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thermodynamic simulations & feynman quantum path integration 417It is not surprising then that the sum over paths in Green’s function has each pathweighted by the Boltzmann factor P = e −εE usually associated with thermodynamics.We make the connection complete by identifying the temperature with theinverse time step:P = e −εE = e −E/k BT⇒ k B T = 1 ε ≡ ¯h ε . (15.69)Consequently, the ε → 0 limit, which makes time continuous, is a “hightemperature”limit. The τ →∞limit, which is required to project the ground-statewave function, means that we must integrate over a path that is long in imaginarytime, that is, long compared to a typical time ¯h/ ∆E. Just as our simulation of theIsing model in Unit I required us to wait a long time while the system equilibrated,so the present simulation requires us to wait a long time so that all but the groundstatewave function has decayed. Alas, this is the solution to our problem of findingthe ground-state wave function.To summarize, we have expressed the Green’s function as a path integral thatrequires integration of the Hamiltonian along paths and a summation over all thepaths (15.66). We evaluate this path integral as the sum over all the trajectories in aspace-time lattice. Each trial path occurs with a probability based on its action, andwe use the Metropolis algorithm to include statistical fluctuation in the links, as ifthey are in thermal equilibrium. This is similar to our work with the Ising model inUnit I, however now, rather than reject or accept a flip in spin based on the changein energy, we reject or accept a change in a link based on the change in energy.The more iterations we let the algorithm run for, the more time the deduced wavefunction has to equilibrate to the ground state.In general, Monte Carlo Green’s function techniques work best if we start with agood guess at the correct answer and have the algorithm calculate variations on ourguess. For the present problem this means that if we start with a path in space-timeclose to the classical trajectory, the algorithm may be expected to do a good jobat simulating the quantum fluctuations about the classical trajectory. However, itdoes not appear to be good at finding the classical trajectory from arbitrary locationsin space-time. We suspect that the latter arises from δS/¯h being so large that theweighting factor exp(δS/¯h) fluctuates wildly (essentially averaging out to zero)and so loses its sensitivity.15.8.2.1 A TIME-SAVING TRICKAs we have formulated the computation, we pick a value of x and perform anexpensive computation of line integrals over all space and time to obtain |ψ 0 (x)| 2at one x. To obtain the wave function at another x, the entire simulation must berepeated from scratch. Rather than go through all that trouble again and again, wewill compute the entire x dependence of the wave function in one fell swoop. Thetrick is to insert a delta function into the probability integral (15.66), thereby fixing−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 417