(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 541495, 14257]*) (*NotebookOutlinePosition[ 542165, 14280]*) (* CellTagsIndexPosition[ 542121, 14276]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Numerical solution of the 2D Schr\[ODoubleDot]dinger equation", "Title"], Cell["Bernd Thaller", "Subsubtitle"], Cell[CellGroupData[{ Cell["Preparations", "Section"], Cell["\<\ For the numerical calculation we need the external application \ \"QuantumKernel\":\ \>", "Text"], Cell[CellGroupData[{ Cell["<", "Text"], Cell["<", "Text"], Cell["Off[Arg::indet];", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The initial function", "Section"], Cell[CellGroupData[{ Cell["Definition of a Gaussian function", "Subsection"], Cell["\<\ Our initial function will be a Gaussian wave packet. There are \ several parameters describing the physical properties of the initial state:\ \ \>", "Text"], Cell["\<\ pos = {-3.,-3.}; (* average position at t=0 *) mom = {4.,4.}; (* average momentum *) a = 1. ; (* parameter for uncertainty in position space *)\ \>", \ "Input"], Cell["We define the Gaussian function as", "Text"], Cell["\<\ Gauss[x0_,y0_,kx_,ky_,a_] := Compile @@ {{x,y}, Simplify[(a/Pi)^(1/2) Exp[I(kx x+ky y)]* Exp[-a ((x-x0)^2+(y-y0)^2)/2]]};\ \>", "Input"], Cell["and obtain a compiled version just by calling", "Text"], Cell["f=Gauss[pos[[1]], pos[[2]] ,mom[[1]], mom[[2]], 1];", "Input"], Cell["\<\ Next, we produce a table of complex values, representing the \ initial function through the values at a grid of space points. The grid is a \ rectangular array of points defined by\ \>", "Text"], Cell["\<\ numleft = {-7.,-7.}; (* lower left corner *) numright = {7.,7.}; (* upper right corner *) dx = 0.14; (* step size *)\ \>", "Input"], Cell["\<\ The following command calculates the matrix of values. 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" }], "Text"], Cell[CellGroupData[{ Cell["psi = QNewFunction[Re[psi0], Im[psi0]]", "Input"], Cell[BoxData[ \(QFunctionObject[1]\)], "Output"] }, Open ]], Cell["\<\ The array of values which represents the initial function is passed \ to the \"QuantumKernel\". The quantum kernel returns a \"function object\" \ which is referred to as psi.\ \>", "Text"], Cell[TextData[{ StyleBox["Attention", FontWeight->"Bold"], ": Only numerical values can be sent to QuantumKernel. You have to be sure \ that psi0 contains only numerical values (no integers or symbolic \ expressions)." }], "Text"], Cell["\<\ Note: The result of the numerical calculation will be stored in the \ same function object \"psi\". Whenever we want to repeat the calculation, we \ have to reissue the command above.