{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 56 "Quantum tunneling throu gh an arbitrary potential barrier" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 481 "We are i nterested in solving the Schroedinger wave equation with scattering bo undary conditions for an arbitrary potential function in one dimension . The boundary conditions (incident and reflected and transmitted wave ) require the solution to be complex-valued. Our attitude will be to a pply knowledge about general solution to a second-order ordinary diffe rential equations (two independent solutions are required, and then ma tched to the physically imposed boundary conditions)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "Textbook treatments b egin with simple potential shapes (square barrier) for which the solut ion can be obtained exactly. For a symbolic solution along these lines we refer the reader to our book [Marko Horbatsch: " }{TEXT 257 29 "Q uantum Mechanics using Maple" }{TEXT -1 428 ", chapter 3.2 (Springer 1 995)]. Here we rely on numerical techniques included in Maple, namely \+ to solve initial-value problems for systems of first-order differentia l equations. The tunneling problem requires us to solve a boundary-val ue problem. This will be achieved by generating two independent unphys ical solutions based on inital values, and then to match a linear comb ination of these to the physical boundary conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 468 "Maple6 has added capa bility to dsolve[numeric] to handle complex-valued differential equati ons. This permits to solve the Schroedinger equation for scattering bo undary conditions directly. To remain compatible with Maple 5 we stay, however, with real-valued boundary conditions for the solution of the differential equation, and introduce the complex-valued nature of the physical solution in the step where linear combinations of two fundam ental solutions are formed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 241 "We could begin with simple barrier penetration . However, we assume that the reader is already familiar with the basi cs of tunneling. Therefore, we proceed with the more interesting doubl e-barrier. An example of a double-barrier potential is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "V:=exp(-5*x^2)+exp(-5*(x-2)^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,&-%$expG6#,$*$)%\"xG\"\"#\"\"\" !\"&F.-F'6#,$*$),&F,F.F-!\"\"F-F.F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(V,x=-2..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 694 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7`r7$$!\"#\"\"!$\"3Ie&QCi`61#!#E7$$ !3GLL$3#*>u%=!#<$\"3keekM)p,)Q!#D7$$!3om;z43m9S*)eRzNGQFR7$$!3:***\\7oCA$)*!#=$\"317ORlfcdzFR 7$$!3Ulm\"z%*4(*3*Fen$\"3T?z1:LX1;!#>7$$!3qJLe9_>Z$)Fen$\"3aTOEcu5pIF] o7$$!3'em;athqg(Fen$\"3k5g&e&o#*QbF]o7$$!3-++Dc#Gp'oFen$\"3cP>s)*4Vj%* F]o7$$!3MK$e*)**4c5'Fen$\"3'Hxoh=I1b\"Fen7$$!3lkmmTV%eq (R#Fen7$$!30K$eR(p+4]Fen$\"3q(R4Kqi@&GFen7$$!3M)**\\i?APn%Fen$\"37_2Y0 ,$[N$Fen7$$!3>l;aQuVQVFen$\"3R16M@h(>!RFen7$$!3/KL$3n_J+%Fen$\"3U9)=qj Cw[%Fen7$$!3D)*\\PM9qDOFen$\"3#GI?G5iD=&Fen7$$!3[km\"z>]#[KFen$\"3!f** Gx.]