Why is classical physics not valid for a harmonic oscillator in its lowest energy state? I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum validity for an oscillator at lowest energy.

 
 A: I assume you are familiar with the concepts of eigenstates and eigenfunctions of a system in Quantum Physics. If not, I suggest you take the time to go back to the beginning of that same chapter of Eisberg-Resnick to gain a deeper understanding.
If one studies a harmonic oscillator system by means of Quantum Mechanics, they will find that the energy associated with the n-th eigenstate (let me regard the lowest energy one as the 0th) is
$\ E_n = ( n+ \frac{1}{2}) h \nu$
where $\nu$ is the frequency at which the system is oscillating.
So there is no possible energy level below $\frac{h \nu}{2}$ for a quantum harmonic oscillator, and to stress that peculiar phenomenon this value is called the zero-point energy of the oscillator. Obviously if you study the same system by a classical Physics point of view, this value can be neglected no matter how high the frequency (remember, $h$ is of the order of magnitude of $10^{-34} J s$) so the lowest possible energy becomes zero.
An important property of quantum systems is the Correspondence Principle which states that "the behavior of systems described by the theory of quantum mechanics [...] reproduces classical physics in the limit of large quantum numbers". In this case n is our quantum number: if n were large, we would be allowed to regard the oscillator as classical. But when you are dealing with small values of n, as certainly $n=0$ is, you cannot by any means resort to classical Physics.
This is why the book says: "When the oscillator is in its lower energy state we are very far from the range of validity of classical physics".
