An important aspect of understanding of one's position is to appreciate
where one came from. This is also true of modern science. Though admittedly
more complex, it is possible to follow a thread through the last four centuries
up to the birth of modern quantum mechanics. Rene Descartes is often credited
as the Father of Modern Mathematics. While a mercenary soldier, he experienced
a dream one night that triggered in his mind the idea of modern algebra.
(Some nightmare!) It is helpful to follow a short
tour from Descartes through to Newton, Bernoulli, Euler, Hamilton,
Maxwell, Einstein, Schrodinger, Heisenberg, and Dirac. It is revealing
to see how the mathematical ideas of one generation opened new avenues
for the next.
As mathematical physics reached a plateau one century ago, new vistas were being suggested as new experiments produced unexpected results. Spectroscopy and heat capacities, black body radiation and the photoelectric effect, all presented simple results that stood in stark contrast to accepted theories of nature. Another brief tour outlines these results.
With Planck's bold suggestion of the quantum nature of light and Einstein's
equally impressive explanation of the photoelectric effect in quantum terms,
Bohr, who was working with Rutherford on the nuclear scattering experiments,
took another step and developed a mathematical framework to explain the
spectroscopy of the hydrogen atom. He invoked the ideas of standing waves
to justify the quantal stability of certain levels in the atom and this
simple process provided such startlingly good results that the world was
hooked on the quantum. This came to be known as the "Old Quantum Theory"
since its comparatively excellent results were nevertheless limited in
their scope. For another decade, extensions to this quantum theory proved
unsatisfying.
It was Prince Louis de Broglie who challenged the scientific world with his Ph.D. thesis wherein he suggested that if light could behave as particles, then matter could also behave as waves. He wrote down his famous de Broglie relation which was taken up by many people - often to try to prove that it was wrong. One of these was Erwin Schrödinger who soon recognized the remarkable nature of the idea and wrote down his now famous wave equation for matter. Werner Heisenberg developed a completely different pathway but ultimately identical in results following his matrix mechanics methods. When Dirac wrote down the relativistic wave equation, modern quantum theory had arrived. Another short tour is helpful to put these historical ideas into perspective.
A brief introduction to the mathematical
basics of quantum mechanics and a look at a number of simple,
model systems is effective preparation for the application of quantum
ideas to atomic and molecular systems.