{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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# After rescaling we have to solve the diff equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq1:=E- >diff(y(x),x$2)=(x^4-E)*y(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1Gf*6#%\"EG6\"6$%)operatorG%&arr owGF(/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F3\"\"#*&,&*$)F3\"\"%\"\"\"F=9$! \"\"F=F0F=F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# for d ifferent values of E." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# \+ we can write a procedure to plot the resulting solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot_sol:=proc(E,ran) local sol1,psi;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "sol1:=dsolve(\{eq1(E),y(0)=1,D(y)(0)=0\},numer ic,output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "psi:=e val(y(x),sol1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(psi(x),ran) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot_sol(1.060361945,x=0..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 362 362 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F)7$$\"+m;')=()!#6$ \"+JOsf**!#57$$\"+e'40j\"F2$\"+x]Qf)*F27$$\"+<6m$[#F2$\"+G(4[n*F27$$\" +(>%F2$\"+)Q&=#3*F27$$\"+\">K'*)\\F2$\"+: \\e8()F27$$\"+Dt:5eF2$\"+Zw^u#)F27$$\"+\"fX(emF2$\"+Y\\=lxF27$$\"+DCh/ vF2$\"+$)oa4sF27$$\"+L/pu$)F2$\"+\"RY#)f'F27$$\"+;c0T\"*F2$\"+;G&f.'F2 7$$\"+I,Q+5!\"*$\"+Ww6)Q&F27$$\"+]*3q3\"Fbo$\"+O\"*GNZF27$$\"+q=\\q6Fb o$\"+`VDF27$$\"+`dF!e\"Fbo$ \"+!H@rg\"F27$$\"+sgam;Fbo$\"+PYiR7F27$$\"+Fbo$\"+1A?U^F/7$$\"+Uc-)*>Fbo$\"+Gav3OF/ 7$$\"+f`@'3#Fbo$\"+`eeCCF/7$$\"+nZ)H;#Fbo$\"+>&G8n\"F/7$$\"+Ky*eC#Fbo$ \"+epG(3\"F/7$$\"+S^bJBFbo$\"+'3\"zan!#77$$\"+0TN:CFbo$\"+_DE/TFet7$$ \"+7RV'\\#Fbo$\"+*3._X#Fet7$$\"+:#fke#Fbo$\"+h19O8Fet7$$\"+`4NnEFbo$\" +.qwmu!#87$$\"+],s`FFbo$\"+\\&G;'QFju7$$\"+zM)>$GFbo$\"+g#=90#Fju7$$\" +qfaQ\"Fju7$$\"+2^rZP Fbo$!+jJG3SFju7$$\"+sI@KQFbo$!+E=l98Fet7$$\"+S2lsQFbo$!+O!znO#Fet7$$\" +2%)38RFbo$!+4u)eJ%Fet7$$\"+/UacRFbo$!+Qws]$)Fet7$$\"\"%F)$!+?=OS;F/-% 'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fb\\l-%%VIEWG6$ ;F(Fc[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " plot_sol(7.4557,x=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 362 362 {PLOTDATA 2 "6%-%'CURVESG6$7Y7$$\"\"!F)$\"\"\"F)7$$\"+DJdpK!#6$\"+]`PF2 7$$\"+%>fS*\\F2$\"+9/bf?F27$$\"+>$f%GcF2$\"+$pXrV$F/7$$\"+Dy,\"G'F2$!+ 9)o8V\"F27$$\"+7&)\\\"F`q$!+tf`V%)F27$ $\"+>:mk:F`q$!+Bs3n$F27$$\"+6W%)R>F`q$!+A#)**eHF27$$\"+:K^+?F`q$!+!fl\\R#F27$$\" +7,Hl?F`q$!+mRqv=F27$$\"+4w)R7#F`q$!+Vz,y9F27$$\"+y%f\")=#F`q$!+9b.=6F 27$$\"+/-a[AF`q$!+&fBRW)F/7$$\"+ial6BF`q$!+T.pxhF/7$$\"+i@OtBF`q$!+w$3 ZY%F/7$$\"+fL'zV#F`q$!+rqh7JF/7$$\"+!*>=+DF`q$!+9`Ra@F/7$$\"+E&4Qc#F`q $!+gVcZ9F/7$$\"+%>5pi#F`q$!+`S:Y&*!#77$$\"+bJ*[o#F`q$!+![D6Q'Fiz7$$\"+ r\"[8v#F`q$!+(\\)R=RFiz7$$\"+Ijy5GF`q$!+QP0jCFiz7$$\"+/)fT(GF`q$!+^*Q( Q9Fiz7$$\"+1j\"[$HF`q$!+e&[;$z!#87$$\"\"$F)$!