Quantum Mechanics PHYSICS 137A
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This 1 page Class Notes was uploaded by Marjorie Hahn on Thursday October 22, 2015. The Class Notes belongs to PHYSICS 137A at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/226694/physics-137a-university-of-california-berkeley in Physics 2 at University of California - Berkeley.
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Date Created: 10/22/15
Physics 137A Properties of Dirac delta functions7 Dirac delta functions arent really functions7 they are functionals but this distinction won7t bother us for this course We can safely think of them as the limiting case of certain functionsL without any adverse consequences lntuitively the Dirac 6 function is a very high7 very narrowly peaked function with unit area We may de ne it by the condition f 7wf for any function In particular plugging the function fy E 1 into Eq 1 shows that the 6 function has unit area We can write schematically f96596 y fy596 y 2 and 596 54 3 and 1 6az 76x a gt 0 4 a We de ne derivatives of the 6 function using integration by parts so that d6 7 df ma a 7 7 6xdx 5 since the surface terms are 0 One can use these properties to show for example a eampgt uwemwm i ixia 16zaixai716xia 20 159 a59 al i A on V A 00W V The 6 function can be represented as the limit of several common sorts of functions7 for example a Gaussian with 039 a 0 or the limits of eWzz 62 or sinxe7Tz as E a 0 One representation which will be very useful for us is the Fourier Transform of 17 or T dk e 6Q E lim 1M 9 Taco 7T 27139 Note that when x 0 the exponential is 1 and the integral is in nite the function value is very large while if z 31 0 the integral is highly oscillatory and will evaluate to 0 1See eg http mathwor1dwolfram comDeltaFunctionhtm1 for some examples
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