{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 } {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 0 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 "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 "" 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 "R 3 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 71 "c International Thomson Pu blishing, Bonn 1995 filename: rydb" }}{PARA 0 "" 0 "" {TEXT -1 85 "Autor: Komma \+ Datum: 11.94" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Thema: Quasiklassisches Atom" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Synopsis: Wellenpaket m it Dispersion" }}{PARA 0 "" 0 "" {TEXT -1 60 "Zeitliche Entwicklung im Laborsystem und im SPS des Paketes." }}{PARA 0 "" 0 "" {TEXT -1 65 " \334berlappung des zerflie\337enden Paketes auf Kreisbahn: Interferenz ." }}{PARA 0 "" 0 "" {TEXT -1 37 "Animation des \"Rydberg/Landau-Atoms \"." }}{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 "" {MPLTEXT 1 0 21 "restart; with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Gau\337paket" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "psi:=1/s*exp(-X/(2*a^2*s^2)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG*&-%$expG6#,$*&%\"XG\"\"\" *&)%\"aG\"\"#F,)%\"sGF0F,!\"\"#F3F0F,F2F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "mit komplexer Varianz (vgl. paket1.ms)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s:=sqrt(1+I*t/a^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG*$-%%sqrtG6#,&\"\"\"F**&*&^#F*F*%\"tGF*F**$)%\"aG \"\"#F*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "X:=x^2- 2*I*a^2*k*x+I*a^2*k^2*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,(*$ )%\"xG\"\"#\"\"\"F***^#!\"#F*)%\"aGF)F*%\"kGF*F(F*F***^#F*F*F.F*)F0F)F *%\"tGF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Das Betragsquadrat \+ wird schnell berechnet" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a psi:=simplify(evalc(abs(psi))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%apsiG*&-%$expG6#,$*&*&)%\"aG\"\"#\"\"\"),&%\"xGF/*&%\"kGF/%\"tGF/!\" \"F.F/F/,&*$)F-\"\"%F/F/*$)F5F.F/F/F6F6F/*$-%%sqrtG6#*&F7F/*$F9F/F6F/F 6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Der Realteil ist etwas umfan greicher, (auch mit assume a>0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#assume(a >0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rpsi:=evalc(Re(psi) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%rpsiG,&*&*(-%%sqrtG6#,&*$-F)6 #,&\"\"\"F0*&*$)%\"tG\"\"#F0F0*$)%\"aG\"\"%F0!\"\"F0F0F5F5F0F0-%$expG6 #,&*&*$)%\"xGF5F0F0*&)F8F5F0F/F0F:#F:F5*&*&,&*(FDF0%\"kGF0FBF0F0*&#F0F 5F0*(FDF0)FJF5F0F4F0F0F:F0F4F0F0*&F7F0F/F0F:F0F0-%$cosG6#,&*&FHF0*&FDF 0F/F0F:F0*&*(#F0F5F0FAF0F4F0F0*&F7F0F/F0F:F0F0F0,(F,FXFXF0*(#F0F9F0)-% %csgnG6#,&F4F0*&^#F:F0FDF0F0F5F0,&F,F5F5F:F0F0F:FX*&*,FXF0FhnF0-F)6#F^ oF0F;F0-%$sinGFRF0F0FZF:F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 338 "Die Vereinfachung dauert \+ recht lange und man bekommt das \374berfl\374ssige (?) csgn nicht weg. Au\337erdem h\344ngt die Reaktion von simplify davon ab, ob das .m-fi le mit eingelesen wurde!! (Restart \344ndert daran nichts). Das Ergebn is der Vereinfachung wurde deshalb via lprint in das Worksheet \374ber nommen, um den Term schnell zur Verf\374gung zu haben." }}{PARA 0 "" 0 "" {TEXT -1 21 "rpsi:=simplify(rpsi);" }}{PARA 0 "" 0 "" {TEXT -1 44 "Sicherung des fl\374chtigen vereinfachten rpsi:" }}{PARA 0 "" 0 " " {TEXT -1 13 "lprint(rpsi);" }}{PARA 2 "" 0 "" {TEXT -1 247 "-1/2*(-( 2*((a^4+t^2)/a^4)^(1/2)+2)^(1/2)*cos(1/2*(-2*a^4*k*x+a^4*k^2*t-x^2*t)/ (\na^4+t^2))+csgn(t-I*a^2)*(2*((a^4+t^2)/a^4)^(1/2)-2)^(1/2)*sin(1/2*( -2*a^4*k*x+a\n^4*k^2*t-x^2*t)/(a^4+t^2)))*exp(-1/2*a^2*(-x+k*t)^2/(a^4 +t^2))/((a^4+t^2)/a^4)^\n(1/2)\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:='a': k:='k': x:='x': t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "safe:=-1/2*(-(2*((a^4+t^2)/a^4)^(1/2)+2)^(1/2)*cos(1/ 2*(-2*a^4*k*x+a^4*k^2*t-x^2*t)/(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a^4+t^2))+csgn(t-I*a^2)*(2*((a^4+t^2)/a^4)^(1/2)-2)^(1/2)*sin(1/2*(-2 *a^4*k*x+a" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "^4*k^2*t-x^2*t)/(a^4+ t^2)))*exp(-1/2*a^2*(-x+k*t)^2/(a^4+t^2))/((a^4+t^2)/a^4)^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "(1/2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% %safeG,$*&*&,&*&-%%sqrtG6#,&*$-F+6#*&,&*$)%\"aG\"\"%\"\"\"F7*$)%\"tG\" \"#F7F7F7*$F4F7!\"\"F7F;F;F7F7-%$cosG6#,$*&,(*(F4F7%\"kGF7%\"xGF7!\"#* (F4F7)FEF;F7F:F7F7*&)FFF;F7F:F7F=F7F2F=#F7F;F7F=*(-%%csgnG6#,&F:F7*&^# F=F7)F5F;F7F7F7-F+6#,&F.F;F;F=F7-%$sinGF@F7F7F7-%$expG6#,$*&*&FTF7),&F FF=*&FEF7F:F7F7F;F7F7F2F=#F=F;F7F7*$-F+6#F1F7F=F]o" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k:=5: a:=1: t:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(safe,x=-20..20);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "t:='t':" }}{PARA 0 "" 0 "" {TEXT -1 46 "animate(rpsi, x=-20..20,t=-2..2,numpoints=200);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" } }{PARA 0 "" 0 "" {TEXT -1 40 "ENTSTEHUNG und Zerfall des Wellenpaketes " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "t:='t':k:=5:a:=1:fsafe: =evalf(safe);# am schnellsten ca. 2min." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&fsafeG,$*&*&,&*&-%%sqrtG6#,&*$-F+6#,&$\"\"\"\"\"!F3*$)%\"tG\" \"#F3F3F3$F8F4F9F3F3-%$cosG6#,$*&,(%\"xG$!#5F4*&$\"#DF4F3F7F3F3*($F3F4 F3)F@F8F3F7F3!\"\"F3F1FI$\"+++++]FBF3$FIF4*(-%%csgnG6#,&F7F3^#FLF3F3-F +6#,&F.F9$F8F4FIF3-%$sinGF " 0 "" {MPLTEXT 1 0 55 "paket1:=animate(f safe,x=-20..20,t=-2..2,numpoints=200):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "save paket1,`pak1.m`;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#read `pak1.m`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "paket1;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 192 "Wir wollen nun dieses zerflie\337ende Paket auf eine K reisbahn setzen. Dann wird fr\374her oder sp\344ter die Front das Ende einholen und \374berholen und dabei das Paket \"mit sich selbst inter ferieren\". " }}{PARA 0 "" 0 "" {TEXT -1 87 "Die Transformation in das SPS (Schwerpunktsystem) des Paketes erleichtert das Vorgehen:" }} {PARA 0 "" 0 "" {TEXT -1 46 "(x=x+kt nicht in apsi od. rpsi substituie ren!