S ECTION T ITLE Numerical Solution of the Stationary State Schrödinger Equation Using Transparent Boundary Conditions To solve the discrete version of the stationary state Schrödinger equation to Numerov accuracy, the author uses boundary conditions at the limits of the computational domain that mimic an interval of infinite extent. He also describes methods for finding particle energies, scattering coefficients, and partial-wave phase shifts. igenvalue problems in the form of dif- 2 d2ψ ferential equations have widespread ap- −+E Vx()ψψ () x = E () x, 2m 2 EE plication in all fields of science and dx engineering. Their numerical study re- Equires replacing the continuum formulation with a is an example of an eigenvalue problem. The coordi- discrete version, then solving the resulting model nate x extends from –¥ to +¥Ͻ and acceptable wave- ␷ iteratively using appropriate start values. How to forms E(x) must be continuous bounded functions of choose start values in each individual case is a mat- x, which are conditions that generally occur only for ter of some controversy. special values of the particle energy E. The eigenval- In this article, I advocate using discrete trans- ues are these allowed energies, and the corresponding ␷ parent boundary conditions (DTBCs), a technique eigenfunctions are the stationary states E(x). that has proven successful in other contexts. Im- For simplicity, we can recast the problem as plementing DTBCs for eigenvalue problems typi- cally involves an approximation—the slowly dy2 varying approximation—that I present here along +=gxyx()() 0 .(1) 2 with suitable criteria for its validity. As the exam- dx ples make clear, this approach isn’t limited to just finding eigenvalues but also can be adapted to cal- The stationary state Schrödinger equation is re- ␷ culate other properties of physical interest. covered with the obvious identifications y(x) = E(x) and gx()=−2 mE [ V ()]/ x 2. The discrete ver- ® º Background sion of this equation follows by writing x xn nh ® º Schrödinger’s equation for the stationary states of and f (x) f (xn) fn for any function f (x), where n a particle with mass m in a 1D potential V(x), is an integer and h is the size of the spatial grid. The procedure would be straightforward except for the second derivative, which requires some approxi- 1521-9615/06/$20.00 © 2006 IEEE mation. To this end, note that Copublished by the IEEE CS and the AIP ⌬2 ⌬ ⌬ CURT A. MOYER fn–1 = ( fn–1) = f (xn + h) + f (xn – h) – 2f (xn), University of North Carolina, Wilmington ⌬ º where fn–1 fn – fn–1 symbolizes the usual forward 50 COMPUTING IN SCIENCE & ENGINEERING difference operation on any function f (x). Expand- accuracy by using the exact solution function y(x0) ¢ ing the first two terms on the right in a Taylor se- and its slope y (x0) at a single endpoint x0. But those ries gives values, too, are often elusive. Here, I advocate a different approach: instead of ⌬2 2 ¢¢ 4 (iv) 6 fn–1 = h f (xn) + (h /12)f (xn) + O(h ). focusing on the exact solution y(x), we can recover start values from the asymptotic solutions to the ⌬2 » 2 ¢¢ The simplest approximation fn–1 h f (xn) discrete equation for yn. This tactical shift amounts gives the central difference approximation to the to fully embracing the discrete formulation for all second derivative and yields a discrete version of values of n; in effect, we impose boundary condi- Equation 1 that is accurate to O(h3): tions at the limits of the computational domain that mimic an interval of infinite extent. Researchers pi- ⌬2 2 yn–1 + h gnyn = 0. (2) oneered using such DTBCs in connection with solving the time-dependent Schrödinger equa- The Numerov method improves on this by us- tion;3–5 applying DTBCs to stationary state prob- ing the original differential equation for y(x) to lems is actually simpler, as we will see in the 2 (iv) 2 ¢¢ » ⌬2 write h f (xn) = –h (gy) – [gn–1yn–1]. This gives following sections. Although the technique is fa- the Numerov discretization of Equation 1: miliar,6 its connection to DTBCs and a rigorous exposition of the related slowly varying approxi- ⎡⎛ 2 ⎞ ⎤ mation are new here. ∆+2 ⎢ h ⎥ +=2 ⎜1 gynn−−11⎟ hgy nn0 ,(3) ⎣⎢⎝ 12 ⎠ ⎦⎥ Discrete Asymptotic Forms £ If the computational domain corresponds to N1 5 £ ¥ ¥ which is accurate to O(h ). Interestingly, the Nu- n N2, then – < n < N1 and N2 < n < define the merov version is identical to Equation 2, with the exterior regions. Our task is to find suitable analytic replacements solutions to Equation 2 in these exterior regions, from which we will infer the proper boundary con- ⎛ 2 ⎞ ditions to be imposed at the