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["QuantumKernel commands for function objects", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Some ", StyleBox["Mathematica", FontSlant->"Italic"], " commands provided by QuantumKernel" }], "Subsection"], Cell["", "Text"], Cell["QNewFunction[arrays]", "Program", Evaluatable->False], Cell["\<\ Generates a function object in QuantumKernel by passing the values \ from one or several arrays of real numbers.QuantumKernel returns a reference \ to the function object to Mathematica. Here every array is a one-, two-, or \ three-dimensional table of real numbers. \ \>", "Text"], Cell["QDisposeFunction[ref_]", "Program", Evaluatable->False], Cell["\<\ Disposes the numerical data contained in the function object which \ is referred to as 'ref'. \ \>", "Text"], Cell["QGetFunctionInfo[ref_]", "Program", Evaluatable->False], Cell["\<\ Get information about the function object with reference 'ref'. \ \>", "Text"], Cell["QGetArray[ref_]", "Program", Evaluatable->False], Cell[TextData[{ "Transfers the numerical data contained in the function object 'ref' from \ QuantumKernel to ", StyleBox["Mathematica", FontSlant->"Italic"], ".\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Examples:", "Subsection"], Cell["\<\ Above, we have defined the function object psi, using QNewFunction. \ This says that psi is a function object consisting of two arrays of 100x100 \ numerical data:\ \>", "Text"], Cell[CellGroupData[{ Cell["QGetFunctionInfo[psi]", "Input"], Cell[BoxData[ \({100, 100, 2}\)], "Output"] }, Open ]], Cell[TextData[{ "This gives the numerical values stored in psi back to ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"], Cell["psi1=QGetArray[psi];", "Input"], Cell["\<\ psi1 is now a list of two arrays. 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L .66997 .3035 L .67258 .30089 L .67968 .29379 L .68229 .29118 L .68939 .28408 L .692 .28147 L .69911 .27437 L .70172 .27176 L .70882 .26465 L .71143 .26204 L .71853 .25494 L .72114 .25233 L .72824 .24523 L .73085 .24262 L .73796 .23552 L .74057 .23291 L Mistroke .74767 .2258 L .75028 .22319 L .75738 .21609 L .75999 .21348 L .76709 .20638 L .7697 .20377 L .77681 .19667 L .77942 .19406 L .78652 .18695 L .78913 .18434 L .79623 .17724 L .79884 .17463 L .80594 .16753 L .80855 .16492 L .81566 .15781 L .81827 .15521 L .82537 .1481 L .82798 .14549 L .83508 .13839 L .83769 .13578 L .84479 .12868 L .8474 .12607 L .85451 .11896 L .85712 .11636 L .86422 .10925 L .86683 .10664 L .87393 .09954 L .87654 .09693 L .88364 .08983 L .88625 .08722 L .89336 .08011 L .89597 .07751 L .90307 .0704 L .90568 .06779 L .91278 .06069 L .91539 .05808 L .92249 .05098 L .9251 .04837 L .93221 .04126 L .93482 .03866 L .94192 .03155 L .94453 .02894 L .95163 .02439 L .95424 .01923 L Mfstroke .333 g .01923 .93556 m .02257 .93221 L .02894 .92584 L .03229 .92249 L .03866 .91613 L .042 .91278 L .04837 .90642 L .05172 .90307 L .05808 .89671 L .06143 .89336 L .06779 .88699 L .07114 .88364 L .07751 .87728 L .08085 .87393 L .08722 .86757 L .09057 .86422 L .09693 .85786 L .10028 .85451 L .10664 .84814 L .10999 .84479 L .11636 .83843 L .1197 .83508 L .12607 .82872 L .12942 .82537 L .13578 .819 L .13913 .81566 L .14549 .80929 L .14884 .80594 L .15521 .79958 L .15855 .79623 L .16492 .78987 L .16827 .78652 L .17463 .78015 L .17798 .77681 L .18434 .77044 L .18769 .76709 L .19406 .76073 L .1974 .75738 L .20377 .75102 L .20712 .74767 L .21348 .7413 L .21683 .73796 L .22319 .73159 