Abstract In this chapter we develop the stochastic dynamical theory of Rayleigh scattering, from the point of view of a random walk model. Physically this is appropriate to forward and backward scattering experiments where the scattering population is effectively constant over the observation period. From a theoretical point of view, the purpose of the chapter is to present a self-contained derivation that follows essentially from first principles. We shall see how the development, according to a random walk model pertaining to the components of the scattered field, leads to an elegant mathematical derivation of the Ornstein–Uhlenbeck process for the resultant amplitude. This derivation of the scattering dynamics, motivated by a physical model (the coherent superposition of a number of randomly phased scattered wave components) is of inherent pure mathematical interest, as well as being satisfying from a theoretical physics point of view.
Abstract In this chapter we develop the stochastic dynamical theory of Rayleigh scattering, from the point of view of a random walk model. Physically this is appropriate to forward and backward scattering experiments where the scattering population is effectively constant over the observation period. From a theoretical point of view, the purpose of the chapter is to present a self-contained derivation that follows essentially from first principles. We shall see how the development, according to a random walk model pertaining to the components of the scattered field, leads to an elegant mathematical derivation of the Ornstein–Uhlenbeck process for the resultant amplitude. This derivation of the scattering dynamics, motivated by a physical model (the coherent superposition of a number of randomly phased scattered wave components) is of inherent pure mathematical interest, as well as being satisfying from a theoretical physics point of view.
Random processes are of interest not only from the theoretical point of view but also for practical use in algorithms for investigating large combinatorial structures. The theory of quantum computing requires implementation of classical algorithms using quantum-mechanical devices, and random walk is an obvious candidate. We present a model for quantum random walk that is based on an interferometric analogy, can be easily implemented, and is a generalization of a former model of quantum random walk proposed by Aharonov and colleagues.
In this thesis, two new types of solvable stochastic model are proposed and applied to light scattering phenomena. The theoretical framework for each model Is made based on a time-convolutlon-type equation In the damping theory. Each model Is used to develop a theory of line shape, in which the Influence of an environment for a relevant system is idealized so as to be represented with a stochastic process. Next, each model Is used to formulate a light scattering theory, especially focusing on the problem of "Raman versus Luminescence", or on a correlation among incident and scattered photons. For qualitative understanding of a problem like the light scattering, a stochastic approach such as employed here is really powerful. Recently, according to the development in experimental techniques of generating an ultra-short pulse excitation, various kinds of information upon relaxation phenomena have become obtainable in the real time domain. The models considered here may be of good use for phenomena in the ultra-short time domain. Two typical stochastic models are frequently used; one is the two-state-jump Markoff process and the other Is the Gaussian process. They are the opposite extremes with each other, whereas many real processes assume intermediate character between the two. Thus In part I a model (the first model) which bridges smoothly between these two extremes is introduced as a superposition of N two-state-jump Markoff processes. For a single two-state-jump Markoff process, the "partial cumulants" of higher than the second order vanish Identically, so that, its power spectrum can be expressed by the first order continued fraction. Because of this simple property, the general structure of the partial cumulants in case of an. arbitrary number N can be found. Thus, summing up contributions with the aid of a diagrammatic technique, the power spectrum of the proposed model is determined in a form of the N-step continued fraction. For a clear illustration of this model, a simple example of the random frequency modulation is considered, and then numerical calculations for its power spectra and its relaxation functions are given. This model is further adopted to analyze the stationary light scattering; a rigorous expression of the scattering Intensity is obtained. This result generalizes that of the stochastic model proposed by Takagahara, Hanamura and Kubo. According to an approach based on a generalized master (time-convolution-type) equation, It is found that the inhomogeneous term plays an essential role, whereas this term plays no important role In existing treatments of the relaxation. This Is due to the relaxation mechanism of the light scattering phenomena. Behavior of the light scattering spectrum is systematically investigated by the present theory with the aid of numerical calculations. It is shown how the spectrum changes from the "Raman" to the "Luminescence" under the Influence of environment. The multi-peak structure of the luminescence-like spectrum gradually disappears to become a smooth curve when the process is changed from the two-state-jump to the Gaussian. The transient response for a pulse excitation Is also considered with the use of this new model. From the above-mentioned result of the stationary light scattering, the expression for the time-resolved spectrum is readily obtained. This expression can be calculated numerically and a problem of the coherence is discussed. In part II, a further generalized model (the second model) is proposed. While the usual two-state-jump Markoff process assumes equal weight population for the two states, the newly introduced two-state-jump Markoff process Includes an asymmetric population. As a superposition of such constituent processes with asymmetric property, the new model becomes more flexible than the previous (first) model. In other words, the model Is composed of hopping processes among (N+1)-states with the binomial distribution in equilibrium. Namely, the distribution of the n-th state in equilibrium is given by p(n)=_NC_nx^ (1-x)^n, (O≤X≤1). This model is applied to various problems; the random frequency modulation, the stationary light scattering and the transient light scattering. The spectrum of this model is quite analogous to the one due to the phonon side bands, though the direct comparison must be done carefully. Our second model may be powerful in treating various systems except the ordinary light scattering, for Instance, γ-ray or X-ray scattering in solids. Moreover the formulation developed here may have wide range of applicability to various other phenomena. These remain to be solved in future.
