Appendix D: Introduction to Fourier Series, the Fourier Transform, and the Fast Fourier Transform Algorithm

Abstract

Real quantum mechanical systems have the tendency to become mathematically quite complicated and may discourage a novice in the field from pursuing the detailed steps to understand how the mathematical principles apply to physical systems. Thus, a simple scenario is presented here to illustrate the principles of Quantum Mechanics introduced in Section 1.4. The model to be presented is the so-called particle-in-a-box (henceforth referred to as the "PiB") that is an artificial system, yet with wide-ranging analogies to real systems. This model is very instructive, because it shows in detail how the quantum mechanical formalism works in a situation that is sufficiently simple to carry out the calculations step by step. Furthermore, the symmetry (parity) of the PiB wavefunctions is very similar to that of vibrational wavefunctions discussed in Section 1.4. Finally, the concept of transition from one stationary state to another can be demonstrated using the principles of the transition moment introduced in Section 1.5. The PiB model assumes a particle, such as an electron, to be placed into a potential energy well, or confinement shown in Figure A.1. This confinement (the "box") has zero potential energy for 0 ≤ x ≤ L, where L is the length of the box. Outside the box, that is, for x < 0 and for x > L, the potential energy is assumed to be infinite. Thus, once the electron is placed inside the box, it has no chance to escape, and one knows for certain that the electron is in the box. Next, the kinetic and potential energy expression will be defined, which subsequently allows writing the Hamiltonian, or the total energy operator of the system. For any quantum mechanical system, the total energy is written as the sum of the kinetic and potential energies, T and V, respectively: E = T + V (A.1) Modern Vibrational Spectroscopy and Micro-Spectroscopy: Theory, Instrumentation and Biomedical Applications, First Edition. Max Diem.