/!fFen7$$!3qI$e9'*)zqGFen$\"3@SEpZ/vAmFen7$$!3!p****\\sZL\\#Fen$\" 3l)4Z'ouJGtFen7$$!3_(***\\PmM9@Fen$\"3%*e\"o i$z***Fen7$$\"3y7+++&*p$=\"F]o$\"39k<+dn*H***Fen7$$\"3$Q,+]P)))4IF]o$ \"3)\\BGTL0[&**Fen7$$\"3([,++Dxg$[F]o$\"3Is;r2Hu$))*Fen7$$\"3]M$3_]\\p 9\"Fen$\"3=K/Wk&=MO*Fen7$$\"3^nmT&G\"H5=Fen$\"3#es$Fen$\"3`X$RT\"[`-]Fen7$$\"3)eLLeRWg0%Fen$\"3Aumtf5-$R%F en7$$\"3AOLe9M9!R%Fen$\"3zC\"*)[Ph\\\"QFen7$$\"3gOLLLCCCZFen$\"3eY:bM@ @wKFen7$$\"3r-v=9MQ\"F]o7$$\"3 a+]i!o<-1\"F1$\"3i$Qh/ra1d\"F]o7$$\"3#Qe*)f$yr(4\"F1$\"3IJ_lCta[>F]o7$ $\"37nTN\"*z@N6F1$\"3/l'[mJ%GODF]o7$$\"3S](=n9=F<\"F1$\"3anA%HJwzO$F]o 7$$\"3qLL3-$=-@\"F1$\"31HDLW]8([%F]o7$$\"3mLe*)**>4z7F1$\"3I60Rnj?muF] o7$$\"3kL$3xplzM\"F1$\"3'Hc@.JvX>\"Fen7$$\"37+DJ&GbAU\"F1$\"3ioUm&>j[) =Fen7$$\"3gmm\"H([a'\\\"F1$\"3_f[*3VYf\"GFen7$$\"3k;HKuy7N:F1$\"3AYiP? 1B%R$Fen7$$\"3om\"Hd(3rt:F1$\"3#[e4Gqu3.%Fen7$$\"3s;a8xQH7;F1$\"3a2,_+ )Hir%Fen7$$\"3wm;ayo(3l\"F1$\"3+Lz`dygFen7$$\"3[LL$36]!=F1$\"3_uE^-i;g(*Fen7$$\"3[$e9;ue!\\>F 1$\"3k:J*zf(3r)*Fen7$$\"3A+D\"yF'zn>F1$\"3WViii-G[**Fen7$$\"3[e9\"f/lr (>F1$\"3E(R`LBiR(**Fen7$$\"3'pT5S\"Q`')>F1$\"3U=;!zAP4***Fen7$$\"3Av$4 @e-f*>F1$\"3q'[Lfhg\"****Fen7$$\"3ZL$3-Nr_+#F1$\"3%Hv1%e2h)***Fen7$$\" 3%p;/EUYF/#F1$\"3:-JjmK04**Fen7$$\"3)******\\\\@-3#F1$\"3W2CfYuM$o*Fen 7$$\"3')*\\(=UN)o6#F1$\"3)*Qx+t+sR$*Fen7$$\"3=+]P*eXN:#F1$\"3qopO/g,)) ))Fen7$$\"3]+DcOw?!>#F1$\"3A]o7&3K_M)Fen7$$\"3Q++v$opoA#F1$\"30W/L*[\" )4t(Fen7$$\"3?]7`*4UBE#F1$\"3qa6\"3pg%)3(Fen7$$\"3Y+DJ:X\"yH#F1$\"380O h&Qr!=kFen7$$\"3u]P4JpGLBF1$\"3+c'pBm@%QdFen7$$\"3c+](oMf(oBF1$\"3u*p5 p,*em]Fen7$$\"3Q](=nTX\"3CF1$\"3'3=F_Y2yM%Fen7$$\"3?+Dc'[JvW#F1$\"3A\" GudImNn$Fen7$$\"3-]iScv\"p[#F1$\"3'*\\rQ3g5cIFen7$$\"3#)***\\ii.j_#F1$ \"3*3eTPB%H.DFen7$$\"3Km;H(*Q3(f#F1$\"3'e<%GrN4#o\"Fen7$$\"3%GLL$oT'ym #F1$\"3g+_Btb0v5Fen7$$\"3&om;a@PMu#F1$\"35avMR$)42jF]o7$$\"3'3++DE5!>G F1$\"3_(*=iDu\"[\\$F]o7$$\"3gL3_v1\\()GF1$\"3s(\\(fO#G$[>F]o7$$\"3Mm;a )3rf&HF1$\"3CqJq$*QTO5F]o7$$\"3nL3-o#Q3.$F1$\"39$e:N'[WE\\FR7$$\"3*4++ vW0d5$F1$\"3swe;@.29AFR7$$\"3;L$3-\"QfYKF1$\"3P))\\w;W@AUFL7$$\"3C+]PW F'QR$F1$\"3BEiY(fQ:/'FF7$$\"3[LL$e/Xy`$F1$\"3E'HWF1$\"3+QE5C*\\t#>!#I7$$\" 3'pmmmV,&eXF1$\"37Q[+7Q70h!#K7$$\"3<+](o(GP1ZF1$\"3@^;QVU%\\C\"!#L7$$ \"3g+]78Z!z%[F1$\"3#H%y?*R=UW#!#N7$$\"\"&F*$\"3m$R\\0e=D'G!#P-%'COLOUR G6&%$RGBG$\"#5!\"\"$F*F*F^am-%+AXESLABELSG6$Q\"x6\"Q!Fcam-%%VIEWG6$;F( Fb`m%(DEFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "We are interested in discovering so-called transmission resonances. We can safely assum e that the interaction vanishes at " }{TEXT 259 1 "x" }{TEXT -1 8 "=-2 and " }{TEXT 258 1 "x" }{TEXT -1 19 "=4 for our example." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We work in units w here hbar=" }{TEXT 260 1 "m" }{TEXT -1 90 "=1. This simplifies relatio nships between momentum and wavenumber (they become identical)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The wave \+ number (momentum) can be obtained from the non-relativistic energy as: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "kE:=En->sqrt(2*En);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#kEGf*6#%#EnG6\"6$%)operatorG%&arrow GF(-%%sqrtG6#,$9$\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 " We need to propagate two independent solutions, and then to apply matc hing conditions. The Schroedinger wave equation is given as:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "SE:=En->-1/2*diff(u(x),x$2)+ (V-En)*u(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEGf*6#%#EnG6\"6$ %)operatorG%&arrowGF(/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F4\"\"##!