# RadSqWl_PhaseShift - Main Phase Shifting by a Spherical...

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Main Page 1 Phase Shifting by a Spherical Square Well The wavefunction is normalized to unit amplitude outside the barrier. For bound states, such as the one at W = 5, D = 5, E = - 0.794, the normalization can be overridden by typing 1 in cell I 13, sheet 2. (But then you'll need to download a fresh copy to reinstate the original normalization.) Vary D, E, L and watch the phase shift. Energy of particle: E= 1.466200 12331 Max Well Depth: D_Max= 10 1150 Depth of Well: D = 1.5 Width of Well: W= 5 1 Adjust Depth of Well with this Slidebar: Angular Momentum: L= 0 Begin to integrate from x = 0 Initial wavefunction value f(0): 0 1 Numerical step size dx: 0.015 Adjusting the Energy Range: Minimum Energy: E_init= -1 Increment: delta_E= 0.0002 Maximum Energy(automatic): 5.5534 The purpose of this spreadsheet is to illustrate phase shifts in scattering states . Red is the potential. Green is the V = 0 wavefunction. Blue is the wavefunction. Adjust Energy with Slidebar : Don't type in the Cell above! 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 Radial Square Well Wavefunction Potential Wavefunction Effective Potl V=0 Wavefn

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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. Prenormalized Normalized Normalized Position StepPotential Wavefunction 2nd_deriv deriv at 1/2Effective Potl Max f Wavefunction V = 0 Wavefn 2nd_deriv deriv at 1/2 Max V=0 V=0 Wavefn v(x) 0.62459 0.82553 0 -1.5 0 -0 1 -1.5 0 0 -0 1 0 0.015 -1.5 0.015 -0.044493 0.999333 -1.5 0.0240157874 0.015 -0.021993 0.99967 0.0181701822 0.03 -1.5 0.02998999 -0.08896 0.997998 -1.5 0.0480155468 0.02999505 -0.0439787 0.99901 0.0363343702 0.045 -1.5 0.044959963 -0.13336 0.995998 -1.5 0.0719832609 0.04498021 -0.06595 0.998021 0.0544865717 0.06 -1.5 0.05989993 -0.17768 0.993333 -1.5 0.0959029337 0.05995053 -0.0878995 0.996703 0.0726207983 0.075 -1.5 0.07479992 -0.22187 0.990005 -1.5 0.1197586014 0.07490107 -0.1098199 0.995055 0.0907310677 0.09 -1.5 0.08964999 -0.26592 0.986016 -1.5 0.1435343428 0.0898269 -0.1317042 0.99308 0.1088114054 0.105 -1.5 0.10444023 -0.30979 0.981369 -1.5 0.1672142901 0.10472309 -0.153545 0.990777 0.1268558467 0.12 -1.5 0.11916076 -0.35345 0.976067 -1.5 0.1907826394 0.11958474 -0.1753352 0.988147 0.1448584389 0.135 -1.5 0.13380177 -0.39688 0.970114 -1.5 0.2142236613 0.13440694 -0.1970675 0.985191 0.162813243 0.15 -1.5 0.14835348 -0.44005 0.963513 -1.5 0.2375217114 0.1491848 -0.2187348 0.98191 0.1807143359 0.165 -1.5 0.16280618 -0.48292 0.95627 -1.5 0.2606612407 0.16391345 -0.2403299 0.978305 0.198555812 0.18 -1.5 0.17715022 -0.52546 0.948388 -1.5 0.283626806 0.17858802 -0.2618457 0.974377 0.2163317855 0.195 -1.5 0.19137604 -0.56766 0.939873 -1.5 0.3064030802 0.19320367 -0.2832752 0.970128 0.2340363922 0.21 -1.5 0.20547413 -0.60948 0.930731 -1.5 0.3289748625 0.20775559 -0.3046112 0.965559 0.2516637916 0.225 -1.5 0.21943508 -0.65089 0.920967 -1.5 0.3513270887 0.22223897 -0.3258468 0.960671 0.2692081683 0.24 -1.5 0.23324959 -0.69186 0.910589 -1.5 0.3734448409 0.23664903 -0.3469748 0.955466 0.2866637345 0.255 -1.5 0.24690843 -0.73238 0.899604 -1.5 0.3953133578 0.25098103 -0.3679884 0.949947 0.3040247319 0.27 -1.5 0.26040248 -0.77241 0.888017 -1.5 0.4169180446 0.26523022 -0.3888806 0.944113 0.321285433 0.285 -1.5 0.273722744 -0.81192 0.875839 -1.5 0.4382444824 0.27939192 -0.4096444 0.937969 0.3384401436 0.3 -1.5 0.28686032 -0.85089 0.863075 -1.5 0.4592784381 0.29346145 -0.4302732 0.931515 0.3554832045 0.315 -1.5 0.29980646 -0.88929 0.849736 -1.5 0.4800058735 0.30743417 -0.45076 0.924753 0.3724089933 0.33 -1.5 0.3125525 -0.92709 0.83583 -1.5 0.5004129555 0.32130547 -0.4710981 0.917687 0.3892119262 0.345 -1.5 0.32508994 -0.96428 0.821365 -1.5 0.5204860644 0.335070767 -0.4912808 0.910317 0.4058864601 0.36 -1.5 0.33741043 -1.00083 0.806353 -1.5 0.5402118034 0.34872553 -0.5113014 0.902648 0.422427094 0.375 -1.5 0.34950572 -1.0367 0.790803 -1.5 0.5595770078 0.36226525 -0.5311533 0.894681 0.4388283714 0.39 -1.5 0.36136776 -1.07189 0.774724 -1.5 0.5785687533 0.37568546 -0.55083 0.886418 0.4550848815 0.405 -1.5 0.37298862 -1.10636 0.758129 -1.5 0.597174365 0.38898173 -0.570325 0.877863 0.4711912613 0.42 -1.5 0.38436055 -1.14009 0.741027 -1.5 0.6153814254 0.40214968 -0.5896319 0.869019 0.4871421975 0.435 -1.5 0.39547597 -1.17306 0.723432 -1.5 0.6331777833 0.41518496 -0.6087442 0.859888 0.502932428 0.45 -1.5 0.40632744 -1.20525 0.705353 -1.5 0.6505515616 0.42808328 -0.6276557 0.850473 0.5185567435 0.465 -1.5 0.41690773 -1.23663 0.686803 -1.5 0.667491165 0.44084037 -0.6463601 0.840777 0.5340099898 0.48 -1.5 0.42720978 -1.26719 0.667795 -1.5 0.6839852881 0.45345203 -0.6648514 0.830805 0.5492870688 0.495 -1.5 0.43722671 -1.2969 0.648342 -1.5 0.7000229229 0.4659141 -0.6831233 0.820558 0.5643829408 0.51 -1.5 0.44695184 -1.32575 0.628456 -1.5 0.7155933659 0.47822247 -0.7011698 0.81004 0.5792926256 0.525 -1.5 0.45637868 -1.35371 0.60815 -1.5 0.7306862254 0.49037307 -0.718985 0.799255 0.5940112048 0.54 -1.5 0.46550093 -1.38077 0.587439 -1.5 0.7452914286 0.5023619 -0.736563 0.788207 0.6085338226 0.555 -1.5 0.47431251 -1.40691 0.566335 -1.5 0.759399228 0.514185009 -0.7538981 0.776899 0.6228556881 0.57 -1.5 0.48280753 -1.4321 0.544853 -1.5 0.7730002082 0.52583849 -0.7709844 0.765334 0.6369720767 0.585 -1.5 0.49098034 -1.45635 0.523008 -1.5 0.7860852919 0.53731849 -0.7878164 0.753517 0.6508783313 0.6 -1.5 0.49882546 -1.47962 0.500814 -1.5 0.7986457462 0.54862124 -0.8043885 0.741451 0.6645698645 0.615 -1.5 0.50633767 -1.5019 0.478286 -1.5 0.8106731884 0.559743 -0.8206952 0.72914 0.6780421593 0.63 -1.5 0.51351195 -1.52318 0.455438 -1.5 0.8221595913 0.57068011 -0.8367312 0.716589 0.6912907715 0.645 -1.5 0.52034352 -1.54344 0.432286 -1.5 0.833097289 0.58142895 -0.8524911 0.703802 0.7043113303 0.66 -1.5 0.52682781 -1.56268 0.408846 -1.5 0.8434789817 0.59198598 -0.8679698 0.690782 0.7170995403 0.675 -1.5 0.5329605 -1.58087 0.385133 -1.5 0.8532977408 0.60234771 -0.8831622 0.677535 0.7296511827 0.69 -1.5 0.5387375 -1.598 0.361163 -1.5 0.8625470133 0.61251074 -0.8980632 0.664064 0.7419621169 0.705 -1.5 0.54415494 -1.61407 0.336952 -1.5 0.8712206261 0.6224717 -0.912668 0.650374 0.7540282814 0.72 -1.5 0.549209219 -1.62906 0.312516 -1.5 0.8793127907 0.63222731 -0.9269717 0.636469 0.7658456959 0.735 -1.5 0.55389696 -1.64297 0.287871 -1.5 0.8868181063 0.64177435 -0.9409696 0.622355 0.7774104616 0.75 -1.5 0.55821503 -1.65578 0.263035 -1.5 0.893731564 0.65110967 -0.954657 0.608035 0.7887187635 0.765 -1.5 0.56216055 -1.66748 0.238023 -1.5 0.9000485497 0.6602302 -0.9680295
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