The Schroedinger Equation

The fundamental principles in the form of main equations lead us to the following Schroedinger equation. One can (and surely must) try to draw conclusions using heuristic way of thinking and then try to compare a number of solutions of "invented" equation with experimental measurements. You can find one of the derivations of Schroedinger equation here.

We can write down the following differential equation for probability amplitude when process doesn't depend on time:

time independent Schroedinger equation: [− h²/2m∂²/∂x² + V(x)]y = E y

where V(x) is the potential that acts upon particlet. If we substitute p2/2m (that is directly corresponds to the kinetic energy) for (−h²/2m∂²/∂x²) it's immediately clear that we have the energy conservation law here. And if one will also substitute E for ih∂/∂t then it'll be the following

time dependent Schroedinger equation: [− h²/2m∂²/∂x² + V]y = ih ∂/∂ty

We can also obtain the three dimensional (x,y,z) equation, if we substitute ∂²/∂x² for ∂²/∂x²+∂²/∂y²+∂²/∂z². Erwin Schroedinger was the first scientist who solved this equation for hydrogen atom (as three dimensional problem). All experimental verifications show that Schroedinger equation fully holds true in non-relativistic quantum mechanics. Since all chemical processes run with a speed slower than speed of light then one can say that Schroedinger equation for chemist is a torch of truth which gives light to a dark road of chemical interactions between atoms and molecules.
The wavefunction y as it has been mentioned earlier has no direct physical sense. Since the probability to detect particle somewhere in a space is equal to 1 then we can write down the normalization condition:

-¥ò+¥|y(x,y,z)|2dx dy dz = 1

The expression |y(x,y,z)|2dxdydz shows the probability P(x,y,z) to detect particle in a place with coordinates (x,y,z) in a volume unit dV = dxdydz. P(x,y,z) unlike y(x,y,z) has direct physical meaning because the probability can be measured in experiments.

Wavefunction y requirements:

1. The Schroedinger equation has the second derivative that's why the second derivative should exist or y continuous and
2. ¶y/∂x , i.e. the first derivative should also be continuous so far the potential has "reasonable" meanings. (for instance, the 1/r-potential when r=0)
3. And finally according to simple physical reasons one can suppose y to be unambiguous and never unlimited (|y|2dxdydz) < ¥).

The probability amplitude y describes considered system completely and we can obtain all necessary information about our system using it.