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NRQM10_1

NRQM10_1 - Physics 216 Problem Set 1 Spring 2010 DUE...

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Physics 216 Problem Set 1 Spring 2010 DUE: TUESDAY, APRIL 13, 2010 1. Prove that ( x , t | T [ X ( t 1 ) X ( t 2 ) · · · X ( t n )] | x , t ) = integraldisplay D [ x ( t )] x ( t 1 ) x ( t 2 ) · · · x ( t n ) e iS [ x ( t )] / planckover2pi1 , where T is the time-ordered product symbol, S [ x ( t )] is the action [which depends on the path x ( t )], X ( t ) is the position operator in the Heisenberg picture, and x ( t ) is the eigenvalue of X ( t ) when acting on the position eigenstate | x , t ) . Assume that t t i and t t i for all i = 1 , 2 , . . . , n . 2. Using the path integral technique, compute the propagator for a particle in a linear potential, where the corresponding Lagrangian is given by: L ( x, ˙ x ) = 1 2 m ˙ x 2 + fx . 3. You are the creator of a wonderful theory that uniquely predicts the quark- antiquark potential to be: V ( r ) = a | vector r | planckover2pi1 c , (1) with a = 0 . 24 GeV 2 . However, before you can claim The Prize, there is a small formality: a comparison between theory and experiment. (a) Using the potential specified in eq. (1), calculate the energy differences, 1 Δ E n E ns E 1 s , for a few lowest lying s -states ( = 0) of the bound state of a charmed quark and charmed antiquark, called charmonium ( c ¯ c ), and the lowest lying

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NRQM10_1 - Physics 216 Problem Set 1 Spring 2010 DUE...

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