Should the ground state electron density of an atom go to zero at the origin? I have heard from my professor that the particle density of electrons (in the ground state) of an atom should vanish near the nucleus. Hydrogen is an obvious counter-example. So I am trying to work out what he could have meant? Which quantum phenomenon is he thinking of?
 A: 
Hydrogen is an obvious counter-example. 

Indeed it is. The claim as stated is false.

So I am trying to work out what he could have meant?

You'll have to ask him. There's no way for us to read his mind.
A: As per QM, there are three forces that balance out to keep an electron at a stable energy level:


*

*the electron's potential EM energy keeps it close to the nucleus

*the electron's kinetic energy (momentum) keeps it away from the nucleus

*the HUP keeps it away from the nucleus (in case it should get too close)

In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities[1] asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables or canonically conjugate variables such as position x and momentum p, can be known or, depending on interpretation, to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value.

Now the HUP makes sure that there is a very low probability for the electron to exist inside the nucleus (maybe this is what you are referring to).