limits of the actual (fi- h gn y →+⎜1 gyg⎟ , → ,(4)nite) computational interval. Because the exterior nnnn⎝ 12 ⎠ 2 + h regions include the asymptotic realm n ® ±¥, so- 1 gn 12 lutions here can’t diverge for large |n| if they’re to be physically acceptable. so the Numerov method’s improved accuracy Let’s introduce the discrete counterpart of the ␮ comes at negligible computational cost. For the re- logarithmic derivative of y(x), n yn/yn+1, and a ⌬␮ ␮ ␮ mainder of this article, I will adopt Equation 2 as measure of its variability, n–1 = n – n–1. In terms the discrete equation of interest, with the assurance of these, Equation 2 becomes that the results apply equally well to the Numerov µµµ22++∆−= version after suitably redefining the terms. nnnn()()hg −1 10.(5) We can solve Equation 2 recursively, using given values for yn at two successive points. Numerical Throughout each exterior region, we assume for stability requires that in regions in which y(x) is now that gn is unchanged from its value at the in- monotonic, recursion should be performed in the ner boundary, gN or gN . But if gn doesn’t change, 1 ␮ 2⌬␮ direction of increasing |y| to reduce the solution’s then neither does n, so n–1 vanishes identically relative error.1 Oftentimes, this means construct- and Equation 5 reduces to a quadratic form that we ing separate solutions starting from each interval can solve to give the root pair limit, then matching them at some intermediate lo- ± 1 ⎛ 2422⎞ cation. In any event, start values for each recursive µ =±−hg hg4 hg (6) nnnn2 ⎝ ⎠ solution must be known to an accuracy compatible with the algorithm we use. Obtaining those values with the property in practical applications has sparked much discus- −−=µµ+− sion. Of course, yn = y(xn) are immediately available ()()11nn 1.(7) if y(x) is known exactly in the neighborhood of the interval limits, but such is rarely the case. (The index n is superfluous here, but I retain it for A less restrictive alternative was developed by its usefulness in subsequent analysis.) 2 2 J.L.M. Quiroz Gonzalez and D. Thompson, who For gn < 0 (or h gn > 4), both roots of Equation 6 ␮ showed how to select start values up to Numerov are real. Because (1 – n)yn+1 = yn, we must select the MAY/JUNE 2006 51 ␮ tion 6 as the root for which |1 – N2| >1. 3.If gN2 > 0, choose an h small enough that 2 ␽ h gN2 < 4 and take yN2–1 = exp(–i N2–1)yN2; then calculate yN2–2, yN2–3, … recursively from Equation 2. This models the scatter- ing problem with particles incident from the left to give, in the transmitted region, a solution with a phase that increases with x— that is, a rightward traveling wave. Again, with g constant in the right exterior region, ␽ ␽ N2–1 is indistinguishable from N2 and is calculated from Equation 8. In all cases, the values for yN1 and yN2 are arbi- trary, reflecting an overall choice of normalization, and we can set them equal to unity. If steps 1 and 2 both apply, the recursive solutions we obtain must be joined at some intermediate point, say, n = M. For optimal visual effect, we first match the slopes at yM by adjusting the start value yN2. Then, when- Figure 1. Some stationary state waveforms of the quantum ever E is an eigenvalue, both recursions will also oscillator, against the backdrop of the oscillator potential. The give identical values for yM. Otherwise, the solution lowest energy wave (yellow) is nodeless, followed in order of is discontinuous at yM and physically unacceptable. 3 energy by the wave with one node (green), two nodes (violet), and We can find slopes at yM accurate to O(h ) using the so forth. Because the energies are accurate, no discontinuity is centered difference evident in these waveforms. ¢ 3 y(xM + h) – y(xM – h) = 2hy (xM) + O(h ). (9) root that makes |yn+1| > |yn| over the left exterior However, the Numerov implementation is more ¥ region (– < n < N1) and |yn+1| < |yn| on the right demanding and requires points two steps removed ¥ ¢ (N2 < n < ) to generate solutions that approach zero from xM to calculate the slopes y (xM) to the de- ® ¥ 2 ␮ ± as n ± . If 0 < h gn < 4, then n are complex con- sired accuracy: ␮ ± jugates, and Equation 7 implies |1 – n | = 1. It fol- lows that the exterior solutions have constant y(xM + h) – y(xM – h) – (1/8)y(xM + 2h) + ¢ 5 modulus, |yn+1| = |yn|, and are oscillatory. Specifi- (1/8)y(xM – 2h) = (3h/2)y (xM) + O(h ). (10) ␽ × cally, we have for this case yn+1 = exp(±i n) yn, where Although convenient, the requirement that g be ⎛ 242− ⎞ constant in the exterior regions (these regions must 4hgnn hg θ = arctan ⎜ ⎟ .(8)be force free) is unnecessarily restrictive.