{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 255 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 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 0 "" 0 "" {TEXT -1 91 "c International Thomson Pu blishing Bonn 1995 filename: montew.ms" }} {PARA 0 "" 0 "" {TEXT -1 104 "Autor: Komma \+ Datum: 11.10.94" }} {PARA 0 "" 0 "" {TEXT -1 66 "Thema: Ann\344herung an die wirkliche Bah n durch Auslosen von Pfaden." }}{PARA 0 "" 0 "" {TEXT -1 200 "Es wird \+ kein bestimmter Funktionstyp vorausgesetzt, sondern ein zuf\344lliger \+ Pfad vorgegeben. Dieser Pfad wird zuf\344llig ver\344ndert und die \+ \304nderung akzeptiert, wenn sich dadurch die Wirkung verringert. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Die Bewegung wird in n \344quidist ante Zeitschritte dt = t1/n unterteilt. " }}{PARA 0 "" 0 "" {TEXT -1 148 "Von je drei Punkten mit den x-Werten a, x und b wird der mittlere variiert (x+dx), und die sich daraus ergebende Differenz der Wirkung \+ dS berechnet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Differenz der kinetischen Energie mal (!) dt:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Ti entspricht der mittleren kinetischen Energie (mal dt) der Bewegung von a nach b." }} {PARA 0 "" 0 "" {TEXT -1 138 "dT wurde in einer fr\374heren Version di eses Worksheets berechnet und mit lprint hier eingef\374gt (als kleine s Beispiel f\374r " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dT:=proc(a,x,b,dx)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "-m*dx*(-2*x-dx+a+b)/dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d TGR6&%\"aG%\"xG%\"bG%#dxG6\"F+F+,$*&*(%\"mG\"\"\"9'F0,*9%!\"#F1!\"\"9$ F09&F0F0F0%#dtGF5F5F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Differenz der potentiellen E nergie mal dt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dV:=proc( x,dx)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(V(x+dx)-V(x))*dt;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dVGR6$%\"xG%#dxG6\"F)F)*&,&-%\"VG6#,&9$\"\"\"9%F1F1-F-6#F0!\" \"F1%#dtGF1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Differenz de r Wirkung (als Auswahlkriterium):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dS:=proc(a,x,b,dx)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dT(a,x,b,dx)-dV(x,dx);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dSGR6&%\"aG%\"xG%\"bG%#dxG6\"F+F+ ,&-%#dTG6&9$9%9&9'\"\"\"-%#dVG6$F1F3!\"\"F+F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Wirkun g zur Kontrollausgabe bereitstellen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Sa:=proc(ak) local i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "sum(m/2*(xa[ak,i]-xa[ak,i+1])^2/dt-(V(xa[ak,i+1])+V(xa[ak,i])) /2*dt, i=1..n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#SaGR6#%#akG6#%\"iG6\"F*-%$sumG6$,&*&*&%\"mG \"\"\"),&&%#xaG6$9$8$F2&F66$F8,&F9F2F2F2!\"\"\"\"#F2F2%#dtGF=#F2F>*&#F 2F>F2*&,&-%\"VG6#F:F2-FF6#F5F2F2F?F2F2F=/F9;F2%\"nGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "xa();" }}{PARA 0 "" 0 "" {TEXT -1 87 "Anzahl der zu variierenden Punkte (wegen impliziter array-Definition fast \374berfl\374ssig):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x1:='x1': n:='n':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "anz:=proc() global x,xa,x0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#n:=20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x:=array(1..n+1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x0:=array(1..n+1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x[1]:=0; \+ x[n+1]:=x1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$anzGR6\"F&F&F&C&>%\"xG-%&arrayG6#;\"\"\",&%\"nG F.F.F.>%#x0GF*>&F)6#F.\"\"!>&F)6#F/%#x1GF&6%F)%#xaGF2F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "anz();" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "zuf\344llige Anfangsverteilung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ini:=proc() local i; global x,x0,ran,dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#x1:=3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dt:=t1/n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ran:=kr*(1-rand(1..10 00)/500): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 2 to n do " }}{PARA 0 "" 0 "" {TEXT -1 117 "eine Systematik in der \"Zufallsvert eilung\" (z.B. gleichf. Bewegung als nullte N\344herung) verschlechter t das Verfahren:" }}{PARA 0 "" 0 "" {TEXT -1 83 "es bleiben dann h\344 ufig die Pkte. neben den Randpunkten oder sonstige Spitzen stehen" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "#ran:=-rand(1..100)/100*x1+i*dt*x1/ t1: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x[i]:= ran():" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x0[i]:=x[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x0[1]:=x[1]: x0[n+1]:=x 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$iniGR6\"6#%\"iGF&F&C'>%#dtG*&%#t1G\"\"\"%\"nG!\"\">% $ranG*&%#krGF.,&F.F.*&#F.\"$+&F.-%%randG6#;F.\"%+5F.F0F.?(8$\"\"#F.F/% %trueGC$>&%\"xG6#F?-F2F&>&%#x0GFFFD>&FJ6#F.&FEFM>&FJ6#,&F/F.F.F.%#x1GF &6&FEFJF2F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "x(); " }}{PARA 0 "" 0 "" {TEXT -1 5 "x[5];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Potentialtyp" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m:='m': g:='g':k:='k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "V:=proc(x )" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 10 "1/2*k*x^2;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "m*g*x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGR6#%\"xG6\"F(F(*(%\"mG\"\"\"%\"g GF+9$F+F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "dV(s,d);dS(q,w,e ,r);" }}{PARA 0 "" 0 "" {TEXT -1 12 "dS(1,2,3,4);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "zuf\344llige Auswahl des \+ Punktes erh\366ht Konvergenz nicht" }}{PARA 0 "" 0 "" {TEXT -1 17 "ran i:=rand(2..n);" }}{PARA 0 "" 0 "" {TEXT -1 7 "rani();" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 25 "#xa:=array(1..10,1.. n+1);" }}{PARA 0 "" 0 "" {TEXT -1 5 "#x();" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Prozedur zur Iteration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "monte:=proc (pfade) local l,lk,ii,i,kk,mk; global x,xa,dx,ran,dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for l to pfade do # 10 Pakete der Groesse" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " for lk to kn do # kn (wenige r Ausgabe, weniger Kurven) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " for i from 2 to n do " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " for kk to 1 do # soll fuer kk>1 Spitzen abbauen " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " dx:=ran()/l^e x: # 1/l^ex soll zufaellige Variation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " #dem Iterationsstadi um anpassen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " if dS(x[i-1],x[i],x[i+1],dx) <=0 then x[i]:=x[i]+dx ; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " #print(x(),dS(x[i-1],x[i] ,x[i+1],dx));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " \+ fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " #x(); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " fo r mk to n+1 do xa[l,mk]:=x[mk]:od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " print(l*kn,Sa(l)); #Kontrollausgabe" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&monteGR6#%&pfadeG6(%\"lG%#lkG%#iiG% \"iG%#kkG%#mkG6\"F/?(8$\"\"\"F29$%%trueGC%?(8%F2F2%#knGF4?(8'\"\"#F2% \"nGF4?(8(F2F2F2F4C$>%#dxG*&-%$ranGF/F2)F1%#exG!\"\"@$1-%#dSG6&&%\"xG6 #,&F:F2F2FG&FN6#F:&FN6#,&F:F2F2F2FA\"\"!>FQ,&FQF2FAF2?(8)F2F2,&F&%#xaG6$F1FZ&FN6#FZ-%&printG6$*&F1F2F8F2-%#SaG6#F1F/6'FNFhnFAFD%#d tGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Sa(2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$sumG6$,&*&*&%\"mG\"\"\"),&&%#xaG6$\"\"#%\"iGF *&F.6$F0,&F1F*F*F*!\"\"F0F*F*%#dtGF5#F*F0*&#F*F0F**&,&*(F)F*%\"gGF*F2F *F**(F)F*F=F*F-F*F*F*F6F*F*F5/F1;F*%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Wirkung S der wirklichen Bahn" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#diff(y (t),t$2)=-diff(V(z),z)/m;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dgl:=subs(z=y(t),diff(y(t),t$2)=-diff(V(z),z)/m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dglG/-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#,$% \"gG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "soly:=proc() r hs(dsolve(\{dgl,y(0)=0,y(t1)=x1\},y(t))); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solyGR6\"F&F&F&-%$rhsG6#-%'dsolveG6$<%/-%\"yG6#\"\"! F2/-F06#%#t1G%#x1G%$dglG-F06#%\"tGF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "S:=proc () int(m/2*diff(soly(),t)^2-V(soly()),t=0..t1); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGR6\"F&F&F&-%$intG6$,&*&%\"mG\"\"\")-%%diffG6$- %%solyGF&%\"tG\"\"#F-#F-F5-%\"VG6#F2!\"\"/F4;\"\"!%#t1GF&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#S();" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Parame ter und Anfangsverteilung" }}{PARA 0 "" 0 "" {TEXT -1 90 "n: Anzahl de r Punkte, kr: halbe Breite der zuf\344lligen Anfangsverteilung (-kr < \+ x0[i] < kr)" }}{PARA 0 "" 0 "" {TEXT -1 111 "kn: Anzahl der Iteratione n, nach der eine Ausgabe erfolgt, bzw. ein Pfad abgespeichert wird, (G esamtzahl: 10kn)" }}{PARA 0 "" 0 "" {TEXT -1 99 "ex: die Streuung der \+ Zufallszahlen nimmt mit 1/j^ex ab, wo j das j-te Paket von kn Iteratio nen ist." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "n:=4: kr:=20: \+ kn:=1: ex:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Masse, Fallbesch leunigung und Federkonstante" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "m:=2: g:=100: k:=20: # m nicht float" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Endpunkt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "x1:=3: t1:=0.9: # fuer dezimale Aus gabe mindestens eine Groesse als float eingeben" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Vorbereitung" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "_seed:=100: # fuer neuen Anf angspfad deaktivieren" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "an z(): ini():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Iteration mit Kontrollausgabe" }}{PARA 0 "" 0 "" {TEXT -1 11 "_seed:=100:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pfade:=100:\nmonte(pfade);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+@Aqn9!