(I am referring to section 3.1 in Burkhardt's "Foundations of Quantum Physics", if you happen to have the book.) In that book it's pointed out that the apparent contradiction between the pdf's of the QM ground state solution to the harmoinc oscillator with its classical conterpart (at the same energy = ℏω/2) is due to the comparison setup itself; we should compare the corresponding states rather than the corresponding energy. so QM's groud state with E = ℏω/2 and the classical E = 0. But why? shouldnt energy defines the system uniquely so when we make comparison we use the same energy?