Numerical Solution of the Hydrogen Atom
Introduction"In the years before 1920, the picture of space as a three-dimensional space, and of time as a separate thing, was changed by Einstein, first into a combination which we call space-time, and then still further into a curved space-time to reflect gravitation. Then it was also found that the rules for the motion of particles were incorrect. The mechanical rules of inertia and forces are wrong--Newton's laws are wrong--in the world of atoms. Instead it was found that things on a small scale behave nothing like things on a large scale. " The Feynman Lectures on Physics For all the suspension of disbelief that quantum mechanics requires, students approaching this subject should at all times remember that quantum mechanics is possibly sciences greatest achievement in empirical science. The foundations of quantum mechanics are not in theory, but in facts that are observable and repeatable. All particles interact in a way that as of yet we can only describe as wave interaction. The interaction of light with matter occurs in a manner that as of yet we can only describe with discrete energy levels. The quantization of energy levels in hydrogen is observed clearly in hydrogen spectra, and can be seen by as simple a process as exciting hydrogen gas in a glass tube by running current through either end, and observing the light produced through a refraction grating. The hydrogen atom allows us to address our understanding of quantum mechanics at all levels: empirical through the observed spectra of hydrogen gas, theoretical through one of the rare cases in which Schrodinger's equation can be solved exactly, and computational by methods which will be the building blocks of our understanding of large molecules. A numerical solution can be verified by comparing to the theoretical solution, and can be validated by comparing to empirical data. This module will begin with a discussion of the observations that led to quantum theory, through introductory exercises to practice skills required in the solution of the hydrogen atom, and culminating in solving for and visualizing the orbitals of the lowest energy states of a single electron hydrogen atom. |