What do $R(r)$ , $R^2(r)$ & $4\pi r^2 R^2(r) $ represent in accordance to schrodinger's wave equation? What do $R(r)$ , $R^2(r)$ & $4\pi r^2 R^2(r) $ represent in accordance to schrodinger's wave equation?
Being a JEE aspirant I haven't been taught to properly interpret the Schrodinger's wave equation. We have just been trained to solve basic questions without proper understanding as we haven't yet learnt the mathematics required to solve this.
Can you explain just the meaning of the terms $R(r)$ , $R^2(r)$ & $4\pi r^2 R^2(r) $ without going too deep into the maths involved in the equation so that I can at least feel the difference ?
 A: The  Schrodinger wave equation is the first quantum mechanical equation discovered, a differential equation where the potential between two quantum particles can be inserted and a solution found.
It is called a wave equation because the main solutions are functions of  sine and cosine  , and the equations were used in classical mechanics to describe from water waves to acoustic waves to seismic waves. The difference with quantum mechanical representations is that the solutions are by postulate related with the probability of measuring the behavior of particles at (x,y,z,t). This is the $Ψ$ in the link.

If the potential energy and the boundary
conditions are spherically symmetric, it is
useful to transform H into spherical
coordinates and seek solutions to
Schrodinger’s equation which can be written
as the product of a radial portion and an
angular portion: $ψ(r, θ, φ) = R(r)Y (θ, φ), or
even R(r)Θ(θ)Φ(φ)$ .

Note that because $ψ$ is a complex function $ψ^*ψ$ is the probability function , a real function.  R can be treated separately as seen here. look at exercise  8.2.8

the quantity R(r)∗R(r) gives the radial probability density
; i.e., the probability density for the electron to be at a point located the distance r from the proton.

.....

When the radial probability density
for every value of r is multiplied by the area of the spherical surface represented by that particular value of r, we get the radial distribution function


The radial distribution function
gives the probability density for an electron to be found anywhere on the surface of a sphere located a distance r from the proton. Since the area of a spherical surface is 4πr2, the radial distribution function
is given by 4πr2R(r)∗R(r).

This link also may help.

When the wavefunction, ψ, is squared the result is a number that is directly proportional to the probability of finding and electron at specific coordinate in 3D space. The radial portion of the wavefunction really only tells us if there is high or low probability at various distances from the nucleus (possible radii for the electrons). Multiplying this probability by the area available at that distance will give us the Radial Distribution Function for the given electron. The concentric spherical shells have areas equal to the surface area of a sphere which is 4πr2.

