Use of Trotter Theorem in Path Integral Molecular Dynamics

[jelathome]

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

Any help would be much appreciated.

 

[DrDu]

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.