\"\"F8 *&,&%\"VG\"\"\"9$F:F>F1F>F>\"\"!F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The boundary condition states that:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "1) From the left we have \+ an incident free plane wave with wavenumber " }{TEXT 261 1 "k" }{TEXT -1 219 " and of known amplitude, and a reflected free plane wave with \+ unknown amplitude (whose magnitude squared will eventually become the \+ reflection probability; (the choice of side from which the wave comes \+ in is arbitrary)." }}{PARA 0 "" 0 "" {TEXT -1 92 "2) To the right of t he potential region we have an outgoing free plane wave with wavenumbe r " }{TEXT 262 1 "k" }{TEXT -1 153 " and unknown magnitude. The square d magnitude of the transmitted wave becomes the transmission probabili ty (if the incident wave is normalized to unity)." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 53 "We know from (2) that the transmitted flux goes li ke " }{TEXT 263 1 "T" }{TEXT -1 6 "exp(i " }{TEXT 264 2 "kx" }{TEXT -1 101 "). Thus, the ratio of the derivative of the physical solution \+ to the solution itself has to equal i " }{TEXT 265 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 143 "Therefore, the strategy is to gene rate two independent solutions, and to form the linear combination tha t satisfies the requested relationship." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The initial and final values for th e independent varible range in the numerical integration of the differ ential equation are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x_i :=-2; x_f:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_iG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_fG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "We pick an energy value:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "En:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnG#\" \"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "An aside:" }}{PARA 0 "" 0 "" {TEXT -1 97 "Experts can fix an oversight in Maple6.0, if they wish to use complex-valued boundary conditions:" }}{PARA 0 "" 0 "" {TEXT -1 93 "For complex-valued solutions in Maple6 override: (Robert \+ Israel contributed this fix to MUG)." }}{PARA 0 "" 0 "" {TEXT -1 90 "` dsolve/numeric/init_y0`:=subs();numeric=complex(numeric),eval(`dsolve/ numeric/init_y0`));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We pick two intial conditions that provide inde pendent 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#>%$IC1G6$/-% \"uG6#!\"#\"\"\"/--%\"DG6#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#>%$IC2G6$/-%\"uG6#!\"#\"\"\"/--%\"DG6#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 "" {TEXT -1 46 "To demonstrate that it calculates something..." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "u1(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3RPN+.iY-5!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "u1p(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3JRyF2.;E()!#= " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now the second solution:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol2:=dsolve(\{SE(En),IC2\}, u(x),numeric,output=listprocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%sol2G7%/%\"xGf*6#F'6\"6#%aoCopyright~(c)~1993~by~the~University~of ~Waterloo.~All~rights~reserved.GF*9$F*F*F*/-%\"uGF)f*F)6&%$resG%*plist procG%)outpointG%&ndsolG6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc. ~All~rights~reserved.GF*C'>8&-%&evalfG6#F->8%f*6#%(rkf45_xG6.%\"iG%/co mp_soln_dataG%(odeprocG%&icvecG%(LR_caseG%\"YG%$valGF5%#ptG%*stop_proc G%+stop_arrayG%&cplexG6#%aoCopyright~(c)~2000~by~the~University~of~Wat erloo.~All~rights~reserved.GE\\s#Q0soln_proceduresF*7%\"(czR%\"(k(*4& \"(#\\'4&Q(complexF*%&falseGC2@$-%'memberG6$F-7$Q4left_comp_soln_dataF *Q5right_comp_soln_dataF*O\"\"!@$2%'DigitsG-%&truncG6#-%'evalhfG6#F`o> F`oFao>8+F<@$4-%%typeG6$Fio.%(numericG@%-Fgn6$Fio7%Q&startF*Q%leftF*Q& rightF*O$!\"#F]oO-.9!F>>F;f*6&%\"NGF'%\"YG%#YPGF*6#%C[Y[1]~=~u(x),~Y[2 ]~=~diff(u(x),x)]GF*C$>&9'6#\"\"\"&9&6#\"\"#>&FjqF_r,$*&,(-%$expG6#,$* $)9%F`rF\\r$!\"&F]oF\\r-Fgr6#,$*$),&F\\sF\\r$F`rF]o!\"\"F`rF\\rF]sF\\r #F\\rF`rFfsF\\r&F^rF[rF\\rF`rF*F*F*>8'=F*6#;F\\rF`rE\\[l#F\\r$F\\rF]oF `r$F]oF]o>8-F]o>8.F]o>8)-%$mapG6$%$rhsG-%&parseG6#-%#opG6$\"\"$-%%eval G6#F;>8/-F^q6#FX@'/FioFipO7$/F'Fio-%$seqG6$/&Fft6#8$-F=6#&FjsFcv/Fdv;F \\r-%%nopsG6#Fft1FipFio>8(F[o>F_wFjn>F@-F^q6#F_w@%4-F]p6$7#F@7#.%%list GC%Z%>F@-%;dsolve/numeric/SC/IVPsolveG6)F;;FipFioFjsFbtFdtFfu/%'scproc GFYF*C&@$/Ffu%%trueGYF*>FfuFhx>F@F_x>FguFhx>8*&&F@6#Fau6#&F@F_r>FbwF@C $>F^yF_y@$53/F_wF[o2F^yFio3/F_wFjn2FioF^yC&@$5/&F@6#\"\")F\\r2\"#5FczC $>F^yF_y@%FjyY6$QQcannot~evaluate~the~solution~further~right~of~%1F*-& F=Fdz6#F^yY6$QPcannot~evaluate~the~solution~further~left~of~%1F*F^[lZ% >F@-%FfuFhx>F@Ff[l>FguFhx>FbwF@>F^yF_y@&FjyC$@$33/%:_Env_smart_dsolve_nume ricGFhx2FipF^y2-F^q6#FgpF^y>Fj\\lF^y@$F[z@)/&FbwFdzF\\rY6$Qcocannot~ev aluate~the~solution~further~right~of~%1,~probably~a~singularityF*F^[l/ F`]lF`rY6$Qcqcannot~evaluate~the~solution~further~right~of~%1,~maxfun~ limit~exceeded~(see~?dsolve,maxfun~for~details)F*F^[l2FgzF`]lC$@$0%7_E nv_dsolve_nowarnstopGFhx-%(WARNINGG6#-_%,StringToolsG%.FormatMessageG6 %Qhocannot~evaluate~the~solution~further~right~of~%1,~stop~condition~# %2~violatedF*F^[l,&F`]lF\\rFgzFfs>FioF^yYF\\[lF]zC$@$33Ff\\l2F^yFip2F^ y-F^q6#Ffp>F__lF^y@$F^z@)F_]lY6$Qbocannot~evaluate~the~solution~furthe r~left~of~%1,~probably~a~singularityF*F^[lFd]lY6$Qbqcannot~evaluate~th e~solution~further~left~of~%1,~maxfun~limit~exceeded~(see~?dsolve,maxf un~for~details)F*F^[lFh]lC$@$F[^l-F^^l6#-Fa^l6%Qgocannot~evaluate~the~ solution~further~left~of~%1,~stop~condition~#%2~violatedF*F^[lFf^l>Fio F^yYFb[l>F^y-%9dsolve/numeric/SC/IVPvalG6$F@Fio7$F]v-F_v6$/Fbv&F^yFcvF hvF*F*F*@$4-F]p6$F;F_p@)-Fgn6$F;FdpO-F@F>/F;Q.procedurelistF*O-Fcu6#F@ 0F^q%(unknownGOF\\qC$>Fjs-%(pointtoG6#&-F@6#FSF_rO-.FjsF>>Fdv7#-%*trap errorG6#Fdal@%-F]p6$&FdvF[r%'stringGY-F_uFcv-Fjt6#&F`clF_rF*F*F*/-%%di ffG6$F/F'f*F)F2F7F*C'>F;F<>F@FA@$F]al@)FaalOFdalFealOFhalFjalOF\\qC$>F js-F`bl6#&FcblFayOFfbl>FdvFibl@%F^clYFccl-Fjt6#&F`clFayF*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "# not needed as we picked r eal-valued initial conditions #`dsolve/numeric/init_y0`:=subs(numeric= complex(numeric),eval(`dsolve/numeric/init_y0`));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "#sol2:=dsolve(\{SE(En),IC2\},u(x),numeric,ou tput=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 "6#$!3y-Zi " 0 "" {MPLTEXT 1 0 9 "u2p(x_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3w@/HrQ2P=!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We generat ed the two independent solutions using real boundary conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 270 "Now we a ssemble the complex-valued solution that satisfies the physical bounda ry condition of only a transmitted wave existing to the right of x_f. \+ We need a single mixing coefficient, and determine it by imposing that the ratio of derivative over the solution equals i " }{TEXT 266 1 "k " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k:=kE(E n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"\"" }}}{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 "6#>%# c2G^$$!+Ca`pf!#5$!+;!yGK#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "N ow we can form the physical solution, which has arbitrary normalizatio n at this point. We graph the real and imaginary parts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "P1:=plot('Re(u1(x)+c2*u2(x))',x=-12 ..14,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P2:=pl ot('Im(u1(x)+c2*u2(x))',x=-12..14,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plots[display](P1,P2);" }}{PARA 13 "" 1 "" {GLPLOT2D 692 229 229 {PLOTDATA 2 "6&-%'CURVESG6$7eu7$$!#7\"\"!$\"+4OO e?!#57$$!+]=$e=\"!\")$\"**G\"GV&F-7$$!++Pmr6F1$!*z#*o#)*F-7$$!+]b\\d6F 1$!+%Gn*)[#F-7$$!+*RFL9\"F1$!+abOXRF-7$$!+o(**48\"F1$!+4YB\\^F-7$$!+O@ n=6F1$!+m.&\\F'F-7$$!+/XM16F1$!+vfU0tF-7$$!+so,%4\"F1$!+43-D#)F-7$$!+] %*Gm5F1$!+@l0>)*F-7$$!+F?cQ5F1$!+c;Im5!\"*7$$!+AUeJ5F1$!+w%H[2\"F[o7$$ !+=kgC5F1$!+_g7y5F[o7$$!+9'Gw,\"F1$!+W`fRF)*F[o$!+P]L*))*F-7$$!+*F-7$$!+9.h\\&*F[o$!+(=fCL)F-7$$!+6vr5%*F[o$!+BN%\\I(F-7$$!+2Z#=F* F[o$!+]KtOhF-7$$!++L0V\"*F[o$!+3%Qv%\\F-7$$!+#*=G9!*F[o$!+LhTwOF-7$$!+ %[5b)))F[o$!+)=:WM#F-7$$!+v!Rnv)F[o$!*%*Hft*F-7$$!+UPSB')F[o$\"*Hi^i%F -7$$!+3%o+\\)F[o$\"+9gT!*=F-7$$!+uItc$)F[o$\"+Isv%G$F-7$$!+RxRB#)F[o$ \"+!o'y?YF-7$$!+Y@]&3)F[o$\"+GV&f\"fF-7$$!+_lgZzF[o$\"+\\s!))4(F-7$$!+ e4r4yF[o$\"+Z&))o9)F-7$$!+l`\"=n(F[o$\"+U4IS!*F-7$$!+W'3pR(F[o$\"+k-dH 5F[o7$$!+B>+ArF[o$\"+U2zx5F[o7$$!+'3380(F[o$\"+P`6x5F[o7$$!+[Uh!)pF[o$ \"+s!f52\"F[o7$$!+5/#*4pF[o$\"+xAlf5F[o7$$!+rlARoF[o$\"+e?&H/\"F[o7$$! 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To the right of the potential we have obviously a travelling plane wave. It is travelling to the right with momentum " }{TEXT 268 2 "k " } {TEXT -1 62 "(we recognize the periodicity as about 2 Pi, correspondin g to " }{TEXT 269 1 "k" }{TEXT -1 135 "=1, and we see that the imagina ry part precedes the real part, corresponding to motion to the right) \+ and we can read off the amplitude " }{TEXT 267 1 "T" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "On the left we must have the sum of an incident and reflected wave. This is \+ characterized by unequal amplitudes for real and imaginary parts." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "In the m iddle we have a complicated connecting piece. This is where dsolve[num eric] and forming the linear combo has done the work for us!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "So far the inc ident wave has arbitrary normalization. Let us see whether we can figu re the normalization out from the solution in region I (at x=x_i)." }} {PARA 0 "" 0 "" {TEXT -1 112 "We match now at the x_i a solution which has independent amplitudes for the incoming and reflected waves, name d " }{TEXT 271 1 "N" }{TEXT -1 5 " and " }{TEXT 270 1 "R" }{TEXT -1 14 " respectively:" }}}{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 "6#>%$eq1G/,&*&%\"NG\"\"\"-%$expG6#^#!\"#F)F)*&%\"RGF)-F +6#^#\"\"#F)F)^$$\"+wXYIS!#5$!+;!yGK#F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eq2:=I*(k*N*exp(I*k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+ c2*u2p(x_i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/*&^#\"\"\"F(,& *&%\"NGF(-%$expG6#^#!\"#F(F(*&%\"RGF(-F-6#^#\"\"#F(!\"\"F(^$$F(\"\"!$! \"!F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{eq1,e q2\},\{N,R\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%\"RG^$$\" +\"4h " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "We observe that " }{TEXT 272 1 "N" }{TEXT -1 5 " and " }{TEXT 273 1 "R" }{TEXT -1 51 " are amplitudes, and turn out to be \+ complex-valued." }}{PARA 0 "" 0 "" {TEXT -1 59 "The properly normalize d reflection coefficient now becomes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=R/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG^$$!+ &yV;#e!#6$!+kULim!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "and the t ransmission coefficient:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "t:=evalf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG^$$!+#[h1n$!#5$!+OiNlkF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The check whether this all makes sense is to sum the \+ magnitude squared for reflection and transmission:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "abs(r)^2+abs(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-?++5!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 " Within numerical accuracy we preserve the norm; therefore we interpret as reflection and transmission probabilities:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "abs(r)^2,abs(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+?8csW!#5$\"+)peu_&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Now we turn the calculation into a procedure: for given \+ energy it calculates the reflection and transmission coefficients." }} {PARA 0 "" 0 "" {TEXT -1 66 "Our previously developed commands are use d step-by-step as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Tunnel:=proc(En) local IC1,IC2,sol1,sol2,u1,u1p,u2,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=listprocedure);" }}{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),numeric,output=listp rocedure);" }}{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));" }}{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))/(u1(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:=evalf((u1(x_f)+c 2*u2(x_f))/exp(I*k*x_f)/N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "#pri nt(\" 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 -1 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 "6#7$$\" +?8csW!#5$\"+$=du_&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Now we \+ are ready to scan the energy range from zero to above the peak height \+ of the barrier. The results are accumulated in lists for subsequent pl otting." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Tr:=[]: Rf:=[]: for iE from 1 to 60 do: En:=iE/20; res:=Tunnel(En); Tr:=[op(Tr),[En,r es[2]]]: Rf:=[op(Rf),[En,res[1]]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P1:=plot(Rf,color=red): P2:=plot(Tr,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plots[display](P1,P2,labels= [\"E\",\"\{t,r\}\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 299 299 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"3G+++++++]!#>$\"3'******fQ#=l**!#=7 $$\"3/+++++++5F-$\"3M+++F0t3**F-7$$\"3%**************\\\"F-$\"3\\+++fj D=)*F-7$$\"35+++++++?F-$\"3a*******[4Sn*F-7$$\"3++++++++DF-$\"3$****** zuvTW*F-7$$\"3))**************HF-$\"3A+++(\\&3y!*F-7$$\"3w************ *\\$F-$\"3[*****>g\"*y\\)F-7$$\"3A+++++++SF-$\"3k*****>Sicf(F-7$$\"35+ ++++++XF-$\"3s*****H*49fiF-7$$\"3++++++++]F-$\"3u******>8csWF-7$$\"3U+ ++++++bF-$\"3-+++]+S$\\#F-7$$\"3w**************fF-$\"39+++N)G&o()F*7$$ \"3A+++++++lF-$\"3e*****H9$fh!)!#?7$$\"3a**************pF-$\"3U+++\"3x CF)Fdo7$$\"3++++++++vF-$\"3;+++,%=#zaF*7$$\"3U+++++++!)F-$\"3%******4y ?c;\"F-7$$\"3w*************\\)F-$\"3!******4bWXw\"F-7$$\"3A+++++++!*F- $\"3/+++Qs$yF#F-7$$\"3a*************\\*F-$\"3%)*****\\Qm4p#F-7$$\"\"\" \"\"!$\"3E+++@(e,,$F-7$$\"3/++++++]5!#<$\"3#)*****4m)zZKF-7$$\"33+++++ ++6F\\r$\"3y*****4MZmT$F-7$$\"3#*************\\6F\\r$\"3!******zZLz_$F -7$$\"3%**************>\"F\\r$\"3!)*****zNw4f$F-7$$\"3+++++++]7F\\r$\" 3'******4s`Lh$F-7$$\"3/+++++++8F\\r$\"3@+++s>@,OF-7$$\"33++++++]8F\\r$ \"39+++qvefNF-7$$\"3!**************R\"F\\r$\"3-+++xSm#\\$F-7$$\"3'**** *********\\9F\\r$\"3>+++kI+/MF-7$$\"3++++++++:F\\r$\"35+++p:q'H$F-7$$ \"3/++++++]:F\\r$\"3))*****p>?N<$F-7$$\"33+++++++;F\\r$\"3A+++BZ)p.$F- 7$$\"3#*************\\;F\\r$\"3=+++mLW*)GF-7$$\"3%**************p\"F\\ r$\"3t*****>XBJt#F-7$$\"3+++++++]F\\r$\"3#*** ****y!)>\"*=F-7$$\"\"#Ffq$\"30+++fwvBE')z_F*7$$F?F\\r$\"3K+++EiS6W F*7$$\"3#)************\\DF\\r$\"3'******\\$y\\QOF*7$$\"33+++++++EF\\r$ \"39+++RdwdHF*7$$\"3#*************\\EF\\r$\"3)******>?\")[O#F*7$$\"3;+ ++++++FF\\r$\"3%********fhZ&=F*7$$\"3+++++++]FF\\r$\"31+++!\\y$\"3s***** *fpHebF*7$FC$\"3k+++Sf=>#*F*7$FH$\"3*)*****p:9@]\"F-7$FM$\"3+++++GM/CF -7$FR$\"3t*****f:u3u$F-7$FW$\"3m*****H=du_&F-7$Ffn$\"3m*****z%Ri1vF-7$ F[o$\"3W+++=r>B\"*F-7$F`o$\"3s*****zTF%>**F-7$Ffo$\"3[+++s*>t\"**F-7$F [p$\"3!******Hp7@X*F-7$F`p$\"3p******y*4W$))F-7$Fep$\"3M+++LsZN#)F-7$F jp$\"3*)*****p8#=AxF-7$F_q$\"3l******[:04tF-7$Fdq$\"3))******f,')*)pF- 7$Fjq$\"3K+++K+A_nF-7$F`r$\"3_+++e>P$e'F-7$Fer$\"3w*******p$3skF-7$Fjr $\"3h*****RBQ!4kF-7$F_s$\"33+++t*emQ'F-7$Fds$\"3=+++K&*z)R'F-7$Fis$\"3 w*****H^@/W'F-7$F^t$\"3M+++NMM2lF-7$Fct$\"3Y*****z<.gf'F-7$Fht$\"3I+++ ,XI.nF-7$F]u$\"32+++2e[EoF-7$Fbu$\"3[******4e,jpF-7$Fgu$\"3;+++@(\\06( F-7$F\\v$\"3s*****Hh#)oE(F-7$Fav$\"3;+++'em)HuF-7$Ffv$\"3!******p!)fuf (F-7$F[w$\"3_*******y-xw(F-7$F`w$\"3^+++0XuQzF-7$Few$\"3\\+++f#4)3\")F -7$Fjw$\"3Y*****4I`iF)F-7$F_x$\"3L+++U4bR%)F-7$Fdx$\"3g*****R?etf)F-7$ Fix$\"3w*****fdp%[()F-7$F^y$\"3')*****H@2>*))F-7$Fcy$\"3/+++LQ%o-*F-7$ Fhy$\"3-+++u%fE:*F-7$F]z$\"3O+++`G$*o#*F-7$Fbz$\"3/+++AoTv$*F-7$Fgz$\" 3[+++O9/s%*F-7$F\\[l$\"3o*****RV*))e&*F-7$F`[l$\"3A+++fJ=O'*F-7$Fe[l$ \"3<+++udD/(*F-7$Fj[l$\"33+++KQaj(*F-7$F_\\l$\"3$)*****4faX\")*F-7$Fd \\l$\"3=+++mJ&y&)*F-7$Fi\\l$\"3u******\\P.