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498 chapter 18the Taylor expansion,∂ 2 ψ∂x 2 ≃−1 [ψn2 i+1,j + ψi−1,j n − 2ψi,jn ], (18.54)with the corresponding-order expansion of the evolution equation (18.53). Substitutingthe resulting expression for the second derivative into the 2-D timedependentSchrödinger equation results in 2ψ n+1i,j= ψ n−1i,j − 2i [( 4α + 1 2 ∆tV i,j)ψni,j − α ( ψ n i+1,j + ψ n i−1,j + ψ n i,j+1 + ψ n i,j−1)],where α =∆t/2(∆x) 2 . We convert this complex equations to coupled real equationsby substituting in the wave function ψ = R + iI,R n+1i,j = R n−1i,j +2 [( 4α + 1 2 ∆tV i,j)Ini,j − α ( Ii+1,j n + Ii−1,j n + Ii,j+1 n + Ii,j−1)] n ,I n+1i,j = I n−1i,j − 2 [( 4α + 1 2 ∆tV i,j)Rni,j + α ( Ri+1,j n + Ri−1,j n + Ri,j+1 n + Ri,j−1)] n .This is the algorithm we use to integrate the 2-D Schrödinger equation. To determinethe probability, we use the same expression (18.47) used in 1-D.18.8.0.1 EXPLORATION: A BOUND AND DIFFRACTED 2-D PACKET1. Determine the motion of a 2-D Gaussian wave packet within a 2-D harmonicoscillator potential:V (x, y)=0.3(x 2 + y 2 ), −9.0 ≤ x ≤ 9.0, −9.0 ≤ y ≤ 9.0. (18.55)2. Center the initial wave packet at (x, y)=(3.0, −3) and give it momentum(k 0x ,k 0y )=(3.0, 1.5).3. Young’s single-slit experiment has a wave passing through a small slit withthe transmitted wave showing interference effects. In quantum mechanics,where we represent a particle by a wave packet, this means that an interferencepattern should be formed when a particle passes through a small slit.Pass a Gaussian wave packet of width 3 through a slit of width 5 (Figure 18.10)and look for the resultant quantum interference.2 For reference sake, note that the constants in the equation change as the dimension ofthe equation changes; that is, there will be different constants for the 3-D equation, andtherefore our constants are different from the references!−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 498