# Tunnelling through a Dirac potential barrier

I am reading a QM book by Griffiths, which says it is possible for wave particle to tunnel through a barrier formulated by a Dirac function. This function is known to peak at infinity and also infinitesimally narrow. Is there anything special about the Dirac function for this to happen?

If I change the function to a infinitely tall rectangular function with a finite width, would it still be able to tunnel through this barrier

If we try to solve the Time independent Schrodinger equation, I thought the exponential would fall off very quickly since it has a large potential. Note that gamma is a function of Energy and potential. If we use exponential gamma as trial function.

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The area of the potential comes in when we try to integrate the Schrodinger equation to find $\Delta(\frac{d\psi}{dx})$.