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thermodynamic simulations & feynman quantum path integration 419highest for the particle to be near its turning points where its velocity vanishes).However, when t b − t a equals the longer time 20T , the system has had enough timeto decay to its ground state and the wave function looks like the expected Gaussiandistribution. In either case (Figure 15.8 right), the trajectory through space-timefluctuates about the classical trajectory. This fluctuation is a consequence of theMetropolis algorithm occasionally going uphill in its search; if you modify theprogram so that searches go only downhill, the space-time trajectory will be a verysmooth trigonometric function (the classical trajectory), but the wave function,which is a measure of the fluctuations about the classical trajectory, will vanish!The explicit steps of the calculation are1. Construct a grid of N time steps of length ε (Figure 15.9). Start at t =0andextend to time τ = Nε[this means N time intervals and (N +1)lattice pointsin time]. Note that time always increases monotonically along a path.2. Construct a grid of M space points separated by steps of size δ. Use a range of xvalues several time larger than the characteristic size or range of the potentialbeing used and start with M ≃ N.3. When calculating the wave function, any x or t value falling between latticepoints should be assigned to the closest lattice point.4. Associate a position x j with each time τ j , subject to the boundary conditionsthat the initial and final positions always remain the same, x N = x 0 = x.5. Choose a path of straight-line links connecting the lattice points correspondingto the classical trajectory. Observe that the x values for the links of the pathmay have values that increase, decrease, or remain unchanged (in contrast totime, which always increases).6. Evaluate the energy E by summing the kinetic and potential energies for eachlink of the path starting at j =0:E(x 0 ,x 1 ,...,x N ) ≃N∑j=1[m2( ) 2 ( ) ]xj − x j−1 xj + x j−1+ V. (15.73)ε27. Begin a sequence of repetitive steps in which a random position x j associatedwith time t j is changed to the position x ′ j (point C in Figure 15.9). This changestwo links in the path.8. For the coordinate that is changed, use the Metropolis algorithm to weigh thechange with the Boltzmann factor.9. For each lattice point, establish a running sum that represents the value of thewave function squared at that point.10. After each single-link change (or decision not to change), increase the runningsum for the new x value by 1. After a sufficiently long running time, the sumdivided by the number of steps is the simulated value for |ψ(x j )| 2 at eachlattice point x j .11. Repeat the entire link-changing simulation starting with a different seed. Theaverage wave function from a number of intermediate-length runs is betterthan that from one very long run.−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 419