348-problem20 - ized to unity times the remaining constant...

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Chem. 540 Instructor: Nancy Makri PROBLEM 20 To understand an important property of the Dirac δ function consider the integral ikx e dk -∞ . Clearly the integrand does not go to zero at infinity so the integral is not well behaved. By expand- ing the exponential into sine and cosine terms, one concludes that this integral must be real valued. Insert a damping factor in the integrand, i.e., consider the following (modified) integral: 2 ikx ak e e dk - -∞ . This new integral is well behaved, so you can evaluate it. Express the result as a function normal-
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Unformatted text preview: ized to unity, times the remaining constant factors. Now imagine making the parameter a smaller and smaller, such that in the limit a → we recover the original integral. You will see that the normalized function in your result is more and more sharply peaked ab o ut x , while the area under this function remains equal to 1. In the limit a → this function becomes infinitely narrow and thus behaves as a function. Therefore establish the following relation: 2 ( ) ikx e dk x πδ ∞-∞ = ∫ ....
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This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.

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