%*)*F-7$F^]l$\"3U+++!H>P#**F -7$Fc]l$\"3_*****ReKv%**F-7$Fh]l$\"3'******fx(3m**F-7$F]^l$\"3++++X\\) *z**F--Fb^l6&Fd^lFh^lFh^lFe^l-%+AXESLABELSG6%Q\"E6\"Q&|frt,r|hrFfjl-%% FONTG6#%(DEFAULTG-%%VIEWG6$F[[mF[[m" 1 2 0 1 10 3 2 9 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 33 "Observe the interesting features:" }}{PARA 0 "" 0 "" {TEXT -1 128 "1) For sufficiently high energies only transmission occu rs. Note, however, that classically this situation would be reached fo r " }{TEXT 274 1 "E" }{TEXT -1 3 ">1!" }}{PARA 0 "" 0 "" {TEXT -1 7 "2 ) For " }{TEXT 275 1 "E" }{TEXT -1 642 "=0.66 perfect transmission and no reflection is observed. This is the phenomenon of a transmission r esonance. The resonance structure has a width associated with it. Most physical phenomena that have a lifetime associated with them (unstabl e elementary particles, alpha decay, nuclear gamma lines, atomic optic al transitions, excitations in semiconductors, etc.) can be understood as processes where the wave equation describing the matter (or radiat ion) field involves a tunneling process. Broad resonances are associat ed with short lifetimes, sharp resonances with long lifetimes, accordi ng to Heisenberg's energy-time uncertainty relation." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "The transmission reson ance is quite broad. We can make the walls of the two barriers thicker , and observe what the effect is on the transmission probability." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "V:=exp(-8*x^6)+exp(-8*(x-2)^ 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,&-%$expG6#,$*$)%\"xG\"\" '\"\"\"!\")F.-F'6#,$*$),&F,F.\"\"#!\"\"F-F.F/F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "plot(V,x=-2..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 219 219 {PLOTDATA 2 "6%-%'CURVESG6$7ds7$$!\"#\"\"!$\"3'> 0`q.\"\\xV!$S#7$$!3GLL$3#*>u%=!#<$\"3e7XZ**fOAv!$c\"7$$!3om;z43m9!3#*H)et<;!#A7$$!3:***\\7o CA$)*!#=$\"3i]N8)3:=E(!#@7$$!3+mm\"z*ffY'*Ffn$\"3[/^C([:ee\"!#?7$$!3%G L$e9t'4Y*Ffn$\"31&o0)RQkBKF_o7$$!3e)**\\7jQ`F*Ffn$\"3+gC[T)RF8'F_o7$$! 3Ulm\"z%*4(*3*Ffn$\"3],2PJ&3u4\"!#>7$$!3+)**\\7e_%=()Ffn$\"3+\"=`,]r&z HF_p7$$!3qJLe9_>Z$)Ffn$\"3/$yP2*[6!o'F_p7$$!3M*****\\ZGr(zFfn$\"3oy\"e Z)Grs7Ffn7$$!3'em;athqg(Ffn$\"3GZiY\\d-A@Ffn7$$!3o**\\il$G?U(Ffn$\"3QI EnK>cDEFfn7$$!3RKL$e*\\*pB(Ffn$\"315'R%>G`oJFfn7$$!3Am;/E;'>0(Ffn$\"37 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Be careful with the values of x_i, x_f, when changes are made to the potential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "We can include an option in Tun nel to graph the solution. Perhaps this will help to understand why re flection and transmission comes about." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Tunnel:=proc(En,iplot) local IC1,IC2,sol1,sol2,u1,u1 p,u2,u2p,k,c2,N,R,r,t,sol,eq1,eq2; global V,kE,x_i,x_f,PL;" }}{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), numeric,output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u 2:=subs(sol2,u(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u2p:=subs(so l2,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:= evalf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "if iplot=2 then print(\" R= \",abs(r)^2,\" T= \",abs (t)^2,\" R+T= \",abs(r)^2+abs(t)^2,\" norm equal to 1.0 ?\") fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "if iplot=1 then" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "PL:=plot([V,'Re((u1(x)+c2*u2(x))/N)','Im((u1(x)+c2 *u2(x))/N)'],x=x_i-10..x_f+10,color=[red,blue,green],title=cat(\"Energ y= \",convert(evalf(En,3),string))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 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