+S`UeJFc\\l-%'COLOURG6&%$ RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fd]l-%%VIEWG6$;F(Fe\\l%(DEF AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "#Energy eigenvalue E \+ = 1.060361944. We see that the function goes from going to infinity to " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "#minus infinity just b y changing the last decimal so we conclude the eigenvalue is between \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "#those two values of E. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "# We can normalize the wave function by comput ing" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "sqrt(int((psi(x))^2,x=-3.5..3.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*3K6E\"!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "# so that: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int((psi(x)/1.261132089)^2,x=-3.5..3.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# We now do an approximate solution as a gaussian wave-function. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# Gaussian minimizatio n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume(alpha>0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "psi:=x->(2*alpha/Pi)^(1/4)*e xp(-alpha*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(*()\"\"##\"\"\"\"\"%F0)*&%&alphaGF0%#PiG!\"\"F/ F0-%$expG6#,$*&F4F0)9$F.F0F6F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(psi(x)^2,x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "int(psi(x)*(-diff(psi(x),x$2)+x^4*psi(x)),x=-infinity..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%'alpha|irG\"\"\"*(\"\"$F%\"#;! \"\"F$!\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(%,alph a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*(\"\"$F$\"\")!\"\"%' alpha|irG!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(%= 0,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&\"\"#!\"\"\"\"$#\"\" \"F'F),&*&\"\"%F&F'F(F&*&^##F)F,F))F'#\"\"&\"\"'F)F),&*&F,F&F'F(F&*&^# #F&F,F)F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(alpha =1/2*3^(1/3),alpha+3/16/alpha^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*(\"\"$\"\"\"\"\"%!\"\"F%#F&F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yro\" 3\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# To be compared with E = 1.060361944 exact value." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " 100 *( 1.081687178-1.060361944)/1.060361944;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:y76?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# So it has a 2% error. Not so bad. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# We can compare n ow the normalized wave functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol1:=d solve(\{eq1(1.060361944),y(0)=1,D(y)(0)=0\},numeric,output=listprocedu re);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "psi:=eval(y(x),sol1);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plot(\{psi(x)/1.261132089,subs(alp ha=1/2*3^(1/3),(2*alpha/Pi)^(1/4)*exp(-alpha*x^2))\},x=0..4,color=[red ,green]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G7%/%\"xGf*6#F'6\" 6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~rese rved.GF*9$F*F*F*/-%\"yGF)f*F)6&%$resG%)solnprocG%)outpointG%&ndsolG6#% inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF*C(> %8_EnvDSNumericSaveDigitsG%'DigitsG>F<\"#9@%/%-_EnvInFsolveG%%trueG>8& -&%&evalfG6#F;6#F->FD-FGFI>8%f*6#%#mxG6+F5%#dtG%$datG%(odeprocG%#x0G%$ 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#Fd\\l7$$\"+`Q\"GT\"F``l$\"+w6$y(>Fd\\l7$$\"+s]k,:F``l$\"+1#oDe\"Fd\\l 7$$\"+`dF!e\"F``l$\"+:\"[VF\"Fd\\l7$$\"+sgam;F``l$\"+g#f%H)*Fh\\l7$$\" +F``l$ \"+?