[overlord1647]

radial density function(ψ²) states that there is maximum electron density for '1s' orbital at the nucleus.But radial probability function(ψ².4.π.r²) gives maximum probability at 52.9 pm.Can electron exist at nucleus??What these two plots really signify and what is the difference between two??

[AdiDex]

It's the rule , atleast you have to show your attempts .

What these two plots really signify and what is the difference between two??

What's the major difference you see between probability density and probability curves ?
It's answer is in the definition , read it again from your textbook .

Can electron exist at nucleus?

http://www.chemicalforums.com/index.php?topic=88369.0

The answer  given by Enthalpy . Read it.

[Irlanur]

I think the mystery resolves if you think about hot to integrate spherical coordinates. (dxdydz=sin(theta)r^2 dphi dtheta dr)
 
The probability DENSITY might be highest at the point of the nucleus, but the position of the nucleus is also in infinitesimally small volume.

[Enthalpy]

One density is per volume unit, the other per radius unit.

Multiplying a small radius increment by the sphere's area at that volume, you get a volume increment. This tells that the density per radius unit is zero even though the spherical orbitals have non-zero density per volume unit at the nucleus, and also that the both maximums don't need to coincide, as the r² factor can shift the maximum towards bigger radii.

It is the same kind of computation, in the k_xk_yk_z space of wave vectors, for the density of eigenstates per unit of |k|² (hence per unit of E), knowing the density per unit of k_x×k_y×k_z "volume" of wave vectors, for acoustics, electromagnetism, semiconductors.

Only spherical orbitals can (and do) have non-zero density per volume unit at the nucleus. The orbital angular momentum being (with some constants) the number of phase turns of the wave function (or better, its part without e^iωt) over a geometric turn around the nucleus, all orbitals with non-zero angular momentum have several phases at the nucleus, which only the number zero fits. As opposed, spherical orbitals have one single phase at the nucleus, and can be non-zero there.

Electron capture (a radioactive decay mode) needs some electron density at the nucleus, so only electrons from spherical orbitals are candidates, 1s more so than 2s 3s... In the case of ⁷Be, chemical bonds influence the electronic density at the nucleus (from molecular orbitals resulting from 2s) hence the decay half-time
https://en.wikipedia.org/wiki/Electron_capture#Reaction_details