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422 chapter 150.6QMCAnalytic| (z)| 20.40.200 2 4 6zFigure 15.10 The Airy function squared (continuous line) and the Quantum Monte Carlosolution |ψ 0 (q)| 2 (dashed line) after a million trajectories.where N n is a normalization constant and z E is the scaled value of the energy. Theboundary condition ψ(0)=0implies thatψ(0) = N E Ai(−z E )=0, (15.79)which means that the allowed energies of the system are discrete and correspondto the zeros z n of the Airy functions [Pres 00] at negative argument. To simplify thecalculation, we take ¯h =1, g =2, and m = 1 2 , which leads to z = x and z E = E.The time-dependent solution for the quantum bouncer is constructed by formingthe infinite sum over all the discrete eigenstates, each with a time dependenceappropriate to its energy:ψ(z,t)=∞∑C n N n Ai(z − z n )e −iEn t/¯h , (15.80)n=1where the C n ’s are constants.Figure 15.10 shows the results of solving for the quantum bouncer’s ground-stateprobability |ψ 0 (z)| 2 using Feynman’s path integration. The time increment dt andthe total time t were selected by trial and error in such a way as to make |ψ(0)| 2 ≃ 0(the boundary condition). To account for the fact that the potential is infinite fornegative x values, we selected trajectories that have positive x values over all theirlinks. This incorporates the fact that the particle can never penetrate the floor. Ourprogram is given in Listing 15.4, and it yields the results in Figure 15.10 after using10 6 trajectories and a time step ε = dτ =0.05. Both results were normalized via a−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 422