# ps01soln - Physics 651 - Problem Set 1 Solutions 1....

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Physics 651 - Problem Set 1 Solutions 1. Throughout this homework, we will encounter many gaussian integrals. There is only one integral to remember and there it is: Z dxe 1 2 ax 2 + Jx = r 2 π a e - J 2 a (1) This could be applied even when there is an imaginary number in the exponential. You just absorb the factor in a redeﬁnition of ”a” or ”J” (technically, one must do the integral in the complex plane and then show that the answer can be obtained by absorbing the factor of i in a constant). To solve the path integral, we expand x i and J i in terms of orthonormal eigenvectors of K, φ n i with eigenvalues λ n : x i = X n φ n i . 1 Z N Y i =1 dx i e - 1 2 xKx + xJ = Z N Y i =1 det( φ n i ) | {z } =1 dX n e - 1 2 n λ n X 2 n + X n J n = N Y i Z dX n e - 1 2 λ n X 2 n + X n J n = N Y i r 2 π λ n e J 2 n 2 λn = (2 π ) N/ 2 (det K ) 1 2 e 1 2 JK - 1 J 2. To calculate U ( x a ,x b ,T ) for a free non-relativistic particle using traditional me- thods, take its deﬁnition and insert complete sets of orthonormal states: < x b | e - iHT/ ~ | x b > = Z dpdp 0 < x b | p >< p | e - i ˆ p 2 T 2 m ~ | p 0 >< p 0 | x a > = Z dpdp 0 e ix b p/ ~ 2 π ~ e - i p 0 2 T 2 m ~ δ ( p - p 0 ) e - ix a p 0 / ~ 2 π ~ = 1 2 π ~ Z dpe - i p 2 T 2 m ~ e ip ( x b - x a ) / ~ = r - im 2 π ~ T exp[ i m ( x b - x a ) 2 2 ~ T ] 1 I always use the Einstein’s convention that repeated indices are summed 1

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Now, lets calulate the same quantity using path integral method < x b | e - iHT/ ~ | x a > = Z x b x a D q exp( - im 2 ~ Z T 0 ˙ q 2 dt ) A priori, we don’t know what we should use for the measure and so we will leave it as a constant C and then by comparing with our previous calculations we will be able to determine it. To perform the path integral we discretize q into N parts. The
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## This note was uploaded on 09/28/2008 for the course PHYS 651 taught by Professor Tye,henry during the Fall '03 term at Cornell University (Engineering School).

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ps01soln - Physics 651 - Problem Set 1 Solutions 1....

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