Why do quantum and classical particles both transmit through a potential step when E > V?

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Consider a particle moving in 1 dimension and encounters a potential step.

Suppose that the total energy of the particle is higher than the potential barrier height, E > V.

Wouldn't a classical mechanical particle, traveling at velocity v, break through the barrier? And wouldn't a quantum mechanical particle also break through the barrier? I see no difference between quantum and classical behavior in this particular case.

:confused:

(This problem is different than that of quantum tunneling, where E < V and the classical particle reflects back, but the quantum particle "tunnels" through the other side.)
 

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Nevertherless, the wave function is different in regions of higher potentials...

Its amplitude is lesser, meaning there is a lesser chance of observing the particle in a region of higher potential. In classical mechanics, the probability of finding the particle anywhere where its equation of motion allows it to be is the same. So there IS a difference btw what QM is saying about the potential step situation and what CM is saying.
 
Classically, the particle always makes it through (or over) the barrier in this case. Quantum-mechanically, there is a probability that the particle is reflected instead. As the particle's energy increases, the probability of reflection decreases.

Somewhat similarly, water waves are partially reflected by an obstacle that lies just underneath the surface.