http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation
DeBroglie's matter waves weighing heavy upon his mind, Erwin Schrodinger wanted time to ponder, time to consider all the implications. Schrodinger, an Austrian physicist noted for his work on the physics of strings, took flight to a villa in the Swiss Alps in 1925, leaving his wife behind and gathering a former Viennese girlfriend. What would come of this (presumably) quiet period of reflection and thought would forever change the landscape of physics. Indeed, it would change the way we as a species reckons the universe we live in.
A sort of microscopic solar system, with electrons orbiting about the nucleus like planets to stars -- the Bohr atomic model was proving to be of limited utility. For hydrogen atoms, the agreement between predicted and observed behavior was sterling. However, for atoms with more than one electron -- even helium with only two electrons -- predicted and observed behavior radically diverged. Schrodinger desired to develop a model that agreed with the experimental evidence. What came of that illicit vacation to the Swiss Alps was a model that was not derived from any other, a model that can be called an intuitive guess, a leap of imagination, a model that is astonishingly accurate.
In these pages, we will, for the sake of both brevity and simplicity, only consider the time-independent Schrodinger wave equation in one dimension. We will not consider the full equation in all of its gruesome splendor. The time-independent Schrodinger Wave Equation, which could validly be called Schrodinger's law, is given by the differential equation
where j (x) is the is the wave function, m is mass, 
is Planck's constant divided by 2p, E is the total energy of the particle, and U(x) is the potential energy function of the particle. As when one ingests something disagreeable and the natural reaction is nausea, so too is the natural reaction to this equation. However, comfort may be taken if we consider that acceleration is the second derivative of the position function and, therefore, could be written
As surely as acceleration simplifies to something more palatable, The Schrodinger wave equation must simplify (a little, at least).
To find general solutions to this equation, boundary conditions must be established. The principle conditions that it must adhere to are
1.  as  |
2.  if x is in someplace it is physically impossible to be |
3. j (x) is a continuos function |
4. j (x) is a normalized function |
In these pages, we will, again for brevity and simplicity, consider the case of a particle in a one-dimensional box of ideal rigidity, such that its walls are impenetrable. Let the box have length L. As may be seen in the illustration, the potential
energy function has two states:
1. U(x)=0 for  |
2. U(x)= for x<0 or x>L |
Since it is physically impossible for the particle to be outside of the box, it is the first state that is of interest. Indeed, this simplifies the wave equation considerably, with the term U(x) dropping out. Therefore, the wave equation corresponding to the particle in the box is given by
Before assailing this equation with a display of mathematical acumen, let us ask ourselves what function's second derivative is merely some negative constant -- all of the terms on the right-hand side save j (x) -- multiple of itself? To simplify, let
Therefore, the wave equation becomes
It becomes clear, a trigonometric function like sine or cosine would be a good candidate for j (x). Therefore, our guess for the solutions to the wave equations is
By the first above boundary condition, it is known that
Therefore,
where n=1,2,3,... When the smoke clears, we have that
where A is the function's amplitude. To determine the amplitude, recall the fourth boundary condition, j (x) is a normalized function. Mathematically, this means
In words, this states that the probability of finding the particle somewhere on the x-axis is one or 100%. Waving hands a bit to omit the gory details, this gives
Gasping for breath, we at last have unearthed the solution to the wave equation for the particle of the nth quantum state in the rigid box.
The utility of this solution lies primarily in that the probability of finding the particle at some position x is given by the square of j (x).
The importance of this relationship is best illustrated graphically. Consider a particle in the third quantum state.
It can be seen that there are regions where the probability of finding the particle is zero -- so-called nodes. This is not unique to simply the particle in a rigid box model. It is observed in more sophisticated ones such as the model of an electron orbiting a nucleus...