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#$!+yx4[6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$$!+yx4[6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$!+yx4[6! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$!+yx4[6!\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"\"'$!+8!*>7?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"($!+8!*>7?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")$!+8! *>7?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*$!+YB$z!H!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$!+()=qDL!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6$!+()=qDL!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" #7$!+c.w=Z!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#8$!+c.w=Z!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9$!+Ck-o\\!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#:$!+>`W;_!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"# ;$!+q\\a$4'!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#<$!+A7\"p9'!\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#=$!+%*e#[='!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#>$!+xWj)f'!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#?$!+2-irq!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#@$!+yN66t!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#A$!+yN66t!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#B$!+[T^Cu!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"#C$!+gwhzv!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D$!+$oCnj(!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#E$!+$oCnj(!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#F$!+D`4Tw!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"#G$!+D`4Tw!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#H$!+KS)>o(!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#I$!+J7T&y(!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#J$!+d,6#)y!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#K$!+1o/lz!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#L$!+.7z,!)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#M$!+#*p4-!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#N$!+#*p4-!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#O$!+9ljN!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" #P$!+nP/R!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Q$!+nP/R!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#R$!+.8tg!)!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#S$!+R$)[h!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $\"#T$!+R$)[h!)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#U$!+!Q5A1)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#V$!+FZFm!)!\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"#W$!+N9I?\")!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#X$!+N9I?\")!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Y$!+1#o ]7)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z$!+1#o]7)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[$!+iT<^\")!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#\\$!+X&z()=)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#]$!+I53.#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#^$!+'fF(H#)!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#_$!+ScdT#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#`$!+ScdT#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#a$!+ScdT#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b$!+a83V#)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#c$!+Z1BW#)!\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"#d$!+L*)Rk#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#e$!+'H6lE)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#f$!+jHim #)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g$!+jHim#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h$!+Oy$)y#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#i$!+eLL#G)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" #j$!+A?W$G)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#k$!+A?W$G)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#l$!+&f6QG)!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#m$!+;b%QG)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#n$!+vu*QG)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#o$!+(Rs))G)!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#p$!+i:/\"H)!\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"#q$!+i:/\"H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#r$!+i:/\"H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#s$!+i:/ \"H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#t$!+)4MEH)!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#u$!+'\\DNH)!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#v$!+'\\DNH)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $\"#w$!+uk1%H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#x$!+uk1%H)!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#y$!+y*)f%H)!\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"#z$!+iAu%H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#!)$!+iAu%H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#\")$!+L )f\\H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"##)$!+4G.&H)!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#$)$!+4G.&H)!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#%)$!+4G.&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $\"#&)$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#')$!+mP0&H)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#()$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#))$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#*)$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#!*$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#\"*$!+mP0&H)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"##*$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#$*$!+mP0&H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#%*$!+d$z^H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#&*$!+d$z^H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#'*$!+d$z^H)! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#(*$!+d$z^H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#)*$!+d$z^H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#**$!+d$z^H)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"$+\"$!+d$z^H)!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "`exa kt`=S();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&exaktG$!++++v')!\"(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "x0();x();" }}{PARA 0 "" 0 "" {TEXT -1 5 "xa();" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Darstellung des Anfangspfades (ploti), der letzten N\344 herung (plotm) und der wirklichen Bahn (plote)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ploti:=plot ([seq([i*dt-dt,x0[i]],i=1..n+1)],color=green):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "plotm:=plot([seq([i*dt-dt,x[i]],i=1..n+1)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plote:=plot(soly(),t=0..t 1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#plote;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(\{ploti,plotm,plo te\});" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Jeder kn-te Pfad" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "aniplot:=seq(display([plote,ploti,plot([seq([i*d t-dt,xa[s,i]],i=1..n+1)])]),s=1..pfade):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display([aniplot]);" }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Animation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "displ ay([aniplot],insequence=true);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Diskussion:" }}{PARA 0 " " 0 "" {TEXT -1 538 "Beim Experimentieren mit den verschiedenen Parame tern stellt man fest, da\337 sich die Konvergenz der gedachten Pfade g egen die wirkliche Bahn nicht immer erzwingen l\344\337t. Insbesondere kann man bei periodischer Bewegung sogar Divergenz bekommen, wenn t1 \+ gr\366\337er als die halbe Periodendauer wird. Das liegt sicher nicht \+ nur an dem simplen Rechenverfahren, sondern in der *Natur* der Sache: \+ die Natur nimmt es nicht so genau mit *der* Wirklichkeit. Es gibt viel e Pfade, die sich in ihrer Wirkung von der wirklichen Bahn nur wenig u nterscheiden. " }}{PARA 0 "" 0 "" {TEXT -1 196 "F\374r gro\337e n soll te man eine Verbesserung der Konvergenz erwarten. Statt dessen bleibt \+ die Iteration manchmal bei lokalen Minima h\344ngen und liefert z.T. z ur wirklichen Bahn spiegelbildliche Bahnen." }}{PARA 0 "" 0 "" {TEXT -1 756 "Aber es ist ja auch nicht der Zweck dieses einfachen Modells, \+ die wirkliche Bahn zu berechnen. Es soll vielmehr das zeigen, was Feyn man meint, wenn er von Teilchen spricht, die \"schnuppern\", um ihre B ahn zu finden. W\344hrend bei den Feynmanschen Pfadintegralen die Ausw ahl (besser die Gewichtung) der Pfade durch Interferenz geschieht, wir d sie hier schlicht durch Probieren erreicht. So viel Zeit kann sich d ie Natur nat\374rlich nicht nehmen. Kein Elektron w\374rde je ankommen , wenn es st\344ndig schnuppernd unterwegs w\344re. Und diese Interfer enz mu\337 schon ein genial schneller Rechner sein, wenn sie zu jeder \+ Zeit und an jedem Ort in einem einzigen Moment das Ergebnis der \334be rlagerung aller m\366glichen Pfade berechnet und das auch noch f\374r \+ alle Teilchen der Welt." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni-tuebingen.de" }}}}{MARK "0 2 0" 7 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }