated with the particle's behavior appear in moving frames of reference. As seen from relatively moving frames of reference, the time and location of impacts and recoils automatically exhibit wave behavior. This model leads to free particle waves with frequencies proportional to the energy and wavelengths inversely proportional to the momentum. As seen from relatively moving frames of reference, impacts and their associated recoils can appear to an observer to take place simultaneously at multiple locations in space. For superposed free particle waves corresponding to bidirectional motion, an amplitude that varies sinusoidally with distance results. A governing equation identical in form to the Schroedinger equation is developed that describes the behavior of the impacts and their associated recoils. This approach permits many features of quantum mechanics to be examined within an intuitively visualizable framework.
The relationship between classical and quantum mechanics is explored in an intuitive manner by the exercise of constructing a wave in association with a classical particle. Using special relativity, the time coordinate in the frame of reference of a moving particle is expressed in terms of the coordinates in the laboratory frame of reference in order to provide an initial spatiotemporal function to work from in initiating the development of a quantum wave. When temporal periodicity is ascribed to the particle, a provisional spatiotemporal function for a particle travelling at constant velocity manifests itself as an running wave characterized by parameters associated with the moving particle. A wave description for bidirectional motion is generated based on an average time coordinate for a combination of oppositely directed elementary running waves, and the resulting spatiotemporal function exhibits wave behavior characteristic of a standing wave. Ascribing directional orientation to the intrinsic periodicity of the particle introduces directional sub-states; variations in the relative number of sub-states as a function of angle in combined states lead to spatially varying magnitudes for the associated waves. Further analysis leads to full mathematical expression for all waves representing free particle motion. A generalization for particles subject to force fields enables us to develop a governing differential equation identical in form to the Schroedinger equation.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]
Correction June 28, 2022: The classical wave-equation should read d/dt d/dt partial W(x,t) = vv d/dx d/dx partial W(x,t) Historically two types of “free” mechanical motions were considered, that of Newtonian mechanics x=vt (v=constant) and classical waves such as waves on a string which were described in terms of Newton’s laws for a continuous medium. It seems these two were considered as fundamentally different phenomena. Equations of motion were developed for classical mechanics as well as electromagnetic phenomena. Maxwell discovered that electromagnetic equations allowed for a classical wavelike solution which he identified as light. In the early 1900s it was realized that to retain electromagnetic equations invariance as seen from a constant moving frame moving, Lorentz transformations were needed. In 1905 Einstein obtained these transformations in a different manner considering that light has the same vacuum speed when viewed from different constant moving frames. Later in the 1920s quantum mechanics was developed using the idea of a probability wave exp(-iEt+ipx) for a free particle beginning with the work of deBroglie who made use of ideas of special relativity. In this note we suggest that there seems to be one fundamental “free” mechanical motion, namely that associated with a wave exp(-iEt+ipx) whether one considers a classical wave or a quantum free particle. Classical mechanics is a mapping into a continuous x-t space, we argue, and describes energy and momentum motion in this continuous space. This same description also occurs for classical waves. The medium undergoes a somewhat complicated nonlocal change, but the “velocity” of the wave maps to a continuous local space time i.e. v=x/t=dx/dt where E= pv. E=pv, however, is only a description of energy momentum propagation given in a simple form. It does not describe the overall picture. We argue the same is true for the simple x=vt (v=constant). It seems that an overall wave structure is associated with energy and momentum and for free motion this is linked with exp(-iEt+ipx). If this wave is fundamental one may argue that its phase may remain constant when viewed from different frames moving at constant v. Special relativity would then follow from this requirement and so is in a sense a consequence of wave behaviour. We investigate these ideas.
Special relativity is a fascinating and challenging branch of physics. It describes the physics of the high velocity/high energy regime, frequently turning up phenomena that appear paradoxical in view of our everyday experience. In this book we will be quite selective in our presentation of the theory of special relativity: we will concentrate on those features that we consider necessary for the later applications to relativistic quantum chemistry. We do this in good conscience, knowing that there is a vast literature on the subject, catering to a wide range of audiences—from the quite elementary to the very sophisticated. A few examples are listed in the reference list, but a visit to any nearby physics library will provide an ample selection of reading material for those wishing to delve deeper into the matter. In the present chapter we adopt a minimalist approach. We develop some of the basic concepts and formulas of special relativity, building on a rather elementary level of basic physics. The aim is to provide a sufficient foundation for those who want to proceed as quickly as possible to the more quantum chemical parts of the text. In later chapters we will introduce more advanced tools of physics and revisit some of the subjects treated here. The theory of special relativity deals with the description of physical phenomena in frames that move at constant velocity relative to each other. The classroom is one such frame, the car passing at constant speed outside the classroom is another. The trajectory of a ball being thrown up vertically in the car will look quite different whether we describe it relative to the interior of the car or relative to the interior of the classroom. In particular we will be concerned with inertial frames. We define an inertial frame as a frame where spatial relations are Euclidean and where there is a universal time such that free particles move with constant velocities. In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S’ are expressed in terms of the Galilean transformations.
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
We assume that particles are point-like objects even when not observed. We report on the consequences of our assumption within the realm of quantum theory. An important consequence is the necessity of vacuum fields to account for particle dynamics. The zero-point energy is evidence of the reality of vacuum fields. The dynamics of vacuum fields depends on wave equations, boundary conditions and external sources. The wavefunction is an expression of vacuum fields. Particle events are described by probabilistic distribution functions obtained from vacuum fields. We solve the measurement problem by noticing that in our model the wavefunction does not collapse.