Reflection coefficient for delta function potential with small denominator term

[Domnu]

So let's say we have a particle in the delta function potential, $$V = - \alpha \delta(x)$$. I calculated that the reflection coefficient (scattering state) is

$$R = \frac{1}{1 + (2 \hbar^2 E/m\alpha^2)}$$​

Now, clearly, the term $$2 \hbar^2 E/m\alpha^2$$ is very small, as $$\hbar^2$$ has an order of magnitude of $$-68$$. This value is so small that even for classical ordered values for $$E, m,$$ and $$\alpha$$, the reflection coefficient is very close to $$1$$. Is this correct? It seems a bit strange to me that a spike in the potential where a particle's energy is above the potential causes the particle to reflect BACK almost 100% of the time... it seems as if a tank shell going at more than a mile per second goes over a cliff and reflects back...

 

[Avodyne]

The problem is that the potential changes values on a scale that is much smaller (in this case, infinitely small!) than the quantum wavelength of the particle. Classical physics only emerges when the quantum wavelength is much smaller than any classically relevant length scale.