&\\u2%Fh\\l7$$\"+Uc-)*>F``l$\"+73_hGFh\\l7$$\"+f`@'3#F``l$\"+BuaA> Fh\\l7$$\"+nZ)H;#F``l$\"+,4ED8Fh\\l7$$\"+Ky*eC#F``l$\"+:7_@')!#77$$\"+ S^bJBF``l$\"+F<9c`F^el7$$\"+0TN:CF``l$\"+Z1WaKF^el7$$\"+7RV'\\#F``l$\" +*oXo%>F^el7$$\"+:#fke#F``l$\"+9I^f5F^el7$$\"+`4NnEF``l$\"+_wG@f!#87$$ \"+],s`FF``l$\"+EH7jIFhfl7$$\"+zM)>$GF``l$\"+NWeG;Fhfl7$$\"+qfaKHFhfl7$$\"+GUYoOF``l$\"+5C*=7*Fhfl7$$\"+2 ^rZPF``l$\"+XQ!eZF^el7$$\"+sI@KQF``l$\" +_+;w')F^el7$$\"+S2lsQF``l$\"+Xg(>c\"Fh\\l7$$\"+2%)38RF``l$\"+-!3$[GFh \\l7$$\"+/UacRF``l$\"+!GO6^&Fh\\l7$Fb[l$\"+*[rD3\"Fd\\l-Fh[l6&Fj[lF(F[ \\lF(-%+AXESLABELSG6$Q\"x6\"Q!F]]m-%%VIEWG6$;F(Fb[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# Reasonable approximati on but not perfect. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "# C an we improve?. We can consider a function with more degrees of freedo m. Example: " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "as sume(alpha>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "psi:=x->( A+B*x^2)*exp(-alpha*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&%\"AG\"\"\"*&%\"BGF/)9$\"\"#F/F/F /-%$expG6#,$*&%&alphaGF/F2F/!\"\"F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "# Instead of normalizing we can just compute /" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "simplify(int(psi(x)*(-diff( psi(x),x$2)+x^4*psi(x)),x=-infinity..infinity)/int(psi(x)^2,x=-infinit y..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#;!\"\",.**\"$ G\"\"\"\"%\"AGF*)%'alpha|irG\"\"%F*%\"BGF*F&*(\"$c#F*)F+\"\"#F*)F-\"\" &F*F***\"$?\"F*F+F*F/F*F-F*F**(\"$7\"F*)F/F3F*)F-\"\"$F*F**(\"#[F*F2F* )F-F3F*F**&\"$0\"F*F:F*F*F*F-!\"#,(*(F%F*F2F*F?F*F**&F " 0 "" {MPLTEXT 1 0 114 "# \+ We can rescale A,B by the same constant without affecting the result, \+ so we choose A to simplify the expression:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "subs(A=1/alpha,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#;!\"\",.*(\"$G\"\"\"\")%'alpha|irG\"\"$F*%\"BGF*F&*&\"$c#F*F +F*F**&\"$?\"F*F.F*F**(\"$7\"F*)F.\"\"#F*F+F*F*\"#[F**&\"$0\"F*F5F*F*F *F,!\"#,(F%F**&\"\")F*F.F*F**&F-F*F5F*F*F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "E1:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G, $**\"#;!\"\",.*(\"$G\"\"\"\")%'alpha|irG\"\"$F,%\"BGF,F(*&\"$c#F,F-F,F ,*&\"$?\"F,F0F,F,*(\"$7\"F,)F0\"\"#F,F-F,F,\"#[F,*&\"$0\"F,F7F,F,F,F.! \"#,(F'F,*&\"\")F,F0F,F,*&F/F,F7F,F,F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# This we want to minimize with respect to A and B." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(diff(E1,alpha)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\")!\"\",.*(\"#k\"\"\")%'alp ha|irG\"\"$F*%\"BGF*F&*&\"$G\"F*F+F*F**(\"#cF*)F.\"\"#F*F+F*F**&\"$?