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But qualitatively, this is clearly different from the classical result.It means the ball has many quasi-stable orbitals for example, and it doesn't give a precise prediction that the ball will hit the ground in $\sqrt \, \mathrm$.
The exact classical result is recovered only in the limit $\hslash\to 0$.
If one does consider $\hslash$ with its real value, one would get corrections to the classical result, in term of powers of $\hslash$ (such corrections for an object of mass 1kg are extremely small).
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Problem statement Particles with two different masses m and M are located along a linear harmonic chain of infinite length.
The chain has a force constant k (see the picture below).
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The point is that, quantum mechanically, the initial conditions cannot be a position $x_0$ and momentum $\xi_0$ at a fixed time.
In classical (statistical) mechanics, the initial condition is a probability distribution in the phase space (in the case we are considering, it is a delta distribution centered in the initial condition $(x_0,\xi_0)$).