Probability of finding a particle at x=0 for a simple HO

Why do you think it should be zero at x = 0?
Also, just to be pedantic, it is not right to speak of the probability finding a particle at a particular position, because position is a continuous observable. You probably mean the probability per unit length of finding the particle about x = 0.

 

In fact, it cannot be zero - there cannot be any nodes in the probability distribution for a particle in the ground state:
http://scitation.aip.org/content/aapt/journal/ajp/70/8/10.1119/1.1479742

 

IS the ground state explicitly mentioned in the exercise ?

 

Well, then the probability density follows from the wave function. You've done that already in post 1. Seems the best answer to me.
Happily sitting there with a minimum energy that still satisfies the uncertainty principle.

Asking for the probability at x=0 is unphysical :) (post #2)
Expectation value for x in ground state is 0, though.

Re infinite well: if it's a square well, continuity of the wave function forces ##\Psi = 0## at the walls.

 

This probably strays away from the original question already, but since we're on the topic of wavefunction "nodes", the vanishing of the wavefunction at the boundaries of an infinite square well potential are not considered as "nodes".

In the wavefunction picture, a node occurs when the wavefunction "changes sign" and crosses the axis. In terms of the probability density picture, a node would manifest itself as a stationary point (one that touches the axis, of course). There are some interesting theorems concerning the number of nodes and the level of the energy eigenfunctions if you like to investigate further.