\" F*F.F*F&\"#[F&*&\"$0\"F*F3F*F&F*F,!\"$,(\"#;F**&F%F*F.F*F**&F-F*F3F*F* F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "solve(-64*alpha^3*B +128*alpha^3+56*alpha^3*B^2-48-105*B^2-120*B=0,alpha);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,$*(\"\"#!\"\",(*&\"\")\"\"\"%\"BGF*F&\"#;F**&\"\" (F*)F+F%F*F*F&*&,(*&\"$0\"F*F/F*F**&\"$?\"F*F+F*F*\"#[F*F*)F'F%F*#F*\" \"$F*,&*(\"\"%F&F'F&F0F8F&**^##F*F " 0 "" {MPLTEXT 1 0 59 "# We choose alpha real and positive, namely first solution." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "simplify(subs(B=0,1/2/(-8*B+ 16+7*B^2)*((105*B^2+120*B+48)*(-8*B+16+7*B^2)^2)^(1/3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"\"$#\"\"\"F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# So if B=0 we get the same as befo re." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "E2:=simplify(subs(al pha=1/2/(-8*B+16+7*B^2)*((105*B^2+120*B+48)*(-8*B+16+7*B^2)^2)^(1/3),E 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G,$*0\"\"$\"\"\"\"\"%!\" \",(*&\"#NF()%\"BG\"\"#F(F(*&\"#SF(F/F(F(\"#;F(F(,(*&\"\")F(F/F(F*F3F( *&\"\"(F(F.F(F(F0F'#F(F'*&F+F()F4F0F(#!\"#F',(F3F(*&F6F(F/F(F(*&F'F(F. F(F(F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(subs(B=0,E2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"\"\"\"%!\"\"F%#F&F%F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# Again same as before." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "simplify(diff(E2,B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"#;\"\"\"\"\"$#F&F'%\"BGF&,.*&\"$N(F&)F)\"\"&F&F&*&\"%%[\"F &)F)\"\"%F&F&*&\"%g " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "735*B^5+1484*B^4-1760*B^3+5760*B^2-1280*B-1024;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"$N(\"\"\")%\"BG\"\"&F&F&*&\"%%[ \"F&)F(\"\"%F&F&*&\"%g " 0 "" {MPLTEXT 1 0 13 "solve(%= 0,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'#\"\"%\"\"(,&F#\"\"\"*&^#F#F' \"\"'#F'\"\"#F',&F#F'*&^##!\"%F%F'F*F+F',&#\"#G\"#:!\"\"*(F$F'F5F6\"#M F+F',&#F4F5F6*(F$F'F5F6F8F+F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+9dG9d!#5^$F#$\"+ D%3(*R\"!\"*^$F#$!+D%3(*R\"F)$!*hhu6$F)$!+tre@MF)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 116 "subs(B=4/7,12*(35*B^2+40*B+16)*(-8*B+16+7*B ^2)^2*3^(1/3)/((35*B^2+40*B+16)*(-8*B+16+7*B^2)^2)^(2/3)/(16+8*B+3*B^2 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"$;'!\"\"\"\"$#\"\"\"F'\"( KSC$F(\"$V$#\"\"#F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eva lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p?J)p\"!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(B=4/7,E2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#@\"\"\"\"#W!\"\"\"#6#F&\"\"$F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "# Our approximate value is t hen E = 21 * 11^(1/3) /44. Numerically:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V]Wh5 !\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# So now the error \+ is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "100*(1.061445043 - 1. 060361944)/1.060361944;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JFW@5!# 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "# much better!, 0.1%. \+ Notice that we always get a value bigger than the correct one. Why? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "# Our approximate value i s then E = 21 * 11^(1/3) /44. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "simplify(subs(B=4/7,1/2/(-8*B+16+7* B^2)*((105*B^2+120*B+48)*(-8*B+16+7*B^2)^2)^(1/3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"#6#\"\"\"\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "subs(A=1/alpha,B=4/7,alpha=1/2*11^(1/3),psi (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*(\"\"#\"\"\"\"#6!\"\"F(# F&\"\"$F'*(\"\"%F'\"\"(F)%\"xGF&F'F'-%$expG6#,$*(F&F)F(#F'F+F/F&F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sqrt(int(%^2,x=-infinity. .infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"(! \"\"\"\"'#F&F%*&)\"#6#F&F)F&%#PiGF*F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H5$y8 \"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol1:=dsolve(\{eq 1(1.060361944),y(0)=1,D(y)(0)=0\},numeric,output=listprocedure);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "psi:=eval(y(x),sol1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "plot(\{psi(x)/1.261132089,subs(alpha=1/2*3^( 1/3),(2*alpha/Pi)^(1/4)*exp(-alpha*x^2)),subs(A=1/alpha,B=4/7,alpha=1/ 2*11^(1/3),(A+B*x^2)*exp(-alpha*x^2)/1.137831029)\},x=0..4,color=[red, green,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G7%/%\"xGf*6#F '6\"6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~ reserved.GF*9$F*F*F*/-%\"yGF)f*F)6&%$resG%)solnprocG%)outpointG%&ndsol G6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF *C(>%8_EnvDSNumericSaveDigitsG%'DigitsG>F<\"#9@%/%-_EnvInFsolveG%%true G>8&-&%&evalfG6#F;6#F->FD-FGFI>8%f*6#%#mxG6+F5%#dtG%$datG%(odeprocG%#x 0G%$valG%$digG%\"nG%\"iG6#%inCopyright~(c)~2002~by~Waterloo~Maple~Inc. ~All~rights~reserved.GE\\s#Q0soln_proceduresF*=F*6#;\"\"\"\"\"$E\\[l$F [o\")%='y8\"\"#\")GLr8$F->FM=F*FinE \\[l\"F[o=F*6#;F[oF>E\\[l/F[o6%/%)datatypeG&%&floatG6#\"\")/%&orderG%( C_orderG/%(storageG%,rectangularGF_oF^pF\\o7$\"\"!F\\q\"\"%X*%)anythin 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(red curve)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Consi der now a potential lambda x^2000. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Digits: =20: (This is just the precision of the numerical calculation)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# After rescaling we have to solve the diff equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eq1:=E->diff(y(x),x$2) =(x^2000-E)*y(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eq1Gf*6#%\"EG6\"6$%)operatorG%&arrowGF(/-%%di ffG6$-%\"yG6#%\"xG-%\"$G6$F3\"\"#*&,&*$)F3\"%+?\"\"\"F=9$!\"\"F=F0F=F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# for different val ues of E." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# we can write a procedure to plot the resulting solution (same as before)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "plot_sol:=proc(E,ran) local sol1,psi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sol1:=dsolve(\{eq1(E),y(0)=1,D(y)(0)=0\},nume ric,output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "psi:= eval(y(x),sol1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(psi(x),ran );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot_sol(2.43301213,x=0..1.01);" }}{PARA 13 "" 1 "" {GLPLOT2D 624 624 624 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"!F)$\"\"\"F)7$$\"5MLLL$3d 7:?#!#@$\"5;*o5wof/T***!#?7$$\"5nmm;H()o.VCriF/$\"5qvhz5!z%>_**F27$$\"5NLLL3P7tR%)F/$\"5KW>g:&\\uM\"**F27$$ \"5nmm\"z9<\"zf5F2$\"5JSe!y[6yO')*F27$$\"5MLLeRG@))f7F2$\"5u\"4P>HJCv! )*F27$$\"5+++DcCZ1n9F2$\"5`i=sutXJR(*F27$$\"5MLLe*=EL8o\"F2$\"5#3:N@4q u!e'*F27$$\"5+++DJPY\"\\*=F2$\"5ko%[,,&)fjc*F27$$\"5nmmmTM$4Y6#F2$\"5Q &G> \\bHF2$\"5'Q6CYbTwg&*)F27$$\"5MLLe*oX8p9$F2$\"58@9v4$*=G>))F27$$\"5nmm mT<@`uLF2$\"5Hy%3!ekzRY')F27$$\"5nmmmmz\\NnNF2$\"5NC%R86*pS\"\\)F27$$ \"5,++D13Ql\"z$F2$\"5O!\\\"\\cv;Y,$)F27$$\"5nmmm;xi>!*RF2$\"5K!eQ\\:1V [7)F27$$\"5,++D1L)G!3UF2$\"549cy6(32@#zF27$$\"5,++v=RpX:WF2$\"5F?;5oN0 b?xF27$$\"5nmm;H2k)=j%F2$\"5A;!p&Q8bj,vF27$$\"5nmm\"z4AO1$[F2$\"5^)eeA .EnIH(F27$$\"5MLL$3_u9]/&F2$\"5_'pH5lYZ-1(F27$$\"5MLLe*=y$pn_F2$\"5*)p $pBV&)e+\"oF27$$\"5,++v$z`O:Y&F2$\"5B\"))=sL!*pbe'F27$$\"5MLL$e\\-#*3n &F2$\"5#**e\"Q\"yxfjL'F27$$\"5,+++]Gn<()eF2$\"5Tlo-wM'3=2'F27$$\"5,++] 7:\"p()4'F2$\"5i\"oHO>33j!eF27$$\"5,++D1Hc\\.jF2$\"5sKbE[d-SVbF27$$\"5 ,++](y^43`'F2$\"5g$eo8>uv[C&F27$$\"5ommm;d61NnF2$\"5U]_n;5N+r\\F27$$\" 5,+++vyL9`pF2$\"5ZEfa!QF2 7$$\"5,++DJ)QtDy(F2$\"51dj(zli_L\\$F27$$\"5ommmT!G>.*zF2$\"58.*\\\"o(4 Vz=$F27$$\"5,++Dcw*4y?)F2$\"5\\q*e`SMEY'GF27$$\"5NLLLLm$zsT)F2$\"5!Gsw sZ)G6]DF27$$\"5NLL$3<2#\\J')F2$\"5'QlER=2lcA#F27$$\"5omm\"Hx)4$R%))F2$ \"5SCZ@#4L\\9!>F27$$\"5,+++])GS\"R!*F2$\"5:k+]Z!oz;g\"F27$$\"5omm;aw@( GE*F2$\"5)Q^]2b>+jD\"F27$$\"5NLLLLW1)HY*F2$\"5>&))[t5\\*zg%*F/7$$\"5-+ +D13!Qjn*F2$\"5oJST.?:]7F/7$$\"5+Dc^ee0x,5!#>$\"5\\(or\"[eZ.B# )!#A7$$\"5+]PMs0P^/5Fe[l$\"51j*)>3\\bbbRFh[l7$$\"5]7yDHz_)e+\"Fe[l$\"5 Uxo+NIlI7>Fh[l7$$\"5+v=<'G&oD25Fe[l$\"5!)*y(Q*Qegf(Q!#B7$$\"5D1*GY'RE% z+\"Fe[l$\"5)f02F?&=P@t!#C7$$\"5]Pf3VE%G'35Fe[l$\"5w])))z'>r[(o$!#D7$$ \"5voHa@8UJ45Fe[l$\"5*oeUw+G4@r'Fd]l7$$\"$,\"!\"#$\"5f_netp^q4;F2-%'CO LOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F[_l-%%VIEWG6$;F(F [^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#So the ener gy of the ground state is E=2.43301213;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*87IV#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "#Con sider: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi^2/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5[lRBF+6SnC!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "#We see that is is somewhat similar with \+ a " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "100 * (2.46740110027 23396548-2.43301213)/2.43301213;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"5m[y&3'G>V89!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "# i.e . 1.5% difference. Is this a coincidence? In which limit the igenvalue should be exactly Pi^2/4 ? " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "103 0 0" 102 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }