ABSTRACT
CONTENTS 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2 Harmonic Oscillator with Chirp Driving Force . . . . . . . . . . . . . . . . . . . . . 122 4.3 Approximation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Appendix 4.1: Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Appendix 4.2: Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Appendix 4.3: Exact Solution to the Gliding Tone Problem . . . . . . . . . . . . . . . 143 Appendix 4.4: Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Appendix 4.5: Derivation of the Wigner Equation of Motion
for Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Appendix 4.6: Green’s Function for the Wigner Distribution . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
In the field of signal analysis, time-frequency distributions have historically been used as a means of analyzing signals for their time-varying spectra.6,7,11
In physics, however, these distributions have been used to understand the solution of the Schro¨dinger equation, which is a partial differential equation. The idea is to obtain the equation of motion for the Wigner distribution corresponding to the solution of the Schro¨dinger equation. The basic reasons for doing so is that one gains considerable insight into the nature of the solution, and that it leads to new analysis and approximation methods. Wigner, Moyal, Kirkwood, and many others made significant contributions
Methods, and
have been written for both the Wigner distribution and other distributions.
