Why electron can not be found at some node locations in the infinite potential well? Consider electron in an infinite potential well, studied in quantum mechanics.
Position probability density of the electron is 
$$ P_n(x)=\left(\frac{2}{L}\right)\sin^2\left(\frac{n\pi x}{L}\right)$$
where $0\leq x\leq L $ and $L$ is length of the box.
So for $n>1$, probability density & hence probability of finding location of the electron at certain $x$ is $0$. The electron moves from left to right & right to left between the walls of the well. So Mathematics says that electron can not be found at certain $x$ node locations within the box; which is very strange. But is there any experimental evidence for this? While crossing these special $x$ node locations, as if electron disappears from the box. This is very absurd.
My question is: Is this a just mathematical result (without any reality) or a physical reality/actuality?
 A: You are imagining the particle in the well as a classical system i.e. a point particle moving to and fro in the well. However this is not a good description of the system. A quantum particle does not have a position. By this I mean that it is meaningless to ask what the position of the particle is because position, in the sense we normally use the term, is an emergent property of a macroscopic system.
Instead the particle has a probability distribution that tells us the probability of finding the particle in some infinitesimal volume element. This probability distribution falls to zero at some places, but that doesn't mean the particle disappears as it passes through those places.
As for experimental evidence, the obvious example of a particle in a (finite) potential well is an electron in a hydrogen atom. The atomic orbitals have nodal planes where the probability distribution falls to zero just like the hypothetical particle in an infinite potential well. These nodal planes were directly imaged in 2013 by Stodolna et al. A description of the paper can be found here.
