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rowGF%F%-I%sqrtGF%6#,$*&\"\"#\"\"\"F$F/F/F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 130 "We need to propagate two independent solutions, an d then to apply matching conditions. The Schroedinger wave equation is given as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "SE:=En->-1/2* diff(u(x),x$2)+(V-En)*u(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I#E nG6\"F%6$I)operatorGF%I&arrowGF%F%/,&*&#\"\"\"\"\"#F--I%diffG%*protect edG6$-I\"uGF%6#I\"xGF%-I\"$GF16$F6F.F-!\"\"*&,&I\"VGF%F-F$F:F-F3F-F-\" \"!F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 35 "The boundary condit ion states that:" }}{PARA 0 "" 0 "" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 69 "1) From the left we have an incident free plane wave wit h wavenumber " }{TEXT 211 1 "k" }{TEXT 246 219 " and of known amplitud e, and a reflected free plane wave with unknown amplitude (whose magni tude squared will eventually become the reflection probability; (the c hoice of side from which the wave comes in is arbitrary)." }}{PARA 0 " " 0 "" {TEXT 246 92 "2) To the right of the potential region we have a n outgoing free plane wave with wavenumber " }{TEXT 212 1 "k" }{TEXT 246 153 " and unknown magnitude. The squared magnitude of the transmit ted wave becomes the transmission probability (if the incident wave is normalized to unity)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 53 "We kno w from (2) that the transmitted flux goes like " }{TEXT 213 1 "T" } {TEXT 246 6 "exp(i " }{TEXT 214 2 "kx" }{TEXT 246 100 "). Thus, te rat io of the derivative of the physical solution to the solution itself h as to equal i " }{TEXT 215 1 "k" }{TEXT 246 1 "." }}{PARA 0 "" 0 "" {TEXT 246 143 "Therefore, the strategy is to generate two independent \+ solutions, and to form the linear combination that satisfies the reque sted relationship." }}{PARA 0 "" 0 "" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 125 "The initial and final values for the independent varib le range in the numerical integration of the differential equation are :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x_i:=-2; x_f:=4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 24 "We pick an energy value:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "En:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 65 "We pick two intial conditions that provide independent solutions:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "IC1:=u(x_i)=1,D(u)(x_i)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-I\"uG6\"6#!\"#\"\"\"/--I\"DG6$%*p rotectedGI(_syslibGF&6#F%F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "IC2:=u(x_i)=1,D(u)(x_i)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/- I\"uG6\"6#!\"#\"\"\"/--I\"DG6$%*protectedGI(_syslibGF&6#F%F'\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol1:=dsolve(\{SE(En),IC1\}, u(x),numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u1:=subs(sol1,u(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u1p:=subs(sol1,diff(u(x),x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "u1(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!3_?c 4#HlC+\"!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "u1p(x_f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"3c$oZB\\`hs)!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 24 "Now the second solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol2:=dsolve(\{SE(En),IC2\},u(x),numeric,output=l istprocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%/I\"xG6\"f*6#F$6)I%_ resGF%I%_datGF%I*_solnprocGF%I&_xoutGF%I'_ndsolGF%I&_parsGF%I#_iGF%6#I inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF%F%C +@$2\"\"\"%&nargsGYQBinvalid~input:~too~many~argumentsF%>I8_EnvDSNumer icSaveDigitsGF%I'DigitsGF%>F;\"#9@%/I-_EnvInFsolveGF%I%trueG%*protecte dG>F,-&I&evalfGFB6#F:F'>F,-FFF'>F*-I%evalGFB6#&IF+-_F*I$GetGF%6#Q/soln_procedureF%>F.-I$map GFB6$I$rhsGFB-FU6#Q'paramsF%@$4-I%typeGFB6$F,.I(numericGFBC$@0-I'membe rGFB6$F$78Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q%leftF%.I%leftGF% Q&rightF%.I&rightGF%Q)leftdataF%Q*rightdataF%Q+enginedataF%Q*eventstop F%.I*eventstopGF%Q+eventclearF%.I+eventclearGF%Q,eventstatusF%.I,event statusGF%Q'laxtolF%.I'laxtolGF%Q'numfunF%.I'numfunGF%I%NULLGFBC$>F)-F+ 6#-I(convertGFB6$F$.I'stringGFB@(2F5-I%nopsGFB6#7#F)OF)-F^o6$F).I&arra yGFBO-FL6$F)F50F)Q)procnameF%Ffr-Feo6$F$7'Q%lastF%.I%lastGF%Q(initialF %.I(initialGF%FfqC$Fhq@%-F^o6$F).I%listGFBO&F)6#F5OFfq-Feo6$F$7'Q+para metersF%.I+parametersGF%Q7initial_and_parametersF%.I7initial_and_param etersGF%FfqC%>F,F[r>F)-F+6#F,@%/F,FftO7#-I$seqGFB6$/&F.6#F/&F)Fju/F/;F 5-Fcr6#F.O7$F`t-Ffu6$/Fiu&&FerFQFjuF\\v3-F^o6$F,I\"=GFB-Feo6$-I$lhsGFB F`u7)FfsFgsFftFgtFitFjtFfqC%>F,/-F\\r6$-F^wF'F^r-FgnF'@)/F]wFfs@%-F^o6 $-FgnF`uF]tF^u>F)-F+6#/Ffs7%Q'singleF%F5F\\x4FjwYQfninitial~and/or~par ameter~values~must~be~specified~in~a~listF%3/F]wFit/-Fcr6#F\\x,&F^vF5F 5F5>F)-F+6#/F]w7%FbxF5-I#opGFBFjxF^u@'FhwF_t/F]wFftFcuF`v3Fhv-Feo6$F]w 7+Q-eventdisableF%.I-eventdisableGF%Q,eventenableF%.I,eventenableGF%Q+ eventfiredF%.I+eventfiredGF%Q*directionF%.I*directionGF%FfqO-F+6#Fbw/F ,Q.solnprocedureF%O-FL6#F+/F,Q(sysvarsF%O-FU6#F^[l@%0%)procnameGI(unkn ownGF%O-.Fd[lF'C%>F--I-tools/gensymGFP6#Q\"xF%-FL6#-I*FromInertGFB6#-I 0_Inert_FUNCTIONGF%6$-I,_Inert_NAMEGF%6#Q'assignF%-I._Inert_EXPSEQGF%6 $-I(ToInertGFB6#F--I0_Inert_VERBATIMGF%6#-I(pointtoGFB6#&-FU6#Q0soln_p roceduresF%FatO-Fb\\l6#-Fe\\l6$F^]l-F\\]l6#-F_]lF'Z%C$F^uF`tF%YF%F%F%F %/-I\"uGF%F'f*F'F(F0F%C+F3F9FFJFSFY@$F\\oC$@0FdoFgqF`sC$Fhq@%F[tO&F )FQFbtFctC%F]uF^u@%FbuFcuO7$Fa_lFbvFgvC%Faw@)Fhw@%FjwF^u>F)-F+6#/Ffs7% FbxFRF\\xFcxFdxFfx>F)-F+6#/F]w7%FbxFRFayF^u@'FhwF`_lFdyFcuFd_lFeyFezFh zFjzF][lF_[l@%Fc[lFf[lC%>F--F\\\\l6#Q%u(x)F%-FL6#-Fb\\l6#-Fe\\l6$Fg\\l -F\\]l6$F^]l-Fb]l6#-Fe]l6#&Fh]lFQF[^lZ%C$F^uFa_lF%Fe^lF%F%F%/-I%diffGF B6$Fg^lF$f*F'F(F0F%C+F3F9FFJFSFY@$F\\oC$@0FdoFgqF`sC$Fhq@%F[tO&F)6# \"\"$FbtFctC%F]uF^u@%FbuFcuO7$FeblFbvFgvC%Faw@)Fhw@%FjwF^u>F)-F+6#/Ffs 7%FbxFgblF\\xFcxFdxFfx>F)-F+6#/F]w7%FbxFgblFayF^u@'FhwFdblFdyFcuFjblFe yFezFhzFjzF][lF_[l@%Fc[lFf[lC%>F--F\\\\l6#Q-diff(u(x),x)F%-FL6#-Fb\\l6 #-Fe\\l6$Fg\\l-F\\]l6$F^]l-Fb]l6#-Fe]l6#&Fh]lFfblF[^lZ%C$F^uFeblF%Fe^l F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "# not needed as \+ we picked real-valued initial conditions #`dsolve/numeric/init_y0`:=su bs(numeric=complex(numeric),eval(`dsolve/numeric/init_y0`));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol2:=dsolve(\{SE(En),IC2\}, u(x),numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u2:=subs(sol2,u(x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u2p:=subs(sol2,diff(u(x),x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "u2(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!381t%>/lXk *!#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "u2p(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"3+KE<%zrq$=!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 74 "We generated the two independent solutions using real bo undary conditions." }}{PARA 0 "" 0 "" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 270 "Now we assemble the complex-valued solution that satis fies the physical boundary condition of only a transmitted wave existi ng to the right of x_f. We need a single mixing coefficient, and deter mine it by imposing that the ratio of derivative over the solution equ als i " }{TEXT 216 1 "k" }{TEXT 246 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k:=kE(En);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "c2:='c2': c2:=solve((u1p(x _f)+c2*u2p(x_f))/(u1(x_f)+c2*u2(x_f))=I*k,c2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"/0;5s`pf!#9!\"\"*&$\"/1]]\"yGK#F%\"\"\"^#F*F*F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 246 126 "Now we can form the physical sol ution, which has arbitrary normalization at this point. We graph the r eal and imaginary parts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "P1:=plot('Re(u1(x)+c2*u2(x))',x=-12..14,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P2:=plot('Im(u1(x)+c2*u2(x))',x=-12 ..14,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plots[d isplay](P1,P2);" }}{PARA 13 "" 1 "" {TEXT 247 0 "" }{GLPLOT2D 702 400 400 {PLOTDATA 2 "6&-%'CURVESG6$7eu7$$!$?\"!\"\"$\"/JYp^Oe?!#97$$!/B\\^!#97$$!/voN@n=6!#7$!/Y#\\*p%\\F'!#97$$!/z%R]Wj5\"!#7$!/\"pBLAa I(!#97$$!/$3A(o,%4\"!#7$!/yVBv,D#)!#97$$!/D\")\\%*Gm5!#7$!-s'p_!>)*!#7 7$$!/nTF?cQ5!#7$!/[!pP,j1\"!#87$$!/#zHA%eJ5!#7$!/(\\77H[2\"!#87$$!/1R5!#87$$!/n;>fRF)*!#8$!/XFxPL*))* !#97$$!.Yh6.&)o*!#7$!/Chq\")[*>*!#97$$!.DJJ5'\\&*!#7$!/&)3l\"eCL)!#97$ $!./,^<2T*!#7$!/Zg[I%\\I(!#97$$!/L32Z#=F*!#8$!/v)3XLn8'!#97$$!/l;*H`I9 *!#8$!/N\\_*Qv%\\!#97$$!-D\"*=G9!*!#6$!.w85+Ar!#6$\"/jEq0zx5!#87$$ !.tb3380(!#7$\"/OTr]6x5!#87$$!.'RZUh!)p!#7$\"/`]5*e52\"!#87$$!.>#4/#*4 p!#7$\"/FPP@lf5!#87$$!/:/rlARo!#8$\"/h^\\=&H/\"!#87$$!/vo%*)Qyp'!#8$\" /Rb4VIS**!#97$$!/LL=7Xcl!#8$\".aR/5FD*!#87$$!-vLDQ2j!#6$\"/sR)f**Gg(!# 97$$!/m;\\QJeg!#8$\"/'eN/vQ[&!#97$$!.Dcn<\"=f!#7$\"/#>B[0H8%!#97$$!/N3 -:#zx&!#8$\"/bTYp$3q#!#97$$!.U&G`sPc!#7$\"/ve&Gqd@\"!#97$$!+b\"Hv\\&! 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R!#97$$\"/&e9Ex!\\))!#8$\"/)fsplF`%!#97$$\"-D\"y:!H\"*!#6$\".bW&[()3[! #87$$\"/0w(H>k>*!#8$\".D\"**)G'>[!#87$$\".rU\"G#QE*!#7$\".a1^%\\3[!#87 $$\".\"yIjAJ$*!#7$\"/&*o(GAbx%!#97$$\"/:HZ)H')R*!#8$\".;K&>'3s%!#87$$ \".7.)oVL&*!#7$\"/3Z;tcZX!#97$$\".LL\"RCo'*!#7$\"/W.0^v\"H%!#97$$\".3_ d.h!)*!#7$\"/N)Q)Rl\\R!#97$$\".$3PK'R%**!#7$\"/`_YugKN!#97$$\"/e*)*G#= 35!#7$\"/A\\M*G&[I!#97$$\"/L3c#o>-\"!#7$\"/JgsPg1D!#97$$\"//^n+kN5!#7$ \"/$em&>;A>!#97$$\"/v$*y=J\\5!#7$\"/O;\"[Z=I\"!#97$$\"/YO!p$)H1\"!#7$ \"/$3AmoBd'!#:7$$\"/\"!#97$$\"/z>EWM96!#7$!/p!pvKMx\"!#97 $$\"+,u!p7\"!\")$!.C:B&)4K#!#87$$\"/U&y)fIT6!#7$!/z\\#HfI!H!#97$$\"/%3 Zd/d:\"!#7$!/=.I80DM!#97$$\"/EchJ5q6!#7$!/=J&3eh(Q!#97$$\"/nT[<]%=\"!# 7$!/RpVJ/ZU!#97$$\"/e9;+Q(>\"!#7$!/-#eV\"\\/X!#97$$\".vQGe-@\"!#6$!/.Q #=Oto%!#97$$\"/'RxT(p;7!#7$!/s^(*fy\\Z!#97$$\"/Ug^l8B7!#7$!.<`))[Dz%!# 87$$\"/)oaov&H7!#7$!/Nc.vW:[!#97$$\"/LL>[,O7!#7$!/e)G.(Q=[!#97$$\"/#H# emZj7!#7$!/CsB793Y!#97$$\".Dr\\Q4H\"!#6$!/8`L)\\D0%!#97$$\"/D1R23/8!#7 $!/#HkuPdn$!#97$$\"+\")HA<8!\")$!/(4wDIbB$!#97$$\"/v$HAl.L\"!#7$!/`%[q >&RF!#97$$\".v[Y2NM\"!#6$!/$*=i.E'>#!#97$$\"/il)fIwN\"!#7$!/.3HL]q:!#9 7$$\"/vVKPvr8!#7$!/8Z=qsM\"*!#:7$$\"/)=i'o(eQ\"!#7$!/(fOH>DQ#!#:7$$\"$ S\"!\"\"$\"/w`(QKrT%!#:-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\" \"-%%VIEWG6$;$!$?\"!\"\"$\"$S\"!\"\"%(DEFAULTG-%+AXESLABELSG6'-I#miG6# /I+modulenameG6\"I,TypesettingGI(_syslibG6\"65Q\"x6\"/%'familyGQ(DEFAU LT6\"/%%sizeGQ#106\"/%%boldGQ&false6\"/%'italicGQ%true6\"/%*underlineG Q&false6\"/%*subscriptGQ&false6\"/%,superscriptGQ&false6\"/%+foregroun dGQ([0,0,0]6\"/%+backgroundGQ.[255,255,255]6\"/%'opaqueGQ&false6\"/%+e xecutableGQ&false6\"/%)readonlyGQ&false6\"/%)composedGQ&false6\"/%*con vertedGQ&false6\"/%+imselectedGQ&false6\"/%,placeholderGQ&false6\"/%6s election-placeholderGQ&false6\"/%,mathvariantGQ'italic6\"Q!6\"-%%FONTG 6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZONTALG%+HORIZONTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" }}{TEXT 247 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 246 150 "So far the incident wave has arb itrary normalization. Let us see whether we can figure the normalizati on out from the solution in region I (at x=x_i)." }}{PARA 0 "" 0 "" {TEXT 246 112 "We match now at the x_i a solution which has independen t amplitudes for the incoming and reflected waves, named " }{TEXT 220 1 "N" }{TEXT 246 5 " and " }{TEXT 221 1 "R" }{TEXT 246 14 " respective ly:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq1:=N*exp(I*k*x_i)+ R*exp(-I*k*x_i)=u1(x_i)+c2*u2(x_i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "/ ,&*&I\"NG6\"\"\"\"-I$expG6$%*protectedGI(_syslibGF&6#,$*&\"\"#F'^#F'F' !\"\"F'F'*&I\"RGF&F'-F)6#F/F'F',&$\"/&R)*yi/.%!#9F'*&$\"/1]]\"yGK#F:F' F1F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eq2:=I*(k*N*exp(I *k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+c2*u2p(x_i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "/*&^#\"\"\"F%,&*&I\"NG6\"F%-I$expG6$%*protectedGI(_syslib GF)6#,$*&\"\"#F%F$F%!\"\"F%F%*&I\"RGF)F%-F+6#F1F%F3F%,&$F%\"\"!F%*&$F: F:F%F$F%F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{ eq1,eq2\},\{N,R\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I\"NG6\",&$\"/ ?wE&[Rw%!#9\"\"\"*&$\"/Ag,\"3lR%F)F*^#F*F*F*/I\"RGF%,&$\"/Ao,9w^EF)F** &$\"/pe2(\\)HMF)F*F.F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 16 "We observe t hat " }{TEXT 222 1 "N" }{TEXT 246 5 " and " }{TEXT 223 1 "R" }{TEXT 246 51 " are amplitudes, and turn out to be complex-valued." }}{PARA 0 "" 0 "" {TEXT 246 59 "The properly normalized reflection coefficient \+ now becomes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=R/N;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&$\"/^qAFi@e!#:!\"\"*&$\"/*QR$RLim!#9\" \"\"^#F+F+F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 33 "and the transmis sion coefficient:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "t:=eva lf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"/Zu,tlqO!#9!\"\"*&$\"/-`J'[`Y'F%\"\"\"^#F*F*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 103 "The check whether this all makes sense \+ is to sum the magnitude squared for reflection and transmission:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "abs(r)^2+abs(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/l7V1++5!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 116 "Within numerical accuracy we preserve the norm; therefo re we interpret as reflection and transmission probabilities:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "abs(r)^2,abs(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/q&ejgDZ%!#9$\"/\"3azXu_&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 122 "Now we turn the calculation into a proc edure: for given energy it calculates the reflection and transmission \+ coefficients." }}{PARA 0 "" 0 "" {TEXT 246 66 "Our previously develope d commands are used step-by-step as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Tunnel:=proc(En) local IC1,IC2,sol1,sol2,u1,u1p,u 2,u2p,k,c2,N,R,r,t,sol,eq1,eq2; global V,kE,x_i,x_f;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "IC1:=u(x_i)=1,D(u)(x_i)=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "IC2:=u(x_i)=1,D(u)(x_i)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol1:=dsolve(\{SE(En),IC1\},u(x),numeric,output=listp rocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u1:=subs(sol1,u(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u1p:=subs(sol1,diff(u(x),x));" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol2:=dsolve(\{SE(En),IC2\},u(x),n umeric,output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u2 :=subs(sol2,u(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u2p:=subs(sol 2,diff(u(x),x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k:=kE(En);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "c2:=solve((u1p(x_f)+c2*u2p(x_f))/(u 1(x_f)+c2*u2(x_f))=I*k,c2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N:=' N': R:='R':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq1:=N*exp(I*k*x_i)+ R*exp(-I*k*x_i)=u1(x_i)+c2*u2(x_i);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eq2:=I*(k*N*exp(I*k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+c2*u2p(x_i);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{eq1,eq2\},\{N,R\}); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=R/N;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "t:=ev alf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "#print(\" R= \",abs(r)^2,\" T= \",abs(t)^2,\" R+T= \" ,abs(r)^2+abs(t)^2,\" norm equal to 1.0 ?\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "[abs(r)^2,abs(t)^2]; end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 51 "We check it for the previously tested energy value:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Tunnel(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"/q&ejgDZ%!#9$\"/\"3azXu_&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 154 "Now we are ready to scan the energy range from \+ zero to above the peak height of the barrier. The results are accumula ted in lists for subsequent plotting." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Tr:=[]: Rf:=[]: for iE from 1 to 60 do: En:=iE/20; r es:=Tunnel(En); Tr:=[op(Tr),[En,res[2]]]: Rf:=[op(Rf),[En,res[1]]]: od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P1:=plot(Rf,color=red) : P2:=plot(Tr,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plots[display](P1,P2,labels=[\"E\",\"\{t,r\}\"]);" }}{PARA 13 "" 1 "" {TEXT 247 0 "" }{GLPLOT2D 702 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7 hn7$$\"\"&!\"#$\"/$yv8\">l**!#97$$\"\"\"!\"\"$\".N@9D(3**!#87$$\"#:!\" #$\"/wd.jD=)*!#97$$\"\"#!\"\"$\"/.jN'3Sn*!#97$$\"#D!\"#$\"/0PD(zTW*!#9 7$$\"\"$!\"\"$\"/,4hx3y!*!#97$$\"#N!\"#$\"/9;(p#*y\\)!#97$$\"\"%!\"\"$ \"/z)Hdecf(!#97$$\"#X!\"#$\"/x%\\KS\"fi!#97$$\"\"&!\"\"$\".dejgDZ%!#87 $$\"#b!\"#$\"/:_*p-M\\#!#97$$\"\"'!\"\"$\".v?>:'o()!#97$$\"#l!\"#$\"/0 >l;sh!)!#;7$$\"\"(!\"\"$\"/TA=eTs#)!#;7$$\"#v!\"#$\"/([i')H#za!#:7$$\" \")!\"\"$\"/l.Tcil6!#97$$\"#&)!\"#$\".:!GTak@,O!#97$$\"$N\"!\"#$\"/P3-\\efN!#97$$\"#9!\"\"$\"/d$47jE\\$!#9 7$$\"$X\"!\"#$\"/<-$H-SS$!#97$$\"#:!\"\"$\"/o=b**p'H$!#97$$\"$b\"!\"#$ \"/MfO9_tJ!#97$$\"#;!\"\"$\"/jy\"[%)p.$!#97$$\"$l\"!\"#$\"/K(QOX%*)G!# 97$$\"#!\"\"$\"/WlC!f71#!#97$ $\"$&>!\"#$\"/vc?Y>\"*=!#97$$\"#?!\"\"$\"/b_2ivBkjgC'!#:7$$\"$X#! \"#$\"/Jf!G])z_!#:7$$\"#D!\"\"$\"/LdOYR6W!#:7$$\"$b#!\"#$\"/,/*e$\\QO! #:7$$\"#E!\"\"$\"/n&*f5 !#:7$$\"$&G!\"#$\"/7W#R;4j(!#;7$$\"#H!\"\"$\"/9t%)*z'\\_!#;7$$\"$&H!\" #$\"/Q#=SzTR$!#;7$$\"#I!\"\"$\"/z$R-WX+#!#;-%&COLORG6&%$RGBG$\"#5!\"\" $\"\"!!\"\"$\"\"!!\"\"-%'CURVESG6$7hn7$$\"\"&!\"#$\"/*3lR))3[$!#;7$$\" \"\"!\"\"$\"/#QKXfu7*!#;7$$\"#:!\"#$\"/ir.!Ru\"=!#:7$$\"\"#!\"\"$\"/-e \"z9*fK!#:7$$\"#D!\"#$\".[i.2#eb!#97$$\"\"$!\"\"$\"/&4)R\"H\">#*!#:7$$ \"#N!\"#$\"/9v!y3@]\"!#97$$\"\"%!\"\"$\"/dtVNM/C!#97$$\"#X!\"#$\"/Om[O '3u$!#97$$\"\"&!\"\"$\"/\"3azXu_&!#97$$\"#b!\"#$\".Rf%[g1v!#87$$\"\"'! \"\"$\"/Xg!4[J7*!#97$$\"#l!\"#$\"/fSMPR>**!#97$$\"\"(!\"\"$\"/b9\"3)G< **!#97$$\"#v!\"#$\"/Wjww3_%*!#97$$\"\")!\"\"$\"/lp#=%QM))!#97$$\"#&)! \"#$\"/TNe%oaB)!#97$$\"\"*!\"\"$\"/0@CpEVc5r!#97$$\"#!\"\"$\"/r*Rab(Qz!#97$$\"$&>!\"#$\"/v**R)>)3\")!#97$$\" #?!\"\"$\"/o+\\%fiF)!#97$$\"$0#!\"#$\".$[_*e&R%)!#87$$\"#@!\"\"$\"/,t6 *))!#97$$\"$D #!\"#$\"/KCui%o-*!#97$$\"#B!\"\"$\"/2s\\)fE:*!#97$$\"$N#!\"#$\".UV(4$* o#*!#87$$\"#C!\"\"$\"//h(>8aP*!#97$$\"$X#!\"#$\"/[?f].s%*!#97$$\"#D!\" \"$\"/$=F1#))e&*!#97$$\"$b#!\"#$\"/;[;<1!" }}{PARA 0 "" 0 "" {TEXT 246 7 "2) F or " }{TEXT 229 1 "E" }{TEXT 246 640 "=0.66 perfect transmission and n o reflection is observed. This is the phenomenon of a transition reson ance. The resonance structure has a width associated with it. Most phy sical phenomena that have a lifetime associated with them (unstable el ementary particles, alpha decay, nuclear gamma lines, atomic optical t ransitions, excitations in semiconductors, etc.) can be understood as \+ processes where the wave equation describing the matter (or radiation) field involves a tunneling process. Broad resonances are associated w ith short lifetimes, sharp resonances with long lifetimes, according t o Heisenberg's energy-time uncertainty relation." }}{PARA 0 "" 0 "" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 157 "The transmission reson ance is quite broad. 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