From Schrodinger wave equation, probability functions for electrons orbits about nuclei can be developed.  These are electron shells called probability densities or orbitals, clouds of probability as to where the electron might be or where it is forbidden to be.  This is demonstrated in the above illustration, which shows the 2P_x orbital and a probability wave corresponding to ian electrons likelihood of being found at some x-position.  However, these are not hard shells, but rather soft shells of where a electron is eighty or ninety percent likely to be.  Among the thought-provoking implications this brings is the possibility, however unlikely, that some given electron orbiting some given nucleus -- say one in the eye you are using to read this -- is some great distance away: the other side of the Planet, the outer edges of our solar system, some distant spiral arm of the Milky Way Galaxy...

     No longer do electrons emulate the heavens, with electrons playing the planet to the nucleus' star.  The Bohr model gave way to a new order in the microscopic realm (as seen in the illustration).  A new order not of deterministic electron orbits that can by predicted as surely as the hour and minute hands of a clock.  A new order of probabilities.  A new order were the position of an electron with respect to the nucleus is given by troubling, blurry clouds of the probability, devoid of deterministic certainty, called probability densities.  A new order called quantum physics.

     The hand of Schrodinger and other countless luminaries has reached across time to touch virtually every field of science: physics, chemistry, biology...  Indeed, it has quietly and subtly crept into popular culture.  The word quantum inspires an almost mystical reverence from some.  Certainly no high-brow literature is complete without some cryptic reference to it, some attempt to connect the puzzling dichotomy of the subatomic world to our everyday existence.  But like Einstein puzzling over light quanta for many years to no avail, we are all left only to gratefully ponder, for the quantum world is inherently counterintuitive.  In the words of Richard Feynman,
 

Acknowledgments

     In writing this paper, both Dr. David Mills and David Arnold were invaluable.  Dr. Mills guided me to appropriate resources concerning the Schrodinger wave equation and patiently answered my questions about quantum physics.  And Dave Arnold allowed me the liberty to stray a little bit from the criteria of this assignment for the differential equations class.  The result, I would like to think, is well worth it.  Additionally, the animated GIFs used to illustrate several key points are from the Saunders Interactive Chemistry CD-ROM.  However, in that work they were quicktime movies, which I edited for content and transformed into animated GIFs for this paper.

References

     Feynman, Richard P.  Six Easy Peices: The Feynman Lectures on Physics.  New York, New York: Addison-Wesley, 1994.
 
     Knight, Randall D.  Physics: A Contemporary Perspective.  Preliminary Edition.  New York, New York: Addison-Wesley, 1997.

     Kotz, John C. and Vininng, William J.  Saunders Interactive Chemistry CD-ROM.  CD-ROM.  Harcourt Brace & Co., 1996.

     Kotz, John C. and Treichel, Paul.  Chemistry and Chemical Reactivity.  Third Edition.  Harcourt Brace & Co., 1996.

     Tipler, Paul.  Physics for Scientists and Engineers Volume II.  Third Edition.  New York, New York: Worth Publishing, 1991.