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:='a':k:='k':t:=' t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sapsi:=simplify(eval c(abs(subs(x=x+k*t,psi))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&saps iG*&-%$expG6#,$*&*&)%\"aG\"\"#\"\"\")%\"xGF.F/F/,&*$)F-\"\"%F/F/*$)%\" tGF.F/F/!\"\"#F9F.F/*$)*&F2F/*$F4F/F9#F/F5F/F9" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "srpsi:=evalc(Re(subs(x=x+k*t,rpsi)));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&srpsiG,&*&*(-%%sqrtG6#,&*$-F)6#,&\" \"\"F0*&*$)%\"tG\"\"#F0F0*$)%\"aG\"\"%F0!\"\"F0F0F5F5F0F0-%$expG6#,&*& *$),&%\"xGF0*&%\"kGF0F4F0F0F5F0F0*&)F8F5F0F/F0F:#F:F5*&*&,&*(FGF0FEF0F CF0F0**#F0F5F0FGF0)FEF5F0F4F0F0F0F4F0F0*&F7F0F/F0F:F0F0-%$cosG6#,&*&FK F0*&FGF0F/F0F:F0*&*(FNF0FAF0F4F0F0*&F7F0F/F0F:F0F0F0*$-F)6#F/F0F:FN*&* ,FNF0-%%csgnG6#,&F4F0*&^#F:F0FGF0F0F0-F)6#,&F,F5F5F:F0F;F0-%$sinGFSF0F 0*$-F)6#F/F0F:F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Die vereinfac hte Version:" }}{PARA 0 "" 0 "" {TEXT -1 14 "lprint(srpsi);" }}{PARA 2 "" 0 "" {TEXT -1 285 "-1/2*(-(2*(1+t^2/a^4)^(1/2)+2)^(1/2)*cos((-a^4 *k*x-1/2*a^4*k^2*t-1/2*(x+k*t)^2*\nt)/(a^4+t^2))+1/2*csgn(t-I*a^2)*abs (2*(1+t^2/a^4)^(1/2)-2)^(1/2)*(1+signum(2*(1\n+t^2/a^4)^(1/2)-2))*sin( (-a^4*k*x-1/2*a^4*k^2*t-1/2*(x+k*t)^2*t)/(a^4+t^2)))*\nexp(-1/2*a^2*x^ 2/(a^4+t^2))/(1+t^2/a^4)^(1/2)\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "safe2:=-1/2*(-(2*(1+t^2/a^4)^(1/2)+2)^(1/2)*cos((-a^4 *k*x-1/2*a^4*k^2*t-1/2*(x+k*t)^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "t)/(a^4+t^2))+1/2*csgn(t-I*a^2)*abs(2*(1+t^2/a^4)^(1/2)-2)^(1/2)*( 1+signum(2*(1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "+t^2/a^4)^(1/2)-2) )*sin((-a^4*k*x-1/2*a^4*k^2*t-1/2*(x+k*t)^2*t)/(a^4+t^2)))*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exp(-1/2*a^2*x^2/(a^4+t^2))/(1+t^2/a^4)^( 1/2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&safe2G,$*&*&,&*&-%%sqrtG6# ,&*$-F+6#,&\"\"\"F2*&*$)%\"tG\"\"#F2F2*$)%\"aG\"\"%F2!\"\"F2F2F7F7F2F2 -%$cosG6#*&,(*(F9F2%\"kGF2%\"xGF2F<*&#F2F7F2*(F9F2)FCF7F2F6F2F2F<*&#F2 F7F2*&),&FDF2*&FCF2F6F2F2F7F2F6F2F2F " 0 "" {MPLTEXT 1 0 24 " #srpsi:=simplify(srpsi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "srpsi:=safe2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Entstehung und Zerfall des Wellenpaketes im SPS. Be i den Signum-Entscheidungen wird viel Zeit verbraucht" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "t:='t':a:=1: k:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "animate(sapsi,x=-20..20,t=-10..10,n umpoints=100,color=red);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "paket2:=animate(s rpsi,x=-20..20,t=-10..10,numpoints=100,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "paket2;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "save paket2,`pak2.m`;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# read `pak2.m`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#paket2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "Nach diesen Kontrollplots, die noch einmal sch\366n demonstrieren, wie die Anteile mit kuzer Wellenl \344nge die mit langer \374berholen, k\366nnen wir die \334berlappung \+ behandeln. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Interferenz durch \334berlappung beim Zerfliessen im SPS" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Substitution x=x+k *t in psi selbst (nicht in Re und abs)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "k:='k':a:='a':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "psi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$* &,(*$)%\"xG\"\"#\"\"\"F.**^#!\"#F.)%\"aGF-F.%\"kGF.F,F.F.**^#F.F.F2F.) F4F-F.%\"tGF.F.F.*&F2F.,&F.F.*&*&F6F.F8F.F.*$F2F.!\"\"F.F.F>#F>F-F.*$- %%sqrtG6#F:F.F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "spsi:=si mplify(subs(x=x+k*t,psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%spsiG *&-%$expG6#,$*&,,*$)%\"xG\"\"#\"\"\"!\"\"**F/F0%\"kGF0%\"tGF0F.F0F1*&) F3F/F0)F4F/F0F1**^#F/F0)%\"aGF/F0F3F0F.F0F0**^#F0F0F:F0F6F0F4F0F0F0,&* $F:F0F0*&F=F0F4F0F0F1#F0F/F0*$-%%sqrtG6#*&F>F0*$F:F0F1F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "n:='n':x0:='x0':a:='a':k:='k': t:=' t':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "\"Partialwelle\" zur n-ten \334berlappung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "part:=su bs(x=x+2*n*x0,spsi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%partG*&-%$e xpG6#,$*&,,*$),&%\"xG\"\"\"*(\"\"#F0%\"nGF0%#x0GF0F0F2F0!\"\"**F2F0%\" kGF0%\"tGF0F.F0F5*&)F7F2F0)F8F2F0F5**^#F2F0)%\"aGF2F0F7F0F.F0F0**^#F0F 0F>F0F:F0F8F0F0F0,&*$F>F0F0*&FAF0F8F0F0F5#F0F2F0*$-%%sqrtG6#*&FBF0*$F> F0F5F0F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Blende zum Ausschneid en von -x0 bis x0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x0:='x0 ':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "blende:=Heaviside(x+x 0)-Heaviside(x-x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'blendeG,&-%* HeavisideG6#,&%\"xG\"\"\"%#x0GF+F+-F'6#,&F*F+F,!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x0:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(blende,x=-10..10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Kontrolle der \"Pa rtialwellenbetr\344ge\" (manchmal taucht nach simplify csgn auf)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sapsi;simplif y(evalc(abs(part)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$* &*&)%\"aG\"\"#\"\"\")%\"xGF,F-F-,&*$)F+\"\"%F-F-*$)%\"tGF,F-F-!\"\"#F7 F,F-*$)*&F0F-*$F2F-F7#F-F3F-F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-% $expG6#,$*&*&)%\"aG\"\"#\"\"\")%\"xGF,F-F-,&*$)F+\"\"%F-F-*$)%\"tGF,F- F-!\"\"#F7F,F-*$)*&F0F-*$F2F-F7#F-F3F-F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "(Die Partialwelle wird leicht verschoben gezeichnet, f \374r n=0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n:=0:a:=2:k:= 2:t:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(\{sapsi+0.0 1,evalc(abs(part))\},x=-10..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 13 "\334berlappung: " }} {PARA 0 "" 0 "" {TEXT -1 250 "Wenn Sie sich ein Wellenpaket auf einen \+ Streifen Papier malen (eine Folie w\344re besser), aus diesem Streifen einem Ring machen und den Ring dann platt dr\374cken, sollten Sie etw a folgendes Bild sehen (das nicht aufgewickelte Paket ist mit eingezei chnet)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(\{sapsi,blen de*sapsi,seq(blende*evalc(abs(part)),n=-2..2)\},x=-10..10,color=red); " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "f\374r Abbild ung p1rydb:=\":save p1rydb, `abbryd.m`:" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 31 "Interferenz der 'Partialwellen'" } }{PARA 0 "" 0 "" {TEXT -1 103 "(F\374r die Summenbildung zur Interfere nz kann die Vereinfachung - z.B. von rpsi - nicht verwendet werden)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n:='n':t:='t':a:='a':k:='k ':x0:='x0':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sumpsi:=blen de*sum(part,n=-2..2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'sumpsiG*&, &-%*HeavisideG6#,&%\"xG\"\"\"%#x0GF,F,-F(6#,&F+F,F-!\"\"F1F,,,*&-%$exp G6#,$*&,,*$),&F+F,*&\"\"%F,F-F,F1\"\"#F,F1**F?F,%\"kGF,%\"tGF,FF,F-F, F,F?F,F1**F?F,FAF,FBF,FcqF,F1FCF1**FGF,FHF,FAF,FcqF,F,FJF,F,FLF1FOF,*$ -FR6#FTF,F1F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "sumpsi;" }}{PARA 0 "" 0 "" {TEXT -1 150 "Vermutung: Mit dem Umfang u=2x0 und der Umlaufsdauer T=2x0/k s ollten Wiederholungen zu Vielfachen von T auftreten, wenn man x0=n*l w \344hlt, mit l=2Pi/k." }}{PARA 0 "" 0 "" {TEXT -1 130 "Das Aufsummiere n sollte nach der 2..3-sigma-Regel geschehen, mit dem zeitabh\344ngige n sigma=a*sqrt(1+t^2/a^4) also n=(2..3)sigma/x0." }}{PARA 0 "" 0 "" {TEXT -1 34 "Ein Kontrollplot vor der Animation" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "t:=0:a:=2:k:=2:x0:=2*Pi/k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(evalc(abs(sumpsi)),x=-10..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 82 "Oh ne Termvereinfachung dauert die Berechnung der beiden n\344chsten Plot s recht lange" }}{PARA 0 "" 0 "" {TEXT -1 7 "t:='t':" }}{PARA 0 "" 0 " " {TEXT -1 52 "animate(evalc(abs(sumpsi)),x=-6..6,t=0..5,frames=5);" } }{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 7 "t:='t': " }}{PARA 0 "" 0 "" {TEXT -1 44 "plot3d(evalc(abs(sumpsi)),x=-6..6,t=- 1..20);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "Deshalb lohnt sich die folgende V ereinfachung sehr (auch wenn man bei simplify warten mu\337). Allerdin gs l\344\337t sich dann der Laufbereich der Summation nicht mehr flexi bel programmieren." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:='a ':t:='t':n:='n':k:='k':x0:='x0':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "evalc(abs(sumpsi)):" }}{PARA 0 "" 0 "" {TEXT -1 12 "simplify(\"): " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 11 "asu mpsi:=\":" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 16 "lprint(asumpsi);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Hier ist das Ergebnis, das Sie mit Return \"abarbeiten\" m\374s sen, wenn Sie weiterkommen wollen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "safe3:= 1/(a^4+t^2)^(1/4)*((Heaviside(x+x0)-Heaviside(x-x0))^2*(2*exp(-a^2*(x^ 2+6*x*x0+10*x0^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 ")/(a^4+t^2))*co s(1/2*(2*a^4*k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8*t*x*x0+16*t*x0^2+2*k*t^2 *" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "x+8*k*t^2*x0+k^2*t^3)/(a^4+t^2 ))*cos(1/2*(2*a^4*k*x+4*a^4*k*x0+a^4*k^2*t+x^2*t+4*t*x*" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "x0+4*t*x0^2+2*k*t^2*x+4*k*t^2*x0+k^2*t^3)/(a^4 +t^2))+2*exp(-a^2*(x^2-6*x*x0+10*x0^2)/(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "a^4+t^2))*sin(1/2*(-2*a^4*k*x+4*a^4*k*x0-a^4*k^2*t-x^ 2*t+4*t*x*x0-4*t*x0^2-2*k*t^2*x+4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*k*t^2*x0-k^2*t^3)/(a^4+t^2))*sin(1/2*(-2*a^4*k*x+8*a^4*k*x0-a^4*k ^2*t-x^2*t+8*t*x*x0-" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "16*t*x0^2-2 *k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2))-2*exp(-a^2*(x^2+4*x0^2)/(a^4+t ^2))*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sin(1/2*(2*a^4*k*x+4*a^4*k *x0+a^4*k^2*t+x^2*t+4*t*x*x0+4*t*x0^2+2*k*t^2*x+4*k*t^2*x0+k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*t^3)/(a^4+t^2))*sin(1/2*(-2*a^4*k*x+4* a^4*k*x0-a^4*k^2*t-x^2*t+4*t*x*x0-4*t*x0^2-2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "k*t^2*x+4*k*t^2*x0-k^2*t^3)/(a^4+t^2))+2*exp(-a^2*(x^ 2+2*x*x0+10*x0^2)/(a^4+t^2))*cos(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "1/2*(2*a^4*k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8*t*x*x0+16*t*x0^2+2*k*t ^2*x+8*k*t^2*x0+k^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^3)/(a^4+t ^2))*cos(1/2*(-2*a^4*k*x+4*a^4*k*x0-a^4*k^2*t-x^2*t+4*t*x*x0-4*t*x0^2- 2*k*t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*x+4*k*t^2*x0-k^2*t^3)/( a^4+t^2))+2*exp(-a^2*(x^2+4*x*x0+8*x0^2)/(a^4+t^2))*sin(1/2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "(2*a^4*k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8*t *x*x0+16*t*x0^2+2*k*t^2*x+8*k*t^2*x0+k^2*t^3)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "/(a^4+t^2))*sin(1/2*(2*a^4*k*x+a^4*k^2*t+2*k*t^2*x+k^ 2*t^3+x^2*t)/(a^4+t^2))-2*exp(-a^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "2*(x^2-2*x*x0+10*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4*k*x+4*a^4*k*x0+a^ 4*k^2*t+x^2*t+4*t*x*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "x0+4*t*x0^2 +2*k*t^2*x+4*k*t^2*x0+k^2*t^3)/(a^4+t^2))*sin(1/2*(-2*a^4*k*x+8*a^4*k* x0-a^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "4*k^2*t-x^2*t+8*t*x*x0-16* t*x0^2-2*k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2))+2*exp(-a^2*(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "x^2-2*x*x0+10*x0^2)/(a^4+t^2))*cos(1/2*(2 *a^4*k*x+4*a^4*k*x0+a^4*k^2*t+x^2*t+4*t*x*x0+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "4*t*x0^2+2*k*t^2*x+4*k*t^2*x0+k^2*t^3)/(a^4+t^2))*cos (1/2*(-2*a^4*k*x+8*a^4*k*x0-a^4*k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*t-x^2*t+8*t*x*x0-16*t*x0^2-2*k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t ^2))+2*exp(-a^2*(x^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "-2*x*x0+2*x 0^2)/(a^4+t^2))*cos(1/2*(2*a^4*k*x+a^4*k^2*t+2*k*t^2*x+k^2*t^3+x^2*t)/ (a^4+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^2))*cos(1/2*(-2*a^4*k*x+ 4*a^4*k*x0-a^4*k^2*t-x^2*t+4*t*x*x0-4*t*x0^2-2*k*t^2*x+4*k*t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*x0-k^2*t^3)/(a^4+t^2))-2*exp(-a^2*(x^2 -4*x*x0+8*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4*k*x" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "+a^4*k^2*t+2*k*t^2*x+k^2*t^3+x^2*t)/(a^4+t^2))*sin(1/ 2*(-2*a^4*k*x+8*a^4*k*x0-a^4*k^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t-x^2*t+8*t*x*x0-16*t*x0^2-2*k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2) )+2*exp(-a^2*(x^2+2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "x*x0+2*x0^2 )/(a^4+t^2))*cos(1/2*(2*a^4*k*x+4*a^4*k*x0+a^4*k^2*t+x^2*t+4*t*x*x0+4* t*x0^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "2+2*k*t^2*x+4*k*t^2*x0+k^2 *t^3)/(a^4+t^2))*cos(1/2*(2*a^4*k*x+a^4*k^2*t+2*k*t^2*x+k^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^3+x^2*t)/(a^4+t^2))+2*exp(-a^2*(x^2+4*x 0^2)/(a^4+t^2))*cos(1/2*(2*a^4*k*x+4*a^4*k*x0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "+a^4*k^2*t+x^2*t+4*t*x*x0+4*t*x0^2+2*k*t^2*x+4*k*t^2* x0+k^2*t^3)/(a^4+t^2))*cos(1/2*(-" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "2*a^4*k*x+4*a^4*k*x0-a^4*k^2*t-x^2*t+4*t*x*x0-4*t*x0^2-2*k*t^2*x+4 *k*t^2*x0-k^2*t^3)/(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "a^4+t^2))+2 *exp(-a^2*(x^2+2*x*x0+2*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4*k*x+4*a^4*k*x0 +a^4*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "k^2*t+x^2*t+4*t*x*x0+4*t*x 0^2+2*k*t^2*x+4*k*t^2*x0+k^2*t^3)/(a^4+t^2))*sin(1/2*(2*a^4*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "k*x+a^4*k^2*t+2*k*t^2*x+k^2*t^3+x^2*t)/(a ^4+t^2))+2*exp(-a^2*(x^2+4*x*x0+8*x0^2)/(a^4+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^2))*cos(1/2*(2*a^4*k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8 *t*x*x0+16*t*x0^2+2*k*t^2*x+8*k*t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*x0+k^2*t^3)/(a^4+t^2))*cos(1/2*(2*a^4*k*x+a^4*k^2*t+2*k*t^2*x+k ^2*t^3+x^2*t)/(a^4+t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2))+2*exp( -a^2*(x^2-6*x*x0+10*x0^2)/(a^4+t^2))*cos(1/2*(-2*a^4*k*x+4*a^4*k*x0-a^ 4*k^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*t-x^2*t+4*t*x*x0-4*t*x0^2 -2*k*t^2*x+4*k*t^2*x0-k^2*t^3)/(a^4+t^2))*cos(1/2*(-2*a^4*k*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "x+8*a^4*k*x0-a^4*k^2*t-x^2*t+8*t*x*x0-16* t*x0^2-2*k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "))+2*exp(-a^2*(x^2+16*x0^2)/(a^4+t^2))*cos(1/2*(2*a^4 *k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*t*x*x0+16*t*x0^2+2*k*t^2*x+8*k*t^2*x0+k^2*t^3)/(a^4+t^2))*cos(1/2 *(-2*a^4*k*x+8*a^4*k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*x0-a^4*k^2 *t-x^2*t+8*t*x*x0-16*t*x0^2-2*k*t^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2))+2 *exp(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "-a^2*(x^2-4*x*x0+8*x0^2)/( a^4+t^2))*cos(1/2*(2*a^4*k*x+a^4*k^2*t+2*k*t^2*x+k^2*t^3+x^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*t)/(a^4+t^2))*cos(1/2*(-2*a^4*k*x+8*a^4* k*x0-a^4*k^2*t-x^2*t+8*t*x*x0-16*t*x0^2-2*k*t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*x+8*k*t^2*x0-k^2*t^3)/(a^4+t^2))-2*exp(-a^2*(x^2+1 