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Main Page 1 A Resonance inside a Radial Square Barrier Click the "Using this Spreadsheet" Tab at the bottom for some ideas on exploring wave functions. Energy of particle: E= 1.617100 17171 Depth of Well: D = 0 Width of Well: W= 4 1500 Max Barrier Height: H_Max= 10 Height of Barrier: H= 5 Adjust Height of Barrier with This Slidebar: Breadth of Barrier: B= 1 Angular Momentum: L= 0 Optional Wavefn Scaledown Factor: 1 1 Numerical step size dx: 0.02 Adjusting the Energy Range: Minimum Energy: E_init= -0.1 Increment: delta_E= 0.0001 Maximum Energy(automatic): 3.1767 Use the slider (or click and hold on its ends) to see how the phase changes by π on going through the resonance! Adjust Energy with Slidebar : Don't type in the Cell above! 0 2 4 6 8 10 12 14 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Radial Square Barrier Wavefunction Potential Wavefunction

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Table Page 2 NUMERICAL SOLUT ION OF SCHRODINGER'S EQUAT ION The wavefunction and its second derivative are found at intervals n*delta_x, where n is an integer, and the first derivative is found at the "leapfrog" points, (n + 0.5)*delta_x. So the numerical step finding the change in the wavefunction over an interval delta_x assumes it is delta_x multiplied by the value of the derivative at the midpoint of the interval. This is much more accurate than using the value of the derivative at the beginning of the interval. Similarly, to find the change in the derivative over an interval the method used the value of the second derivative in the middle of that interval. Position StepPotential Wavefunction 2nd_deriv deriv at 1/2 Potential Max f/Scale Wavefunction v(x) 0.14083 0 0 0 -0 1 0 0 0.02 0 0.02 -0.032342 0.999353 0 0.1420147331 0.04 0 0.03998706 -0.06466 0.99806 0 0.2839376054 0.06 0 0.05994826 -0.09694 0.996121 0 0.4256768155 0.08 0 0.07987068 -0.12916 0.993538 0 0.5671406808 0.1 0 0.09974144 -0.16129 0.990312 0 0.7082376969 0.12 0 0.11954768 -0.19332 0.986446 0 0.8488765964 0.14 0 0.13927659 -0.22522 0.981941 0 0.9889664087 0.16 0 0.15891542 -0.25698 0.976801 0 1.1284165179 0.18 0 0.17845145 -0.28857 0.97103 0 1.2671367221 0.2 0 0.19787205 -0.31998 0.96463 0 1.4050372917 0.22 0 0.21716466 -0.35118 0.957607 0 1.5420290269 0.24 0 0.23631679 -0.38215 0.949964 0 1.678023316 0.26 0 0.25531607 -0.41287 0.941707 0 1.8129321926 0.28 0 0.2741502 -0.44333 0.93284 0 1.9466683921 0.3 0 0.292807 -0.4735 0.92337 0 2.0791454086 0.32 0 0.3112744 -0.50336 0.913303 0 2.2102775507 0.34 0 0.32954046 -0.5329 0.902645 0 2.3399799969 0.36 0 0.34759335 -0.56209 0.891403 0 2.4681688504 0.38 0 0.365421409 -0.59092 0.879584 0 2.5947611935 0.4 0 0.3830131 -0.61937 0.867197 0 2.7196751414 