In the wikipedia article about quantum harmonic oscillator, this image describes the probability density of finding the particle at certain points for the different eigenstates :
I am confused that the probability density is zero at certain points. Does that really mean that it is impossible to measure the particle at those points ? Can we experimentally verify this ?
The wave function of the quantum harmonic oscillator $\psi_n(x)$ for the system at the $n$th energy level can be used to construct a probability density function,
$$\rho_n(x) := |\psi_n(x)|^2$$
such that the probability of finding the particle in an interval, $x \in[a,b]$ is given by,
$$P(a \leq x \leq b; n) = \int_a^b \rho_n(x) \, dx.$$
Thus, it follows that the probability of finding the particle at any one point, $P(x = x_0)$ is in fact zero since it has measure zero. Thus the probability of finding it at the nodes is indeed zero, but it is a meaningless statement in the sense that the probability vanishes for any individual point.
It should be stressed that, in general, for a measurable space $(\Omega, \mathcal F)$, it is not true that $P$ vanishes for continuous variables evaluated at a single $\omega \in \Omega$ as it depends on the choice of dominating measure, as explained in the statistics SE.
