Managing the Messy Mathematics

Abstract

Simulation yields understanding. If you can simulate a phenomenon in nature, then perhaps you understand the parts that make up the natural phenomenon. This chapter begins with the phenomenon of the blackbody emission spectrum represented by Planck's Distribution Law. This is the data. We simulate this spectrum using a one-dimensional particle in a box system. The simulation matches the general shape of the Planck distribution. The rest of the chapter takes you progressively deeper into the equations that produce the wavelengths of the transitions, the spectral line shapes, the temperature dependence of the spectrum, and the intensity of the transitions. These equations all depend upon the wave functions and the interactions of the wave functions with light. This chapter provides a roadmap for spectroscopy by tracing the path from the data to the wave function. All the equations are presented in this chapter, but they are solved in the next three chapters. After reading the Analytical Calculus Chaps. (2–5), you will understand in detail that everything in the spectrum depends upon the wave function, the boundary conditions, the mass of the particle, and the temperature. When confronted with the first application of wave mechanics to the hydrogen atom, Schrödinger exclaimed to his colleague Wilhelm Wein, “If only I knew more mathematics!” (1) This chapter provides a roadmap to help you manage the messy mathematics showing the equations that connect absorption, emission, and scattering spectra to the quantum world through wave mechanics. These equations are solved using two independent approaches—analytical calculus and group theory. A third approach using numerical calculus is developed in a second book which teaches the practical skill of simulating spectroscopic phenomena. The ultimate goal of the text is to produce the math required to compute realistic simulated spectra of the particle in a box and particle on a ring systems. The mathematical results have been implemented in a spectral simulation spreadsheet to illustrate the success of quantum mechanical theory and to develop in the reader an intuition of everything that feeds into the features of the spectra of real molecular systems. The analytical approach is useful for computing the general solutions for spectroscopic transition intensities and energies. From these equations, one can draw stick spectra similar to Schrödinger's (Fig. 1.2) and can readily see the dependence upon mass, temperature, and system size. Inclusion of the line-shape functions allows the reader to experiment with the influence of spectral resolution in the spectral simulations. The group theoretical approach allows a Post-It-Note® computation of the selection rules, which is sufficient in most cases for the assignment of quantum numbers to the experimental spectral transitions.