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:
[− |
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:
[− |
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: