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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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There is a possibility of tunneling even if you change the potential to a very tall (but not infinite) rectangular with finite width, but it would very much depend on the width and height of the barrier (the area). If the barrier is broad, the exponential in the wave function will have enough time to fall to almost zero before reaching the the other side and the possibility of it getting through will be so small that you can consider it to be zero.
If the area of the potential in a section is infinite, the wave function in that section is zero. You might want to check out problem 2.31 in Griffiths (mine is the second edition)- it's not exactly the same thing, but it will surly help, and it shows the special thing about delta functions is the finite area they have, so making the barrier high the width will be tiny.
The area of the potential comes in when we try to integrate the Schrodinger equation to find $\Delta(\frac{d\psi}{dx})$.

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I'm asking what would happen if we have a infinite potential but with finite width well. Thanks – el psy Congroo Oct 19 '13 at 21:17
You can't get through if it's an infinite potential. The wave function in the region of an infinite potential is zero, even if the width is finite. – Fatemeh Oct 19 '13 at 22:03
page 76 Griffiths 2nd edition, it says it can climb over delta function. Do you think a wave packet can pass through a Infinitely high and infinitesimally thin wall? As you said the area is finite, I would take area of a diract function as 1? triangle dphi/dx is the area – el psy Congroo Oct 19 '13 at 23:10
It says "the particle is just as likely to pass through the barrier as to cross over the well "(not wall). It's not talking about climbing the barrier, it just says that it can go through the barrier with the same probability as a particle can cross over a well. I can show you it can't go through an infinite barrier with infinite area. – Fatemeh Oct 20 '13 at 18:38

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