Abstract This book provides a thorough introduction to the fascinating world of phase transitions as well as many related topics, including random walks, combinatorial problems, quantum field theory and S-matrix. Fundamental concepts of phase transitions, such as order parameters, spontaneous symmetry breaking, scaling transformations, conformal symmetry, and anomalous dimensions, have deeply changed the modern vision of many areas of physics, leading to remarkable developments in statistical mechanics, elementary particle theory, condensed matter physics and string theory. This self-contained book provides an excellent introduction to frontier topics of exactly solved models in statistical mechanics and quantum field theory, renormalization group, conformal models, quantum integrable systems, duality, elastic S-matrix, thermodynamics Bethe ansatz and form factor theory. The clear discussion of physical principles is accompanied by a detailed analysis of several branches of mathematics, distinguished for their elegance and beauty, such as infinite dimensional algebras, conformal mappings, integral equations or modular functions. Besides advanced research themes, the book also covers many basic topics in statistical mechanics, quantum field theory and theoretical physics. Each argument is discussed in great detail, paying attention to an overall coherent understanding of physical phenomena. Mathematical background is provided in supplements at the end of each chapter, when appropriate. The chapters are also followed by problems of different levels of difficulty. Advanced undergraduate and graduate students will find a rich and challenging source for improving their skills and for accomplishing a comprehensive learning of the many facets of the subject.
Quantum Mechanics (QM) represents the currently best mathematical theory and computational tool available to quantitatively model the outcomes of measurements in the microscopic world. However, the various interpretations of the theory, including the currently dominant Copenhagen Interpretation, remain actively debated and contested in mainstream scientific journals. This puts us in the curious position of having a theory which evidently maps a significant portion of physical reality, but we still do not fully understand the relationship between its mathematical constituents and physical reality. This thesis develops a more intuitive understanding of QM based on the physical interaction processes occurring within our detectors. Using a semi- classical point of view, I first present a process-based derivation of the black-body spectrum, the mathematically accurate description of which, as first accomplished by Planck, led to the development of modern QM. I further demonstrate that ignoring interpretations like the Copenhagen Interpretation and focusing on measurable light-matter interaction processes provides an alternative productive way forward. I focus specifically on superposition effects created by multiple electro- magnetic waves simultaneously stimulating appropriate detectors, by mapping the interaction processes that result in the reported data. As a result I present a simple model of the detector signal that incorporates some of the important physical processes which do date have suffered from a lack of attention. One key observation of this discussion corresponds to the fact that in the linear domain wave amplitudes by themselves do not interact to generate observable data - the response to the joint amplitude stimulation of a detector and the ensuing energy absorption on the other hand do represent observables. Much of the confusion as well as many of QM’s paradoxes and contentious issues inherent in various interpretations of the theory resolve automatically when we remain focused on the interaction processes that give rise to the measurable superposition effect, rather than the unobservable superposition principle that the standard interpretations struggle to give meaning to. Demanding, as I do in this thesis, a close correspondence between our mathematical symbols and physical quantities, as well as between mathematical operations and physical processes demonstrates that QM maps more of physical reality than what the Copenhagen Interpretation allows us to extract from the QM formalism.
The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.