Hi Awesumsingh, welcome here!

Schrödinger's equation tells how electrons (sometimes others) behave. It contains both the wave nature of electrons and the equivalent of d

^{2}x/dt

^{2}=qE/m.

Variants depend on if one puts importance on relativistic corrections, influences through the magnetic moment... which get included in the hamiltonian H or the mass.

The Hamiltonian pre-existed quantum mechanics. It relates the energy of a particle in some condition with its tendency to let the condition evolve over time. In classical mechanics, it told that a particle accelerates towards a location or condition where it has a smaller energy. QM generalizes it because a wave has a volume, so the electron exists also at locations less favourable - tunnel effect is nothing more.

The Laplacian is just the one used for any wave: acoustic, electromagnetic... and here an electron. Without potential energy, the electron propagates as any free wave does; with a potential, the electron concentrates where the energy is more favourable, and is almost always trapped near a nucleus, on Earth.

Being waves, electrons have no point-like positions. At most, one can compute a probability to find one in a volume around some position - and observed only by measurement means that are more accurate than an atom's size. And since electrons are identical, the experiment can't usually even tell which one was detected. The expression "electron cloud" reminds both.

The radial probability function is a probability to detect an electron in a spherical volume of dR thickness as a function of the distance. It differs from Ψ

^{2} because the spherical volume increases as R

^{2}, so for instance the 1s orbital decreases over the distance, but the probability in a thin spherical volume first increases before decreasing.

http://www.chemistry.mcmaster.ca/esam/Chapter_3/section_2.htmlAnd anyway, only s orbitals depend only on the distance and not on the direction, so this "radial probability function" gets less simple for other orbitals.

p, d, f... orbitals depend on the direction and have angular nodes in addition to radial ones. Best meditate the nice drawings there:

(keep this precious address)

http://winter.group.shef.ac.uk/orbitron/click on 1s, 2p, 4d... in the left panel, the nodal surfaces are the transitions between red and blue, observe that some are radial and others angular. The number of angular nodal surfaces tells the maximum value of the orbital angular momentum for an orbital.