Computational Physics

Abstract

While the first edition of this textbook was based on a one-year course in computational physics with a rather limited scope, its extent has been increased substantially in the third edition, offering the possibility to select from a broader range of computer experiments and to deepen the understanding of the important numerical methods. The computer experiments have always been a central part of my concepts for this book. Since Java applets, which are very convenient otherwise, have become more or less deprecated and their usage in a browser is no longer recommended for security issues, I decided to use standalone Java programs instead and to rewrite all of the old examples. These can also been edited and compiled with the “netbeans” environment and offer the same possibilities to generate a graphical user interface in short time. The major changes in the third edition are as follows. In the first part, a new chapter is devoted to the time-frequency analysis of experimental data. While the classical Fourier transform allows the calculation of the spectrum of a stationary signal, it is not so useful for nonstationary signals with significant variation of the momentaneous frequency distribution. Application of the Fourier transformation to short time windows, a method which is known as short-time Fourier transformation (STFT), allows analyzing the frequency content of a signal as a function of time. Good time resolution, of course, always comes together with a loss in frequency resolution (this is well known as “uncertainty principle”). The STFT method uses the same window for the whole spectrum, therefore the absolute time and frequency resolution is the same for low- and high-frequency components and the time resolution is limited by the period of the lowest frequencies of interest. Analysis of a signal with wavelets, on the other hand, uses shorter windows for the higher frequencies and keeps the relative frequency resolution constant while increasing the time resolution of the high-frequency components. The continuous wavelet transform can be very time consuming since it involves a convolution integral and is highly redundant. The discrete wavelet transform uses a finite number of orthogonal basis function and can be performed much faster by calculating scalar products. It is closely related to multiresolution analysis which analyzes a signal in terms of a basic approximation and details of increasing resolution. Such methods are very popular in signal processing, especially of audio and image data but also in medical physics and seismology. The principles of the construction of orthogonal wavelet families are explained in detail, but without too many mathematical proofs. Several popular kinds of wavelets are discussed, like those by Haar, Meyer and Daubechies and their application is explored in a series of computer experiments.