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Unformatted text preview: Physics 651 Problem Set 4 (due 9/29/03) Read Chapter 2. 1. Suppose a real function f ( x ) has a unique global minimum at x = 0. Define the integral I ( ) = Z- dx exp- 1 f ( x ) . a) Using a Taylor expansion of f ( x ) in the exponent, show that for 0 the function I ( ) obeys the asymptotic expansion I ( ) = e- f / v u u t 2 f (2) 1 + 5 24 ( f (3) ) 2 ( f (2) ) 3- 3 24 f (4) ( f (2) ) 2 + O ( 2 ) , where f f (0), and f ( n ) denotes the n th derivative of f ( x ) evaluated at x = 0. (Hint: Reduce the answer to Gaussian integrals of the form R dx x n e- ax 2 . Explain why this is possible.) This approximation technique is called the method of steepest descent. b) Repeat the above derivation with the replacement i (method of stationary phase), and explain why the expansions are still valid in this case. c) Recall the quantum-mechanical amplitude for a free particle to propagate from x to x is U ( t ) = h x | e- iHt | x i . Using the relativistic expression for the energy, H = E =...
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