L .22654 .72824 L .23291 .72188 L .23625 .71853 L .24262 .71217 L .24597 .70882 L .25233 .70245 L .25568 .69911 L .26204 .69274 L .26539 .68939 L .27176 .68303 L .2751 .67968 L .28147 .67332 L .28482 .66997 L .29118 .6636 L .29453 .66026 L .30089 .65389 L .30424 .65054 L .31061 .64418 L .31395 .64083 L .32032 .63447 L .32367 .63112 L .33003 .62475 L .33338 .62141 L .33974 .61504 L .34309 .61169 L .34946 .60533 L .3528 .60198 L .35917 .59562 L .36252 .59227 L .36888 .5859 L .37223 .58256 L .37859 .57619 L .38194 .57284 L .38831 .56648 L .39165 .56313 L .39802 .55677 L .40137 .55342 L .40773 .54705 L .41108 .54371 L .41744 .53734 L .42079 .53399 L .42716 .52763 L .4305 .52428 L .43687 .51792 L .44022 .51457 L .44658 .5082 L .44993 .50486 L .45629 .49849 L .45964 .49514 L .46601 .48878 L .46935 .48543 L .47572 .47907 L .47907 .47572 L .48543 .46935 L .48878 .46601 L .49514 .45964 L .49849 .45629 L .50486 .44993 L .5082 .44658 L .51457 .44022 L .51792 .43687 L .52428 .4305 L .52763 .42716 L .53399 .42079 L .53734 .41744 L .54371 .41108 L .54705 .40773 L .55342 .40137 L .55677 .39802 L .56313 .39165 L .56648 .38831 L .57284 .38194 L .57619 .37859 L .58256 .37223 L .5859 .36888 L .59227 .36252 L .59562 .35917 L .60198 .3528 L .60533 .34946 L .61169 .34309 L .61504 .33974 L .62141 .33338 L .62475 .33003 L .63112 .32367 L .63447 .32032 L .64083 .31395 L .64418 .31061 L .65054 .30424 L .65389 .30089 L .66026 .29453 L .6636 .29118 L .66997 .28482 L .67332 .28147 L .67968 .2751 L .68303 .27176 L .68939 .26539 L .69274 .26204 L .69911 .25568 L .70245 .25233 L .70882 .24597 L .71217 .24262 L .71853 .23625 L .72188 .23291 L .72824 .22654 L .73159 .22319 L .73796 .21683 L .7413 .21348 L .74767 .20712 L .75102 .20377 L .75738 .1974 L .76073 .19406 L .76709 .18769 L .77044 .18434 L .77681 .17798 L .78015 .17463 L .78652 .16827 L .78987 .16492 L .79623 .15855 L .79958 .15521 L .80594 .14884 L .80929 .14549 L .81566 .13913 L .819 .13578 L .82537 .12942 L .82872 .12607 L .83508 .1197 L .83843 .11636 L .84479 .10999 L .84814 .10664 L .85451 .10028 L .85786 .09693 L .86422 .09057 L .86757 .08722 L .87393 .08085 L .87728 .07751 L .88364 .07114 L .88699 .06779 L .89336 .06143 L .89671 .05808 L .90307 .05172 L .90642 .04837 L .91278 .042 L .91613 .03866 L .92249 .03229 L .92584 .02894 L .93221 .02257 L .93556 .01923 L .01923 .01923 L F 0 g .01923 .93556 m .02257 .93221 L .02894 .92584 L .03229 .92249 L .03866 .91613 L .042 .91278 L .04837 .90642 L .05172 .90307 L .05808 .89671 L .06143 .89336 L .06779 .88699 L .07114 .88364 L .07751 .87728 L .08085 .87393 L .08722 .86757 L .09057 .86422 L .09693 .85786 L .10028 .85451 L .10664 .84814 L .10999 .84479 L .11636 .83843 L .1197 .83508 L .12607 .82872 L .12942 .82537 L .13578 .819 L .13913 .81566 L .14549 .80929 L .14884 .80594 L .15521 .79958 L .15855 .79623 L .16492 .78987 L .16827 .78652 L .17463 .78015 L .17798 .77681 L .18434 .77044 L .18769 .76709 L .19406 .76073 L .1974 .75738 L .20377 .75102 L .20712 .74767 L .21348 .7413 L .21683 .73796 L .22319 .73159 L .22654 .72824 L .23291 .72188 L .23625 .71853 L .24262 .71217 L .24597 .70882 L .25233 .70245 L .25568 .69911 L Mistroke .26204 .69274 L .26539 .68939 L .27176 .68303 L .2751 .67968 L .28147 .67332 L .28482 .66997 L .29118 .6636 L .29453 .66026 L .30089 .65389 L .30424 .65054 L .31061 .64418 L .31395 .64083 L .32032 .63447 L .32367 .63112 L .33003 .62475 L .33338 .62141 L .33974 .61504 L .34309 .61169 L .34946 .60533 L .3528 .60198 L .35917 .59562 L .36252 .59227 L .36888 .5859 L .37223 .58256 L .37859 .57619 L .38194 .57284 L .38831 .56648 L .39165 .56313 L .39802 .55677 L .40137 .55342 L .40773 .54705 L .41108 .54371 L .41744 .53734 L .42079 .53399 L .42716 .52763 L .4305 .52428 L .43687 .51792 L .44022 .51457 L .44658 .5082 L .44993 .50486 L .45629 .49849 L .45964 .49514 L .46601 .48878 L .46935 .48543 L .47572 .47907 L .47907 .47572 L .48543 .46935 L .48878 .46601 L .49514 .45964 L .49849 .45629 L Mistroke .50486 .44993 L .5082 .44658 L .51457 .44022 L .51792 .43687 L .52428 .4305 L .52763 .42716 L .53399 .42079 L .53734 .41744 L .54371 .41108 L .54705 .40773 L .55342 .40137 L .55677 .39802 L .56313 .39165 L .56648 .38831 L .57284 .38194 L .57619 .37859 L .58256 .37223 L .5859 .36888 L .59227 .36252 L .59562 .35917 L .60198 .3528 L .60533 .34946 L .61169 .34309 L .61504 .33974 L .62141 .33338 L .62475 .33003 L .63112 .32367 L .63447 .32032 L .64083 .31395 L .64418 .31061 L .65054 .30424 L .65389 .30089 L .66026 .29453 L .6636 .29118 L .66997 .28482 L .67332 .28147 L .67968 .2751 L .68303 .27176 L .68939 .26539 L .69274 .26204 L .69911 .25568 L .70245 .25233 L .70882 .24597 L .71217 .24262 L .71853 .23625 L .72188 .23291 L .72824 .22654 L .73159 .22319 L .73796 .21683 L .7413 .21348 L Mistroke .74767 .20712 L .75102 .20377 L .75738 .1974 L .76073 .19406 L .76709 .18769 L .77044 .18434 L .77681 .17798 L .78015 .17463 L .78652 .16827 L .78987 .16492 L .79623 .15855 L .79958 .15521 L .80594 .14884 L .80929 .14549 L .81566 .13913 L .819 .13578 L .82537 .12942 L .82872 .12607 L .83508 .1197 L .83843 .11636 L .84479 .10999 L .84814 .10664 L .85451 .10028 L .85786 .09693 L .86422 .09057 L .86757 .08722 L .87393 .08085 L .87728 .07751 L .88364 .07114 L .88699 .06779 L .89336 .06143 L .89671 .05808 L .90307 .05172 L .90642 .04837 L .91278 .042 L .91613 .03866 L .92249 .03229 L .92584 .02894 L .93221 .02257 L .93556 .01923 L Mfstroke .01923 .90812 m .02422 .90307 L .02894 .89841 L .03399 .89336 L .03866 .8887 L .04371 .88364 L .04837 .87898 L .05342 .87393 L .05808 .86927 L .06313 .86422 L .06779 .85956 L .07284 .85451 L .07751 .84985 L .08256 .84479 L .08722 .84013 L .09227 .83508 L .09693 .83042 L .10198 .82537 L .10664 .82071 L .11169 .81566 L .11636 .811 L .12141 .80594 L .12607 .80128 L .13112 .79623 L .13578 .79157 L .14083 .78652 L .14549 .78186 L .15054 .77681 L .15521 .77215 L .16026 .76709 L .16492 .76243 L .16997 .75738 L .17463 .75272 L .17968 .74767 L .18434 .74301 L .18939 .73796 L .19406 .7333 L .19911 .72824 L .20377 .72358 L .20882 .71853 L .21348 .71387 L .21853 .70882 L .22319 .70416 L .22824 .69911 L .23291 .69445 L .23796 .68939 L .24262 .68473 