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Use of Trotter Theorem in Path Integral Molecular Dynamics

  1. Sep 10, 2014 #1
    I am unable to prove step 8.3 in this proof of the path integral formulation of molecular dynamics
    https://files.nyu.edu/mt33/public/jpc_feat/node11.html [Broken]

    Any help would be much appreciated.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Sep 10, 2014 #2

    DrDu

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    The exponentials containing the U's are clear, I suppose? The potential operators are diagonal in the x basis so you are left with the exponential of the T operator between x(s) and x(x+1). Insert momentum eigenstates. The T operator in the exponent becomes proportional to p^2 and ##<p|x>\propto \exp(ipx)##. So you have to evaluate something like ##\int dp \exp(Cp^2+ip(x(s)-x(s+1)))##. Complete the square and integrate over the shifted p. You get the Gaussian ## \exp(-C'(x(s)-x(s+1))^2)##. You are free to work out all the constants I left open.
     
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