{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 "" -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 0 1 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 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Text Outp ut" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helv etica" 1 10 0 0 255 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "c International Thomson Pu blishing 1995 filename: wiwurf" }}{PARA 0 "" 0 "" {TEXT -1 106 "Autor: Komma \+ Datum: Oktober 94" }}{PARA 0 "" 0 "" {TEXT -1 32 "Thema: Wirkungswellen beim Wurf " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Wirkungsw ellen am Beispiel des Wurfes (senkrecht und schief)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;w ith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changeco ords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 16 "Senkrechter Wurf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Wir ben\366tigen den Imp uls, die kinetische Energie und die Lagrangefunktion als Funktionen de r Gesamtenergie H und der H\366he y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Energie-Impuls-Beziehung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ep:=p^2/(2*m)=T;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#EpG/,$*&*$)%\"pG\"\"#\"\"\"F,%\"mG!\"\"#F,F+%\"TG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Energieerhaltung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Ees:=H=T+V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EesG/%\"HG,&%\"TG\"\"\"%\"VGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Lagrange-Funktion:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " Lagr:=L=T-V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LagrG/%\"LG,&%\"TG \"\"\"%\"VG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "potentielle E nergie" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "V:=m*g*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG*(%\"mG\"\"\"%\"gGF'%\"yGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sol:=solve(\{Ep,Ees,Lagr\},\{p,T,L \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<%/%\"TG,&%\"HG\"\"\"*( %\"mGF*%\"gGF*%\"yGF*!\"\"/%\"LG,&F)F***\"\"#F*F,F*F-F*F.F*F//%\"pG-%' RootOfG6$,(*$)%#_ZGF4F*F**(F4F*F,F*F)F*F/**F4F*)F,F4F*F-F*F.F*F*/%&lab elG%$_L1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "allvalues(%); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "allvalues(p);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$< %/%\"TG,&%\"HG\"\"\"*(%\"mGF(%\"gGF(%\"yGF(!\"\"/%\"LG,&F'F(**\"\"#F(F *F(F+F(F,F(F-/%\"pG*$-%%sqrtG6#,&*&F*F(F'F(F2**F2F()F*F2F(F+F(F,F(F-F( <%F$F./F4,$F5F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%%sqrtG6#,&*&%\" mG\"\"\"%\"HGF*\"\"#**F,F*)F)F,F*%\"gGF*%\"yGF*!\"\"F*,$F#F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "py:=op(1,[%]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#pyG*$-%%sqrtG6#,&*&%\"mG\"\"\"%\"HGF,\"\"#**F .F,)F+F.F,%\"gGF,%\"yGF,!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Senkrechter Wurf mit d er Gesamtenergie H und der Fallbeschleunigung g." }}{PARA 0 "" 0 "" {TEXT -1 27 "charakteristische Funktion:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w:=int(py,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" wG,$*&*$),&*&%\"mG\"\"\"%\"HGF,\"\"#**F.F,)F+F.F,%\"gGF,%\"yGF,!\"\"# \"\"$F.F,F,*&F0F,F1F,F3#F3F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "O rtsabh\344ngiger Teil der Wirkungswelle und ihr Realteil" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "psi:=exp(I*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG-%$expG6#*&*&^##!\"\"\"\"$\"\"\"),&*&%\"mGF.%\"HGF.\"\"#* *F4F.)F2F4F.%\"gGF.%\"yGF.F,#F-F4F.F.*&F6F.F7F.F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "pr:=Re(psi):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Darstellung des Realteils " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "m:=1/2:g:=10:H:=50: ys:=H/(m*g) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG\"#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "plr:=plot(evalc(pr),y=0..15,numpoints=1000): \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "plr;" }}{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 -1 271 "Zun\344chst erkennt man den Zusammenhang gro\337er Impul s - kleine Wellenl\344nge. Oberhalb der klassischen Steigh\366he ys = \+ H/(m*g) verschwindet die Amplitude aber nicht. Der Impuls wird dort im agin\344r, und das hat ein exponentielles Abklingen der Amplitude zur \+ Folge: Tunneleffekt!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Imagin\344rteil" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "pli:=plot(evalc(Im(psi)),y=0..15,numpoints=1000, color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "pli;" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 " Er verschwindet oberhalb ys und ist unterhalb ys um 90\260 phasenversc hoben, wie es sich geh\366rt!" }}{PARA 0 "" 0 "" {TEXT -1 9 "Wirklich? " }}{PARA 0 "" 0 "" {TEXT -1 45 "plot(evalc(Im(psi)),y=9.99..10.01,col or=red);" }}{PARA 0 "" 0 "" {TEXT -1 22 "Gemeinsame Darstellung" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(\{plr,pli\});" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "3d-Darstellung des Realteils:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot3d(evalc(pr),x=0..1,y=0..15,axe s=boxed,grid=[10,100],orientation=[-40,10]);" }}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Interferenz der aufsteigenden mit der absteigenden Welle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 543 "Wenn die Wirkungswellen echte \+ Wellen sind, mu\337 sich wegen der \"Reflexion\" im Scheitelpunkt eine stehende Welle ergeben, allerdings mit ortsabh\344ngiger Wellenl\344n ge. Im Gegensatz zu den gewohnten Wellen k\366nnen wir aber die Laufri chtung nicht durch das Vorzeichen von Ht umkehren, denn es gilt immer \+ S=W-Ht (und nie S=W+Ht). Die Laufrichtung wird vielmehr durch den Impu lsvektor bestimmt, der im eindimensionalen Fall das Vorzeichen von W \+ \344ndert. Das ist unterhalb des Scheitels unproblematisch. Aber oberh alb des Scheitels wir w imagin\344r positiv." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&\"#] \"\"\"*&\"\"&F(%\"yGF(!\"\"#\"\"$\"\"#F(#!\"#\"#:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "m:=1/2:g:=10:H:=50:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(\{w,Im(w)\},y=-ys..2*ys);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Wir d\374rfen also oberhalb von y s das Vorzeichen nicht \344ndern, sonst bekommen wir eine unphysikalis che (exponentiell ansteigende) L\366sung." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "m:='m':g:='g':H:='H':t:='t':ys:='ys':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Sp:=w-H*t; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SpG,&*&*$),&*&%\"mG\"\"\"%\"HGF,\"\"#**F.F,)F+F.F,% \"gGF,%\"yGF,!\"\"#\"\"$F.F,F,*&F0F,F1F,F3#F3F5*&F-F,%\"tGF,F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Herausfiltern des unphysikalischen Teils mit conjugate " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sm:=-conju gate(w)-H*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SmG,&-%*conjugateG6 #*&*$),&*&%\"mG\"\"\"%\"HGF/\"\"#**F1F/)F.F1F/%\"gGF/%\"yGF/!\"\"#\"\" $F1F/F/*&F3F/F4F/F6#F/F8*&F0F/%\"tGF/F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Interferenz" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "psiint: =exp(I*Sp)+exp(I*Sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'psiintG,&- %$expG6#*&^#\"\"\"F+,&*&*$),&*&%\"mGF+%\"HGF+\"\"#**F4F+)F2F4F+%\"gGF+ %\"yGF+!\"\"#\"\"$F4F+F+*&F6F+F7F+F9#F9F;*&F3F+%\"tGF+F9F+F+-F'6#*&F*F +,&-%*conjugateG6#F-#F+F;F>F9F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Steigh\366he (kann in Bereichsangaben von Plotbefehlen verwendet w erden)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ys:=H/(m*g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG*&%\"HG\"\"\"*&%\"mGF'%\"gGF'!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Realteil der stehenden Welle" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "rpsiint:=evalc(Re(psiint));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(rpsiintG,&*&-%$expG6#,$*&*&)-%$absG6#,&*&%\"mG\"\"\" %\"HGF4\"\"#**F6F4)F3F6F4%\"gGF4%\"yGF4!\"\"#\"\"$F6F4-%$sinG6#*&,&#F= \"\"%F4*&#F=FDF4-%'signumGF0F4F;F4%#PiGF4F4F4*&F8F4F9F4F;#F4F=F4-%$cos G6#,&*&*&F-F4-FMF@F4F4*&F8F4F9F4F;FK*&F5F4%\"tGF4F4F4F4*&F'F4-FM6#,&FP #F;F=FTF4F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "m:=1/2:g:= 10:H:=50:t:=0.01:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(r psiint,y=-5..12,a=-2..2,numpoints=1000);" }}{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 63 "Eine Momentaufnahme steht nat\374rlich. Aber steht auch die Welle?" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "animate(rpsiint,y=0..12,t=0..2*Pi/H,numpoints=500); " }}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191 "Realist ischer als die unendlich ausgedehnten Wellen ist ein halbwegs lokalisi ertes Teilchen, das den Umkehrpunkt zur Zeit t = 0 erreicht. Wir brauc hen zur Simulation der Reflexion zwei davon:" }}{PARA 0 "" 0 "" {TEXT -1 169 "Reflexion = \334berlagerung zweier gegenl\344ufiger Pakete (je nseits des Umkehrpunktes werden die Pakete durch die Amplitude der Wir kungswelle automatisch ausgeblendet s.u.). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Test der Paketfunktionen" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "m:='m':g:='g':H:='H':y01:='y 01':y02:='y02':t:='t':s:='s':ys:='ys':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Paketmittelpunkte" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y01:=ys+1/2*g*t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y01G,&%#ysG\"\"\"*(#F'\"\"#F'%\"gGF')%\"tGF*F'F '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y02:=ys-1/2*g*t^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y02G,&%#ysG\"\"\"*&#F'\"\"#F'*&%\"g GF')%\"tGF*F'F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "pak et1:=exp(-(y-y01)^2/s^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'paket1 G-%$expG6#,$*&*$),(%\"yG\"\"\"%#ysG!\"\"*&#F.\"\"#F.*&%\"gGF.)%\"tGF3F .F.F0F3F.F.*$)%\"sGF3F.F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "paket2:=exp(-(y-y02)^2/s^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'paket2G-%$expG6#,$*&*$),(%\"yG\"\"\"%#ysG!\"\"*(#F.\"\"#F.%\"gGF.)%\" tGF3F.F.F3F.F.*$)%\"sGF3F.F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "total:=paket1+paket2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&to talG,&-%$expG6#,$*&*$),(%\"yG\"\"\"%#ysG!\"\"*&#F/\"\"#F/*&%\"gGF/)%\" tGF4F/F/F1F4F/F/*$)%\"sGF4F/F1F1F/-F'6#,$*&*$),(F.F/F0F1*(#F/F4F/F6F/F 7F/F/F4F/F/*$F:F/F1F1F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ys:=H/m/g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG*&%\"HG\"\"\"*&%\"mGF'%\"gGF'!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "m:=1/2:g:=10:H:=50:s:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "total;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG 6#,$*$),(%\"yG\"\"\"\"#5!\"\"*&\"\"&F,)%\"tG\"\"#F,F.F3F,#F.\"\"%F,-F% 6#,$*$),(F+F,F-F.*&F0F,F1F,F,F3F,F4F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate(total,y=-ys..2*ys,t=-2..2,frames=100,numpoint s=500);" }}{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 60 "In diese Einh\374llenden setzen wi r die Wirkungswellen von oben" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m: ='m':g:='g':H:='H':s:='s':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Sp:=w-H*t; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SpG,&*&*$),&*&% \"mG\"\"\"%\"HGF,\"\"#**F.F,)F+F.F,%\"gGF,%\"yGF,!\"\"#\"\"$F.F,F,*&F0 F,F1F,F3#F3F5*&F-F,%\"tGF,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sm:=-conjugate(w)-H*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SmG ,&-%*conjugateG6#*&*$),&*&%\"mG\"\"\"%\"HGF/\"\"#**F1F/)F.F1F/%\"gGF/% \"yGF/!\"\"#\"\"$F1F/F/*&F3F/F4F/F6#F/F8*&F0F/%\"tGF/F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Interferenz (nach Multiplikation mit den \+ Paketfunktionen)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "psiint:=exp(I*S p)*paket1+exp(I*Sm)*paket2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'psii ntG,&*&-%$expG6#*&^#\"\"\"F,,&*&*$),&*&%\"mGF,%\"HGF,\"\"#**F5F,)F3F5F ,%\"gGF,%\"yGF,!\"\"#\"\"$F5F,F,*&F7F,F8F,F:#F:F<*&F4F,%\"tGF,F:F,F,-F (6#,$*&*$),(F9F,*&F4F,*&F3F,F8F,F:F:*&#F,F5F,*&F8F,)F@F5F,F,F:F5F,F,*$ )%\"sGF5F,F:F:F,F,*&-F(6#*&F+F,,&-%*conjugateG6#F.#F,F " 0 "" {MPLTEXT 1 0 27 "rpsiint:=evalc(Re(psiint));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(rpsiintG,&*(-%$expG6#,$*&*&)-%$absG 6#,&*&%\"mG\"\"\"%\"HGF4\"\"#**F6F4)F3F6F4%\"gGF4%\"yGF4!\"\"#\"\"$F6F 4-%$sinG6#*&,&#F=\"\"%F4*&#F=FDF4-%'signumGF0F4F;F4%#PiGF4F4F4*&F8F4F9 F4F;#F4F=F4-%$cosG6#,&*&*&F-F4-FMF@F4F4*&F8F4F9F4F;FK*&F5F4%\"tGF4F4F4 -F(6#,$*&*$),(F:F4*&F5F4*&F3F4F9F4F;F;*&#F4F6F4*&F9F4)FUF6F4F4F;F6F4F4 *$)%\"sGF6F4F;F;F4F4*(F'F4-FM6#,&FP#F;F=FTF4F4-F(6#,$*&*$),(F:F4FgnF;* (#F4F6F4F9F4F\\oF4F4F6F4F4*$F^oF4F;F;F4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Bitte warten ..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m: =1/2:g:=10:H:=50:s:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a nimate(rpsiint,y=-ys..2*ys,t=-2..2,numpoints=400,frames=100);" }} {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 22 "Die Sache klappt also " }{TEXT 258 10 "im Prinz ip" }{TEXT -1 159 " f\374r den senkrechten Wurf, d.h. wir haben soeben mit geringem Aufwand die \"WKB-N\344herung f\374r das im Schwerefeld \+ tanzende Elektron\" erzeugt und sichtbar gemacht!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 13 "Schiefer Wurf" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "F\374r de n schiefen Wurf m\374ssen wir in x-Richtung eine Bewegung mit konstant em Impuls px \374berlagern " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "m:='m':g:='g': H:='H':px:='px':V:='V':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "ys:=(H-1/2*px^2/m)/m/g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG*&,&%\"HG\"\"\"*&#F(\"\"#F(*&*$)%#pxGF+F(F(%\"mG! \"\"F(F1F(*&F0F(%\"gGF(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "py:=sqrt(2*m*(H-V)-px^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#py G*$-%%sqrtG6#,(*&%\"mG\"\"\"%\"HGF,\"\"#*(F.F,F+F,%\"VGF,!\"\"*$)%#pxG F.F,F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "V:=m*g*y;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG*(%\"mG\"\"\"%\"gGF'%\"yGF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w:=int(py,y)+px*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG,&*&*$),(*&%\"mG\"\"\"%\"HGF,\"\"#**F. F,)F+F.F,%\"gGF,%\"yGF,!\"\"*$)%#pxGF.F,F3#\"\"$F.F,F,*&F0F,F1F,F3#F3F 8*&F6F,%\"xGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Zur Kontrolle die Bahn, die man aus der Wirkungsfunktion am einfachsten so erh\344l t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ypl:=diff(w,px);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yplG,&*&*&-%%sqrtG6#,(*&%\"mG\"\"\"%\"HGF.\" \"#**F0F.)F-F0F.%\"gGF.%\"yGF.!\"\"*$)%#pxGF0F.F5F.F8F.F.*&F2F.F3F.F5F .%\"xGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ypl:=solve(ypl, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yplG,$*&,(*(%\"mG\"\"\"%\"HG F*)%#pxG\"\"#F*F.*$)F-\"\"%F*!\"\"*()%\"xGF.F*)F)F1F*)%\"gGF.F*F2F**() F)F.F*F8F*F,F*F2#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "m: =1/2:g:=10:H:=50:px:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " plot(ypl,x=-8..8);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "(Ende Bahn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "m:='m':g:='g': H:='H':px:='px':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#w;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rpsiS:=evalc(Re(exp(I*(w-H*t))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rpsiSG*&-%$expG6#,$*&*&)-%$absG6#,(*&%\"mG\"\"\"%\"HGF3\"\"#**F5 F3)F2F5F3%\"gGF3%\"yGF3!\"\"*$)%#pxGF5F3F:#\"\"$F5F3-%$sinG6#*&,&#F?\" \"%F3*&#F?FFF3-%'signumGF/F3F:F3%#PiGF3F3F3*&F7F3F8F3F:#F3F?F3-%$cosG6 #,(*&*&F,F3-FOFBF3F3*&F7F3F8F3F:FM*&F=F3%\"xGF3F:*&F4F3%\"tGF3F3F3" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "#ys;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "Im Gegensatz zum senkrechten Wurf m\374ssen wir jet zt im Umkehrpunkt die Wellenfronten wechseln, die zu positivem bzw. ne gativem py geh\366ren. Hier sind z.B. die Wirkungswellen zum aufsteige nden Ast (py positiv)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "px :=6:H:=50:g:=10:m:=1/2:n:=40:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "animate3d(rpsiS,x=0..2,y=0..1.5*ys,t=0..2*Pi/H*(1-1/n),axes=box ed,grid=[50,20],orientation=[-40,20],frames=n,style=wireframe);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "animate3d(rpsiS,x =0..2,y=0..1.5*ys,t=0..2*Pi/H*(1-1/n),axes=boxed,grid=[50,20],orientat ion=[-40,20],frames=n,style=patchcontour);" }}{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 -1 375 " Man darf sich vom Bild nicht t\344uschen lassen: Wenn Sie die Paramete r \344ndern, werden Sie feststellen, da\337 die Wellen unerwartete Mus ter aufweisen k\366nnen. Das liegt an der Interpolation, die von den M aple-Plotroutinen gemacht wird, also auch an der Orientierung von 3d-P lots. In solchen F\344llen mu\337 man die Aufl\366sung erh\366hen oder \344ndern (grid), bzw. einen Contourplot erstellen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t:=0:px:=6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "contourplot(rpsiS,x=0..2,y=0..1.5*ys,axes=boxed,grid= [60,50],contours=5);" }}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 816 "Ein altes Problem des \334bergangs v on der klassischen Physik zu Quantenphysik, mit dem sich schon Schr \366dinger und Sommerfeld geplagt haben, ist: wie sind die Randbedingu ngen bzw. Anfangsbedingungen zu w\344hlen? In der Quantenmechanik = We llenmechanik gibt es keine wirkliche Bahn mehr. Es gibt nur noch Erhal tungsgr\366\337en. Ein Zustand wird durch Interferenz gebildet. Aber I nterferenz wovon? F\344llt das Teilchen gerade oder steigt es? (Es mac ht zu jeder Zeit beides.) Von wo aus steigt es oder f\344llt es? Welch er Ast der (mindestens zweiwertigen) Funktion der Wellenfronten ist zu nehmen? Diese Fragen f\374hren im Falle der periodischen Bewegungen z u den Wirkungs- und Winkelvariablen. In unserem Beispiel des schiefen \+ Wurfes (aperiodischer Grenzfall = Parabel) k\366nnen wir diesen \334be rlegungen schon fast spielerisch nachgehen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 29 "Schiefer \+ Wurf mit Interferenz" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "m:= 'm':g:='g': H:='H':px:='px':t:='t':w;s:='s':ys:='ys':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*$),(*&%\"mG\"\"\"%\"HGF*\"\"#**F,F*)F)F,F*%\" gGF*%\"yGF*!\"\"*$)%#pxGF,F*F1#\"\"$F,F*F**&F.F*F/F*F1#F1F6*&F4F*%\"xG F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Wir lassen wieder zwei Pa kete gegen den Umkehrpunkt laufen (und dar\374ber hinaus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y01:=ys+1/2*g*t^2*signum(t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y01G,&%#ysG\"\"\"**#F'\"\"#F'%\"gGF ')%\"tGF*F'-%'signumG6#F-F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y02:=ys-1/2*g*t^2*signum(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$y02G,&%#ysG\"\"\"*&#F'\"\"#F'*(%\"gGF')%\"tGF*F'-%'signumG6#F.F'F'! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ys:=(H-1/2*px^2/m)/ m/g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysG*&,&%\"HG\"\"\"*&#F(\"\" #F(*&*$)%#pxGF+F(F(%\"mG!\"\"F(F1F(*&F0F(%\"gGF(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "paket1:=exp(-((y-y01)^2+(x-px/m*t)^2)/s^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'paket1G-%$expG6#,$*&,&*$),(%\"y G\"\"\"*&,&%\"HGF/*&#F/\"\"#F/*&*$)%#pxGF5F/F/%\"mG!\"\"F/F;F/*&F:F/% \"gGF/F;F;*&#F/F5F/*(F=F/)%\"tGF5F/-%'signumG6#FBF/F/F;F5F/F/*$),&%\"x GF/*&*&F9F/FBF/F/F:F;F;F5F/F/F/*$)%\"sGF5F/F;F;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "paket2:=exp(-((y-y02)^2+(x-px/m*t)^2)/s^2):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "F\374r die Plots ist es ganz g \374nstig, wenn wir die Steigzeit und die Wurfweite zur Verf\374gung h aben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ts:=sqrt(2*ys/g); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tsG*&-%%sqrtG6#\"\"#\"\"\"-F'6# *&,&%\"HGF**&#F*F)F**&*$)%#pxGF)F*F*%\"mG!\"\"F*F7F**&F6F*)%\"gGF)F*F7 F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xw:=abs(px)/m*ts;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xwG*&*(-%$absG6#%#pxG\"\"\"-%%sqrtG 6#\"\"#F+-F-6#*&,&%\"HGF+*&#F+F/F+*&*$)F*F/F+F+%\"mG!\"\"F+F;F+*&F:F+) %\"gGF/F+F;F+F+F:F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Nun noch d ie charakteristischen Funktionen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "wp:=w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wpG,&*&*$ ),(*&%\"mG\"\"\"%\"HGF,\"\"#**F.F,)F+F.F,%\"gGF,%\"yGF,!\"\"*$)%#pxGF. F,F3#\"\"$F.F,F,*&F0F,F1F,F3#F3F8*&F6F,%\"xGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "wm:=conjugate(-w)+2*px*x;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#wmG,&-%*conjugateG6#,&*&*$),(*&%\"mG\"\"\"%\"HGF0 \"\"#**F2F0)F/F2F0%\"gGF0%\"yGF0!\"\"*$)%#pxGF2F0F7#\"\"$F2F0F0*&F4F0F 5F0F7#F0F<*&F:F0%\"xGF0F7F0*(F2F0F:F0F@F0F0" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "Und die Amplituden der Wirkungswellen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rpsip:=evalc(Re(exp(I*wp)));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&rpsipG*&-%$expG6#,$*&*&)-%$absG6#,(*&%\"mG\"\"\"% \"HGF3\"\"#**F5F3)F2F5F3%\"gGF3%\"yGF3!\"\"*$)%#pxGF5F3F:#\"\"$F5F3-%$ sinG6#*&,&#F?\"\"%F3*&#F?FFF3-%'signumGF/F3F:F3%#PiGF3F3F3*&F7F3F8F3F: #F3F?F3-%$cosG6#,&*&*&F,F3-FOFBF3F3*&F7F3F8F3F:#F:F?*&F=F3%\"xGF3F3F3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rpsim:=evalc(Re(exp(I*w m)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rpsimG*&-%$expG6#,$*&*&)-% $absG6#,(*&%\"mG\"\"\"%\"HGF3\"\"#**F5F3)F2F5F3%\"gGF3%\"yGF3!\"\"*$)% #pxGF5F3F:#\"\"$F5F3-%$sinG6#*&,&#F?\"\"%F3*&#F?FFF3-%'signumGF/F3F:F3 %#PiGF3F3F3*&F7F3F8F3F:#F3F?F3-%$cosG6#,&*&*&F,F3-FOFBF3F3*&F7F3F8F3F: FM*&F=F3%\"xGF3F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Kontrollplots" }}{PARA 0 "" 0 "" {TEXT -1 19 "m:=1/2:g:=10:H:=50:" }}{PARA 0 "" 0 "" {TEXT -1 6 "px:=2: " }}{PARA 0 "" 0 "" {TEXT -1 10 "t:=0:s:=2:" }}{PARA 0 "" 0 "" {TEXT -1 6 "rpsim;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 103 "plot3d(rpsip,x=-xw..xw,y=0..2*ys,axes=boxed,grid=[50,50] ,orientation=[-90,0],style=contour,contours=4);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 103 "plot3d(rpsim,x=-xw..xw ,y=0..2*ys,axes=boxed,grid=[50,50],orientation=[-90,0],style=contour,c ontours=5);" }}{PARA 0 "" 0 "" {TEXT -1 44 "m:='m':g:='g':H:='H':px:=' px':t:='t':s:='s':" }}{PARA 0 "" 0 "" {TEXT -1 32 "Wirkungsfunktionen, zeitabh\344ngig" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Sp:=wp- H*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SpG,(*&*$),(*&%\"mG\"\"\"% \"HGF,\"\"#**F.F,)F+F.F,%\"gGF,%\"yGF,!\"\"*$)%#pxGF.F,F3#\"\"$F.F,F,* &F0F,F1F,F3#F3F8*&F6F,%\"xGF,F,*&F-F,%\"tGF,F3" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "Sm:=wm-H*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#SmG,(-%*conjugateG6#,&*&*$),(*&%\"mG\"\"\"%\"HGF0\"\"#**F2F0)F/F2F 0%\"gGF0%\"yGF0!\"\"*$)%#pxGF2F0F7#\"\"$F2F0F0*&F4F0F5F0F7#F0F<*&F:F0% \"xGF0F7F0*(F2F0F:F0F@F0F0*&F1F0%\"tGF0F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Interferenz" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "wurf:=paket1*exp(I*Sp)+paket2*exp(I*Sm);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%wurfG,&*&-%$expG6#,$*&,&*$),(%\"yG\"\"\"*&,&%\"HGF1* &#F1\"\"#F1*&*$)%#pxGF7F1F1%\"mG!\"\"F1F=F1*&F " 0 "" {MPLTEXT 1 0 23 "Rwurf:=evalc(Re(wurf)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&RwurfG,&*(-%$expG6#*&,&*$),*%\" yG\"\"\"*&%\"HGF0*&%\"mGF0%\"gGF0!\"\"F6*&*&#F0\"\"#F0)%#pxGF:F0F0*&)F 4F:F0F5F0F6F0*&#F0F:F0*(F5F0)%\"tGF:F0-%'signumG6#FCF0F0F6F:F0F6*$),&% \"xGF0*&*&FF0F5F0F/F0F6*$F;F0F6#\"\"$F:F0-%$sinG6#*&,&#Fhn \"\"%F0*&#FhnF_oF0-FEFXF0F6F0%#PiGF0F0F0*&F>F0F5F0F6#F0FhnF0-%$cosG6#, (*&*&FUF0-FgoF[oF0F0*&F>F0F5F0F6Feo*&FF0F5F0F6F0**F9F0F5F0FBF0FDF0F0F:F0F 6FGF6F0*$FNF0F6F0FPF0-Fgo6#,(Fjo#F6FhnF^pF6F_pF0F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Zahlenwerte" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m:=1/2:g:=10:H:=50:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "px:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "t :=-1/2*ts:s:=2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Alle Parameter gesetzt?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#Rwurf;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Momentaufnahme" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot3d(Rwurf,x=-xw..xw,y=0..2*ys,axes=boxed,grid= [50,50],orientation=[-90,0],style=contour,contours=5);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot3d (Rwurf,x=-xw..xw,y=0..2*ys,axes=boxed,grid=[50,20],orientation=[-40,2] ,style=wireframe);" }}{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 9 "Animation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "n:=40:t:='t':s:=2:px:=1:#Digits:=2: \344ndert nichts \+ an Geschwindigkeit und Interpolation" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Bitte warten ... oder kleineres n w\344hlen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "animate3d(10^10*Rwurf,x=-xw-s..xw+s,y=0..1 .5*ys,t=-ts..ts,axes=boxed,orientation=[-90,0],grid=[50,50],frames=n,s tyle=contour,shading=z,contours=5);" }}{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 -1 343 "Leider \+ Interpolationsfehler bei linearer Darstellung (ln(10+Rwurf) bzw. Quadr ieren beheben das, beseitigen aber auch die interessanten Wellenfronte n S=0, auf denen das Teilchen zu reiten scheint -- mit stroboskopische n Effekten. Quadrat (od. gerade Hochzahl) hat unerw\374nschten Nebenef fekt im Scheitel, weil Max u. Min. gleich bewertet werden)." }}{PARA 0 "" 0 "" {TEXT -1 145 "Man kann aber mit 10^10 multiplizieren (dann a ber nicht 1:1 w\344hlen), um die Interpolationskreuze zum Verschwinden zu bringen (10^6 tut es nicht)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Abb.p5..7wiwurf.ps Postscript-Fans k\366nnen jetzt ihre ps-files a m laufenden Band produzieren:" }}{PARA 0 "" 0 "" {TEXT -1 17 "t:='t':v xlab:=``:" }}{PARA 0 "" 0 "" {TEXT -1 21 "for t from -1 to 1 do" }} {PARA 0 "" 0 "" {TEXT -1 57 "vtitle:=`t=`.t: `Name`:=p.(6+t).wiwurf.`. `.ps:pspl(Name):" }}{PARA 0 "" 0 "" {TEXT -1 142 "plot3d(10^10*Rwurf,x =-xw-s..xw+s,y=0..1.5*ys,axes=framed,orientation=[-90,0],grid=[100,100 ],style=contour,shading=z,contours=5,opt3d);winpl():" }}{PARA 0 "" 0 " " {TEXT -1 3 "od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "n:=50:t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "animate3d(Rwurf,x=-xw-s..xw+s,y=0. .1.5*ys,t=-ts..ts,axes=boxed,frames=n,grid=[20,20],orientation=[-40,55 ],style=wireframe,scaling=constrained);" }}{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 -1 102 "Wir treiben die *Simulation* noch einen Schritt weiter und lassen das Tei lchen entstehen und vergehen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s: =1+abs(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&\"\"\"F&-%$absG6 #%\"tGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Bitte warten, bis das Teilchen entstanden ist ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "animate3d(1 0^10*Rwurf,x=-xw-2..xw+2,y=0..1.5*ys,t=-ts..ts,axes=boxed,orientation= [-90,0],grid=[50,50],frames=n,style=contour,shading=z,contours=5);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "W\344hrend das Elektron im Scheitel eine Schr\366dingersche Zitterbew egung macht, taucht unten Diracs See auf ... verkehrte Welt." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Amplitude umgekehrt prop ortional zur Zeit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=100: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "animate3d(Rwurf/s,x=-xw -3..xw+3,y=0..1.5*ys,t=-ts..ts,frames=n,grid=[30,20],orientation=[-65, 65]);" }}{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 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 }