L .24767 .67968 L .25233 .67502 L .25738 .66997 L .26204 .66531 L .26709 .66026 L .27176 .6556 L .27681 .65054 L .28147 .64588 L .28652 .64083 L .29118 .63617 L .29623 .63112 L .30089 .62646 L .30594 .62141 L .31061 .61675 L .31566 .61169 L .32032 .60703 L .32537 .60198 L .33003 .59732 L .33508 .59227 L .33974 .58761 L .34479 .58256 L .34946 .5779 L .35451 .57284 L .35917 .56818 L .36422 .56313 L .36888 .55847 L .37393 .55342 L .37859 .54876 L .38364 .54371 L .38831 .53905 L .39336 .53399 L .39802 .52933 L .40307 .52428 L .40773 .51962 L .41278 .51457 L .41744 .50991 L .42249 .50486 L .42716 .5002 L .43221 .49514 L .43687 .49048 L .44192 .48543 L .44658 .48077 L .45163 .47572 L .45629 .47106 L .46134 .46601 L .46601 .46134 L .47106 .45629 L .47572 .45163 L .48077 .44658 L .48543 .44192 L .49048 .43687 L .49514 .43221 L .5002 .42716 L .50486 .42249 L .50991 .41744 L .51457 .41278 L .51962 .40773 L .52428 .40307 L .52933 .39802 L .53399 .39336 L .53905 .38831 L .54371 .38364 L .54876 .37859 L .55342 .37393 L .55847 .36888 L .56313 .36422 L .56818 .35917 L .57284 .35451 L .5779 .34946 L .58256 .34479 L .58761 .33974 L .59227 .33508 L .59732 .33003 L .60198 .32537 L .60703 .32032 L .61169 .31566 L .61675 .31061 L .62141 .30594 L .62646 .30089 L .63112 .29623 L .63617 .29118 L .64083 .28652 L .64588 .28147 L .65054 .27681 L .6556 .27176 L .66026 .26709 L .66531 .26204 L .66997 .25738 L .67502 .25233 L .67968 .24767 L .68473 .24262 L .68939 .23796 L .69445 .23291 L .69911 .22824 L .70416 .22319 L .70882 .21853 L .71387 .21348 L .71853 .20882 L .72358 .20377 L .72824 .19911 L .7333 .19406 L .73796 .18939 L .74301 .18434 L .74767 .17968 L .75272 .17463 L .75738 .16997 L .76243 .16492 L .76709 .16026 L .77215 .15521 L .77681 .15054 L .78186 .14549 L .78652 .14083 L .79157 .13578 L .79623 .13112 L .80128 .12607 L .80594 .12141 L .811 .11636 L .81566 .11169 L .82071 .10664 L .82537 .10198 L .83042 .09693 L .83508 .09227 L .84013 .08722 L .84479 .08256 L .84985 .07751 L .85451 .07284 L .85956 .06779 L .86422 .06313 L .86927 .05808 L .87393 .05342 L .87898 .04837 L .88364 .04371 L .8887 .03866 L .89336 .03399 L .89841 .02894 L .90307 .02422 L .90812 .01923 L .01923 .01923 L F .01923 .90812 m .02422 .90307 L .02894 .89841 L .03399 .89336 L .03866 .8887 L .04371 .88364 L .04837 .87898 L .05342 .87393 L .05808 .86927 L .06313 .86422 L .06779 .85956 L .07284 .85451 L .07751 .84985 L .08256 .84479 L .08722 .84013 L .09227 .83508 L .09693 .83042 L .10198 .82537 L .10664 .82071 L .11169 .81566 L .11636 .811 L .12141 .80594 L .12607 .80128 L .13112 .79623 L .13578 .79157 L .14083 .78652 L .14549 .78186 L .15054 .77681 L .15521 .77215 L .16026 .76709 L .16492 .76243 L .16997 .75738 L .17463 .75272 L .17968 .74767 L .18434 .74301 L .18939 .73796 L .19406 .7333 L .19911 .72824 L .20377 .72358 L .20882 .71853 L .21348 .71387 L .21853 .70882 L .22319 .70416 L .22824 .69911 L .23291 .69445 L .23796 .68939 L .24262 .68473 L .24767 .67968 L .25233 .67502 L .25738 .66997 L Mistroke .26204 .66531 L .26709 .66026 L .27176 .6556 L .27681 .65054 L .28147 .64588 L .28652 .64083 L .29118 .63617 L .29623 .63112 L .30089 .62646 L .30594 .62141 L .31061 .61675 L .31566 .61169 L .32032 .60703 L .32537 .60198 L .33003 .59732 L .33508 .59227 L .33974 .58761 L .34479 .58256 L .34946 .5779 L .35451 .57284 L .35917 .56818 L .36422 .56313 L .36888 .55847 L .37393 .55342 L .37859 .54876 L .38364 .54371 L .38831 .53905 L .39336 .53399 L .39802 .52933 L .40307 .52428 L .40773 .51962 L .41278 .51457 L .41744 .50991 L .42249 .50486 L .42716 .5002 L .43221 .49514 L .43687 .49048 L .44192 .48543 L .44658 .48077 L .45163 .47572 L .45629 .47106 L .46134 .46601 L .46601 .46134 L .47106 .45629 L .47572 .45163 L .48077 .44658 L .48543 .44192 L .49048 .43687 L .49514 .43221 L .5002 .42716 L Mistroke .50486 .42249 L .50991 .41744 L .51457 .41278 L .51962 .40773 L .52428 .40307 L .52933 .39802 L .53399 .39336 L .53905 .38831 L .54371 .38364 L .54876 .37859 L .55342 .37393 L .55847 .36888 L .56313 .36422 L .56818 .35917 L .57284 .35451 L .5779 .34946 L .58256 .34479 L .58761 .33974 L .59227 .33508 L .59732 .33003 L .60198 .32537 L .60703 .32032 L .61169 .31566 L .61675 .31061 L .62141 .30594 L .62646 .30089 L .63112 .29623 L .63617 .29118 L .64083 .28652 L .64588 .28147 L .65054 .27681 L .6556 .27176 L .66026 .26709 L .66531 .26204 L .66997 .25738 L .67502 .25233 L .67968 .24767 L .68473 .24262 L .68939 .23796 L .69445 .23291 L .69911 .22824 L .70416 .22319 L .70882 .21853 L .71387 .21348 L .71853 .20882 L .72358 .20377 L .72824 .19911 L .7333 .19406 L .73796 .18939 L .74301 .18434 L Mistroke .74767 .17968 L .75272 .17463 L .75738 .16997 L .76243 .16492 L .76709 .16026 L .77215 .15521 L .77681 .15054 L .78186 .14549 L .78652 .14083 L .79157 .13578 L .79623 .13112 L .80128 .12607 L .80594 .12141 L .811 .11636 L .81566 .11169 L .82071 .10664 L .82537 .10198 L .83042 .09693 L .83508 .09227 L .84013 .08722 L .84479 .08256 L .84985 .07751 L .85451 .07284 L .85956 .06779 L .86422 .06313 L .86927 .05808 L .87393 .05342 L .87898 .04837 L .88364 .04371 L .8887 .03866 L .89336 .03399 L .89841 .02894 L .90307 .02422 L .90812 .01923 L Mfstroke .667 g .98077 .04576 m .97561 .04837 L .97106 .05547 L .96845 .05808 L .96134 .06518 L .95874 .06779 L .95163 .0749 L .94902 .07751 L .94192 .08461 L .93931 .08722 L .93221 .09432 L .9296 .09693 L .92249 .10403 L .91989 .10664 L .91278 .11375 L .91017 .11636 L .90307 .12346 L .90046 .12607 L .89336 .13317 L .89075 .13578 L .88364 .14288 L .88104 .14549 L .87393 .1526 L .87132 .15521 L .86422 .16231 L .86161 .16492 L .85451 .17202 L .8519 .17463 L .84479 .18173 L .84219 .18434 L .83508 .19145 L .83247 .19406 L .82537 .20116 L .82276 .20377 L .81566 .21087 L .81305 .21348 L .80594 .22058 L .80333 .22319 L .79623 .2303 L .79362 .23291 L .78652 .24001 L .78391 .24262 L .77681 .24972 L .7742 .25233 L .76709 .25943 L .76448 .26204 L .75738 .26915 L .75477 .27176 L .74767 .27886 L .74506 .28147 L .73796 .28857 L .73535 .29118 L .72824 .29828 L .72563 .30089 L .71853 .308 L .71592 .31061 L .70882 .31771 L .70621 .32032 L .69911 .32742 L .6965 .33003 L .68939 .33713 L .68678 .33974 L .67968 .34685 L 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.15987 .91278 L .15521 .91744 L .15015 .92249 L .14549 .92716 L .14044 .93221 L .13578 .93687 L .13073 .94192 L .12607 .94658 L .12102 .95163 L .11636 .95629 L .1113 .96134 L .10664 .96601 L .10159 .97106 L .09693 .97578 L .09188 .98077 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{300, 300}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False, ImageRangeCache->{{{0, 287}, {287, 0}} -> {-8.10566, -8.02857, 0.0536322, \ 0.0536322}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] ContourGraphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ As before, we generate a table of numerical values by evaluating \ the potential at the grid points:\ \>", "Text"], Cell["\<\ V0 = Table[potfnc[x,y], {y,numleft[[2]]+dx/2,numright[[2]]-dx/2,dx}, {x,numleft[[1]]+dx/2,numright[[1]]-dx/2,dx}];\ \>", "Input"], Cell["\<\ Finally, we send these values to QuantumKernel by generating a \ function object\ \>", "Text"], Cell["V = QNewFunction[Re[V0],Im[V0]];", "Input"], Cell[TextData[{ StyleBox["Attention", FontWeight->"Bold"], ": Since QuantumKernel accepts complex potentials, the values have to be \ passed as an array of complex numbers. In our case the imaginary part is \ equal to zero." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Schr\[ODoubleDot]dinger operator", "Section"], Cell["\<\ Now we can define an operator object, which defines the generator \ of the time evolution. In QuantumKernel, we can generate several types of \ operator objects. Here we discuss the Schr\[ODoubleDot]dinger equation in two \ dimensions\ \>", "Text"], Cell[BoxData[ \(\[ImaginaryI] \(\[PartialD]\/\[PartialD]t\) \[Psi] \((x, t)\)\ = \ \(-\(1\/\(2 m\)\)\) \((\[PartialD]\^2\/\[PartialD]x\^2 + \[PartialD]\^2\/\[PartialD]y\^2)\) \[Psi] \((x, t)\)\ + \ e\ V \((x, t)\) \[Psi] \((x, t)\)\)], "NumberedEquation"], Cell["\<\ where m is the mass and e is the charge of the particle. Here\ \>", "Text"], Cell[BoxData[ \(H\ = \ \(-\(1\/\(2 m\)\)\) \((\[PartialD]\^2\/\[PartialD]x\^2 + \[PartialD]\^2\/\[PartialD]y\^2)\)\ + \ e\ V \((x, t)\)\)], "NumberedEquation"], Cell["is the Schr\[ODoubleDot]dinger operator.", "Text"], Cell["\<\ The QuantumKernel command for defining a two-dimensional Schr\ \[ODoubleDot]dinger operator is \ \>", "Text"], Cell["\<\ QSchroedinger2D[scalar_potential, vector_potential, domain_mask, \ mass, charge, step_size]\ \>", "Program"], Cell["\<\ Here the arguments mean the following: scalar_potential is a \ reference to a function object representing the scalar potential in the Schr\ \[ODoubleDot]dinger equation. Similarly, vector_potential represents a magnetic vector potential which can \ be used to describe an additional magnetic field. Insert \"None\" if there is \ no vector potential. domain_mask is a function object describing a domain \ with Dirichlet boundary conditions. The domain mask is a function object \ defined by a single two-dimensional array of real numbers. In this array, \ positive values define the region of the numerical simulation, with a \ Dirichlet boundary condition at the border of that region. The solution is \ always set equal to zero in the region where the domain mask has, negative \ values. This can be used to describe particles in a region surrounded by \ infinitely high potential barriers. If there is no domain mask, insert \"None\ \". In this case, Dirichlet boundary conditions are assumed at the borders of \ the rectangular numerical domain. The parameters mass, charge and dx can be \ used for scaling purposes. Here we always assume that mass = 1., charge = 1., \ and for the step size we simply insert dx.\ \>", "Text"], Cell["For our example, we define", "Text"], Cell[CellGroupData[{ Cell["H = QSchroedinger2D[V,None,None,1.,1.,dx]", "Input"], Cell[BoxData[ \(QOperatorObject[1]\)], "Output"] }, Open ]], Cell[TextData[{ "QuantumKernel returns an \"operator object\", which gets the name \"H\" in \ ", StyleBox["Mathematica", FontSlant->"Italic"], ". The data structure in QuantumKernel represents a space-discretization of \ the differential operator." }], "Text"], Cell[CellGroupData[{ Cell["QGetOperatorInfo[H]", "Input"], Cell[BoxData[ \({Type \[Rule] Schroedinger2D, ScalarPotential \[Rule] QFunctionObject[2], VectorPotential \[Rule] None, Domain \[Rule] None, Mass \[Rule] 1.`, Charge \[Rule] 1.`, Units \[Rule] 0.14000000059604645`}\)], "Output"] }, Open ]], Cell["\<\ The above command provides some information about the just defined \ operator object. We can also obtain a summary about the state of \ QuantumKernel with all objects defined so far:\ \>", "Text"], Cell[CellGroupData[{ Cell["QInfo[]", "Input"], Cell[BoxData[ \({{{QFunctionObject[1], {100, 100, 2}}, {QFunctionObject[2], {100, 100, 2}}}, {{QOperatorObject[1], {Type \[Rule] Schroedinger2D, ScalarPotential \[Rule] QFunctionObject[2], VectorPotential \[Rule] None, Domain \[Rule] None, Mass \[Rule] 1.`, Charge \[Rule] 1.`, Units \[Rule] 0.14000000059604645`}}}, {}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The time evolution", "Section"], Cell["\<\ Having defined the generator of the time evolution, we can now \ calculate a time-step. QuantumKernel provides the command QTimeEvolution, \ which needs several parameters.\ \>", "Text"], Cell["\<\ QTimeEvolution[operator_object, function_object, time_step, order, \ repetitions]\ \>", "Program"], Cell[TextData[{ "This command performs the time evolution generated by the Hamiltonian \ operator \"operator_object\". It evolves the initial function \ \"function_object\" for the time repetitions*time_step. The parameter \"order\ \" is a positive integer describing the order of the method. The continuous \ time-evolution exp[i(dt)H] for a small time interval dt is approximated by \ discrete steps giving an approximation of order O(", Cell[BoxData[ \(TraditionalForm\`\((dt)\)\^order\)]], ") for the continuous time-evolution. Higher orders of course mean a better \ approximation, but also take a longer time to execute (because the time step \ dt is subdivided into smaller units). 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