A problem with a Dirac delta function potential

[arenaninja]

Homework Statement

An ideal particle of energy E is incident upon a rectangular barrier of width $$2a$$ and height $$V_{0}$$. Imagine adjusting the barrier width and height so that it approaches $$V(x)=\alpha \delta(x)$$. What is the relationship between V₀, alpha and a?

Homework Equations

The Attempt at a Solution

I must be thinking of this incorrectly because the only thing that occurs to me is that in this limit,
$$V(x) = \alpha \delta (x), x = 0; V(x) = 0, elsewhere$$
and so
$$V_{0} = V(0) = \alpha \delta (0)$$
My issue is that this doesn't include the width at all (but I'm not sure why it should since the width goes to zero). Any insights are greatly appreciated.

 

[cepheid]

You could think of the Dirac delta function as the limiting case of a square pulse as the width of that pulse goes to zero, but the height goes to infinity (in such a way that the product of the two, which is the area of the pulse, remains constant and is always equal to unity). Does that help? It implies that the area under the scaled Dirac delta function in your problem must be the same as the area under the original square pulse.