0.42 0 0.40035704 -0.64742 0.854249 0 2.8428298946 0.44 0 0.41744201 -0.67505 0.840748 0 2.9641457916 0.46 0 0.43425697 -0.70224 0.826703 0 3.0835443607 0.48 0 0.450791027 -0.72897 0.812124 0 3.2009483698 0.5 0 0.4670335 -0.75524 0.797019 0 3.3162818776 0.52 0 0.48297387 -0.78102 0.781398 0 3.4294702815 0.54 0 0.49860184 -0.80629 0.765273 0 3.540440367 0.56 0 0.51390729 -0.83104 0.748652 0 3.6491203539 0.58 0 0.52888033 -0.85525 0.731547 0 3.7554399439 0.6 0 0.54351127 -0.87891 0.713969 0 3.8593303651 0.62 0 0.55779064 -0.902 0.695928 0 3.960724417 0.64 0 0.57170921 -0.92451 0.677438 0 4.0595565139 0.66 0 0.58525797 -0.94642 0.65851 0 4.1557627273 0.68 0 0.59842817 -0.96772 0.639155 0 4.2492808272 0.7 0 0.61121128 -0.98839 0.619388 0 4.3400503222 0.72 0 0.62359903 -1.00842 0.599219 0 4.4280124991 0.74 0 0.63558342 -1.0278 0.578663 0 4.5131104604 0.76 0 0.64715668 -1.04652 0.557733 0 4.5952891613 0.78 0 0.65831134 -1.06456 0.536442 0 4.6744954454 0.8 0 0.66904017 -1.0819 0.514804 0 4.7506780788 0.82 0 0.679336246 -1.09855 0.492833 0 4.8237877837 0.84 0 0.6891929 -1.11449 0.470543 0 4.8937772696 0.86 0 0.69860375 -1.12971 0.447948 0 4.9606012646 0.88 0 0.70756272 -1.1442 0.425064 0 5.0242165444 0.9 0 0.71606401 -1.15795 0.401906 0 5.0845819599 0.92 0 0.72410212 -1.17095 0.378487 0 5.1416584644 0.94 0 0.73167185 -1.18319 0.354823 0 5.1954091385 0.96 0 0.73876831 -1.19466 0.33093 0 5.2457992142 0.98 0 0.7453869 -1.20537 0.306822 0 5.2927960971 1 0 0.75152335 -1.21529 0.282517 0 5.3363693879 1.02 0 0.75717368 -1.22443 0.258028 0 5.3764909014 1.04 0 0.76233424 -1.23277 0.233373 0 5.4131346855 1.06 0 0.76700169 -1.24032 0.208566 0 5.4462770377 1.08 0 0.77117302 -1.24706 0.183625 0 5.4758965199 1.1 0 0.77484552 -1.253 0.158565 0 5.5019739733 1.12 0 0.77801682 -1.25813 0.133402 0 5.5244925298 1.14 0 0.780684863 -1.26245 0.108153 0 5.5434376236 1.16 0 0.78284793 -1.26594 0.082835 0 5.5587970002 1.18 0 0.78450462 -1.26862 0.057462 0 5.5705607245 1.2 0 0.78565386 -1.27048 0.032052 0 5.5787211874 1.22 0 0.78629491 -1.27152 0.006622 0 5.5832731102 1.24 0 0.78642736 -1.27173 -0.01881 0 5.5842135486 1.26 0 0.786051105 -1.27112 -0.04423 0 5.5815418944 1.28 0 0.78516641 -1.26969 -0.06963 0 5.5752598756 1.3 0 0.78377383 -1.26744 -0.09498 0 5.5653715557 1.32 0 0.78187428 -1.26437 -0.12027 0 5.5518833308 1.34 0 0.77946898 -1.26048 -0.14547 0 5.5348039258 1.36 0 0.77655948 -1.25577 -0.17059 0 5.5141443881 1.38 0 0.77314768 -1.25026 -0.1956 0 5.4899180814 1.4 0 0.76923578 -1.24393 -0.22047 0 5.462140676 1.42 0 0.7648263 -1.2368 -0.24521 0 5.4308301395 1.44 0 0.7599221 -1.22887 -0.26979 0 5.3960067249 1.46 0 0.75452636 -1.22014 -0.29419 0 5.3576929572 1.48 0 0.74864255 -1.21063 -0.3184 0 5.3159136195 1.5 0 0.7422745 -1.20033 -0.34241 0 5.2706957362 1.52 0 0.73542631 -1.18926 -0.36619 0 5.222068556 1.54 0 0.72810242 -1.17741 -0.38974 0 5.1700635331 1.56 0 0.72030756 -1.16481 -0.41304 0 5.1147143062 1.58 0 0.71204678 -1.15145 -0.43607 0 5.0560566776 1.6 0 0.70332542 -1.13735 -0.45882 0 4.9941285892 1.62 0 0.69414912 -1.12251 -0.48127 0 4.9289700987 1.64 0 0.68452382 -1.10694 -0.5034 0 4.8606233532 1.66 0 0.67445574 -1.09066 -0.52522 0 4.7891325621 1.68 0 0.66395139 -1.07368 -0.54669 0 4.7145439685 1.7 0 0.65301757 -1.05599 -0.56781 0 4.6369058192 1.72 0 0.64166136 -1.03763 -0.58856 0 4.5562683338 1.74 0 0.62989009 -1.0186 -0.60894 0 4.4726836718 1.76 0 0.61771139 -0.9989 -0.62891 0 4.386205899 1.78 0 0.60513312 -0.97856 -0.64848 0 4.2968909529 1.8 0 0.59216343 -0.95759 -0.66764 0 4.2047966058 1.82 0 0.57881071 -0.93599 -0.68636 0 4.1099824281 1.84 0 0.56508359 -0.9138 -0.70463 0 4.0125097493 1.86 0 0.55099095 -0.89101 -0.72245 0 3.9124416187 1.88 0 0.5365419 -0.86764 -0.73981 0 3.8098427644 1.9 0 0.5217458 -0.84372 -0.75668 0 3.7047795514 1.92 0 0.50661222 -0.81924 -0.77306 0 3.5973199388 1.94 0 0.49115093 -0.79424 -0.78895 0 3.4875334358 1.96 0 0.47537195 -0.76872 -0.80432 0 3.3754910566 1.98 0 0.45928548 -0.74271 -0.81918 0 3.2612652748 2 2.5 0.442901931 0.391038 -0.81136 2.5 3.1449299762 2.02 5 0.42667479 1.443398 -0.78249 5 3.029705341 2.04 5 0.41102501 1.390457 -0.75468 5 2.918580382 2.06 5 0.39593142 1.339396 -0.72789 5 2.8114047291 2.08 5 0.38137358 1.290149 -0.70209 5 2.7080333567 2.1 5 0.3673318 1.242647 -0.67724 5 2.6083263867 2.12 5 0.35378708 1.196826 -0.6533 5 2.5121488996 2.14 5 0.34072109 1.152625 -0.63025 5 2.4193707519 2.16 5 0.32811615 1.109984 -0.60805 5 2.3298664 2.18 5 0.31595521 1.068845 -0.58667 5 2.2435147301 2.2 5 0.3042218 1.029152 -0.56609 5 2.1601988945 2.22 5 0.29290005 0.990852 -0.54627 5 2.0798061537 2.24 5 0.28197465 0.953892 -0.52719 5 2.0022277234 2.26 5 0.2714308 0.918223 -0.50883 5 1.9273586276 2.28 5 0.26125424 0.883797 -0.49115 5 1.8550975564 2.3 5 0.2514312 0.850567 -0.47414 5 1.7853467289 2.32 5 0.24194838 0.818487 -0.45777 5 1.7180117613 2.34 5 0.23279296 0.787515 -0.44202 5 1.6530015384 2.36 5 0.22395255 0.757609 -0.42687 5 1.5902280911 2.38 5 0.21541518 0.728728 -0.41229 5 1.5296064769 2.4 5 0.2071693 0.700833 -0.39828 5 1.471054665 2.42 5 0.19920376 0.673886 -0.3848 5 1.4144934253 2.44 5 0.19150777 0.647852 -0.37184 5 1.3598462216 2.46 5 0.184070917 0.622694 -0.35939 5 1.3070391075 2.48 5 0.17688314 0.598378 -0.34742 5 1.2560006263 2.5 5 0.16993472 0.574872 -0.33592 5 1.206661715 2.52 5 0.16321625 0.552144 -0.32488 5 1.15895561 2.54 5 0.15671864 0.530163 -0.31428 5 1.1128177574 2.56 5 0.15043309 0.5089 -0.3041 5 1.0681857253 2.58 5 0.1443511 0.488325 -0.29433
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