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247 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "kE:=En->sqrt(2*E
n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I#EnG6\"F%6$I)operatorGF%I&ar
rowGF%F%-I%sqrtGF%6#,$*&\"\"#\"\"\"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
+@$2\"\"\"%&nargsGYQBinvalid~input:~too~many~argumentsF%>I8_EnvDSNumer
icSaveDigitsGF%I'DigitsGF%>F;\"#9@%/I-_EnvInFsolveGF%I%trueG%*protecte
dG>F,-&I&evalfGFB6#F:F'>F,-FFF'>F*-I%evalGFB6#&I<dsolve/numeric/data/m
odulesGI(_syslibGF%6#\"\"#>F+-_F*I$GetGF%6#Q/soln_procedureF%>F.-I$map
GFB6$I$rhsGFB-FU6#Q'paramsF%@$4-I%typeGFB6$F,.I(numericGFBC$@0-I'membe
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Q&rightF%.I&rightGF%Q)leftdataF%Q*rightdataF%Q+enginedataF%Q*eventstop
F%.I*eventstopGF%Q+eventclearF%.I+eventclearGF%Q,eventstatusF%.I,event
statusGF%Q'laxtolF%.I'laxtolGF%Q'numfunF%.I'numfunGF%I%NULLGFBC$>F)-F+
6#-I(convertGFB6$F$.I'stringGFB@(2F5-I%nopsGFB6#7#F)OF)-F^o6$F).I&arra
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metersF%.I+parametersGF%Q7initial_and_parametersF%.I7initial_and_param
etersGF%FfqC%>F,F[r>F)-F+6#F,@%/F,FftO7#-I$seqGFB6$/&F.6#F/&F)Fju/F/;F
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F`u7)FfsFgsFftFgtFitFjtFfqC%>F,/-F\\r6$-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$F^uF`tF%YF%F%F%F
%/-I\"uGF%F'f*F'F(F0F%C+F3F9F<F>FJFSFY@$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+F3F9F<F>FJFSFY@$F\\oC$@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 
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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$$\"/q&ejgDZ%!#9$\"/\"3azXu_&F%" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 246 154 "Now we are ready to scan the energy range from \+
zero to above the peak height of the barrier. The results are accumula
ted in lists for subsequent plotting." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 123 "Tr:=[]: Rf:=[]: for iE from 1 to 60 do: En:=iE/20; r
es:=Tunnel(En); Tr:=[op(Tr),[En,res[2]]]: Rf:=[op(Rf),[En,res[1]]]: od
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P1:=plot(Rf,color=red)
: P2:=plot(Tr,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "plots[display](P1,P2,labels=[\"E\",\"\{t,r\}\"]);" }}{PARA 13 "" 1
 "" {TEXT 247 0 "" }{GLPLOT2D 702 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7
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o reflection is observed. This is the phenomenon of a transition reson
ance. The resonance structure has a width associated with it. Most phy
sical phenomena that have a lifetime associated with them (unstable el
ementary particles, alpha decay, nuclear gamma lines, atomic optical t
ransitions, excitations in semiconductors, etc.) can be understood as \+
processes where the wave equation describing the matter (or radiation)
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{TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 157 "The transmission reson
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{PARA 0 "" 0 "" {TEXT 246 36 "We should adjust the right boundary:" }}
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$$\"&v$H!\"%$\"/#[1y0<q*!#97$$\"$&H!\"#$\"/8.p:,<(*!#97$$\"&D'H!\"%$\"
/ENp'==t*!#97$$\"%vH!\"$$\"-\\ts7Y(*!#77$$\"&v)H!\"%$\"/arm%Q*f(*!#97$
$\"#I!\"\"$\"/A8GZDt(*!#9-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"!!\"\"$\"#5!
\"\"-%+AXESLABELSG6'Q\"E6\"Q&|frt,r|hr6\"-%%FONTG6%%(DEFAULTG%(DEFAULT
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0 0 "Curve 1" "Curve 2" }}{TEXT 247 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 246 107 "The transmission resonance is sharper now, and occurs j
ust below the classical border value for the energy." }}{PARA 0 "" 0 "
" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 163 "Exercise: Observe wh
at happens when the gap is widened between the potential humps. Be car
eful with the values of x_i, x_f, when changes are made to the potenti
al." }}{PARA 0 "" 0 "" {TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 246 139 
"We can include an option in Tunnel to graph the solution. Perhaps thi
s will help to understand why reflection and transmission comes about.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Tunnel:=proc(En,iplot)
 local IC1,IC2,sol1,sol2,u1,u1p,u2,u2p,k,c2,N,R,r,t,sol,eq1,eq2; globa
l 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),I
C1\},u(x),numeric,output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT
 1 0 20 "u1:=subs(sol1,u(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u1
p:=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 "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)+c2*u2(x_f))/exp(I
*k*x_f)/N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "if iplot=2 then pri
nt(\" 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 i
plot=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "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=[re
d,blue,green],numpoints=500,title=cat(\"Energy= \",convert(evalf(En,3)
,string))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0
 "" {MPLTEXT 1 0 25 "[abs(r)^2,abs(t)^2]; end:" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 15 "Tunnel(0.47,1);" }}{PARA 11 "" 1 "" {XPPMATH 20
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{MPLTEXT 1 0 19 "plots[display](PL);" }}{PARA 13 "" 1 "" {TEXT 247 0 "
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