6*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*k*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8*t*x*x0+16*t*x0^2+2*k*t^2*x+8*k*t ^2*x0+k^2*t^3)/(a^4+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^2))*sin(1 /2*(-2*a^4*k*x+8*a^4*k*x0-a^4*k^2*t-x^2*t+8*t*x*x0-16*t*x0^2-2*k*t^2*x +8*k*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^2*x0-k^2*t^3)/(a^4+t^2)) +2*exp(-a^2*(x^2+6*x*x0+10*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4*k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*x+8*a^4*k*x0+a^4*k^2*t+x^2*t+8*t*x*x0+16 *t*x0^2+2*k*t^2*x+8*k*t^2*x0+k^2*t^3)/(a^4+t^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "2))*sin(1/2*(2*a^4*k*x+4*a^4*k*x0+a^4*k^2*t+x^2*t+4*t *x*x0+4*t*x0^2+2*k*t^2*x+4*k*t^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "x0+k^2*t^3)/(a^4+t^2))-2*exp(-a^2*(x^2+2*x*x0+10*x0^2)/(a^4+t^2))* sin(1/2*(2*a^4*k*x+8" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "*a^4*k*x0+a ^4*k^2*t+x^2*t+8*t*x*x0+16*t*x0^2+2*k*t^2*x+8*k*t^2*x0+k^2*t^3)/(a^4+t ^2))*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sin(1/2*(-2*a^4*k*x+4*a^4* k*x0-a^4*k^2*t-x^2*t+4*t*x*x0-4*t*x0^2-2*k*t^2*x+4*k*t^2*x0-" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "k^2*t^3)/(a^4+t^2))-2*exp(-a^2*(x^2-2*x*x 0+2*x0^2)/(a^4+t^2))*sin(1/2*(2*a^4*k*x+a^4*k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "^2*t+2*k*t^2*x+k^2*t^3+x^2*t)/(a^4+t^2))*sin(1/2*(-2* a^4*k*x+4*a^4*k*x0-a^4*k^2*t-x^2*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t+4*t*x*x0-4*t*x0^2-2*k*t^2*x+4*k*t^2*x0-k^2*t^3)/(a^4+t^2))+exp(- a^2*(4*x0-x)^2/(a^4+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t^2))+exp(- a^2*(x+4*x0)^2/(a^4+t^2))+exp(-a^2*(x+2*x0)^2/(a^4+t^2))+exp(-a^2*(2*x 0-x)^" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "2/(a^4+t^2))+exp(-a^2*x^2/ (a^4+t^2)))*a^2)^(1/2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&safe3G*& *$-%%sqrtG6#*(),&-%*HeavisideG6#,&%\"xG\"\"\"%#x0GF2F2-F.6#,&F1F2F3!\" \"F7\"\"#F2,T-%$expG6#,$*&*&)%\"aGF8F2)F1F8F2F2,&*$)FA\"\"%F2F2*$)%\"t GF8F2F2F7F7F2**F8F2-F;6#,$*&*&F@F2,(*$FBF2F2*(\"\"'F2F1F2F3F2F2*&\"#5F 2)F3F8F2F2F2F2FCF7F7F2-%$sinG6#,$*&,4*(FEF2%\"kGF2F1F2F8**\"\")F2FEF2F hnF2F3F2F2*(FEF2)FhnF8F2FIF2F2*&FBF2FIF2F2**FjnF2FIF2F1F2F3F2F2*(\"#;F 2FIF2FVF2F2**F8F2FhnF2FHF2F1F2F2**FjnF2FhnF2FHF2F3F2F2*&F\\oF2)FI\"\"$ F2F2F2FCF7#F2F8F2-FX6#,$*&,4FgnF8**FFF2FEF2FhnF2F3F2F2F[oF2F]oF2**FFF2 FIF2F1F2F3F2F2*(FFF2FIF2FVF2F2**F8F2FhnF2FHF2F1F2F2**FFF2FhnF2FHF2F3F2 F2FcoF2F2FCF7FfoF2F2**F8F2-F;6#,$*&*&F@F2,(FQF2*(FFF2F1F2F3F2F7*&FjnF2 FVF2F2F2F2FCF7F7F2-%$cosG6#,$*&,,FgnF8F[oF2**F8F2FhnF2FHF2F1F2F2FcoF2F ]oF2F2FCF7FfoF2-F[q6#,$*&,4Fgn!\"#**FjnF2FEF2FhnF2F3F2F2F[oF7F]oF7**Fj nF2FIF2F1F2F3F2F2*(F`oF2FIF2FVF2F7**F8F2FhnF2FHF2F1F2F7**FjnF2FhnF2FHF 2F3F2F2FcoF7F2FCF7FfoF2F2**F8F2-F;6#,$*&*&F@F2,&FQF2*&F`oF2FVF2F2F2F2F CF7F7F2-F[qFYF2FaqF2F2**F8F2-F;6#,$*&*&F@F2,(FQF2*(F8F2F1F2F3F2F2*&F8F 2FVF2F2F2F2FCF7F7F2FgoF2-FXF\\qF2F2**F8F2FbpF2F^sF2-FXFbqF2F7**F8F2-F; 6#,$*&*&F@F2,(FQF2*(F8F2F1F2F3F2F7*&F8F2FVF2F2F2F2FCF7F7F2F^sF2-FX6#,$ *&,4FgnFfq**FFF2FEF2FhnF2F3F2F2F[oF7F]oF7**FFF2FIF2F1F2F3F2F2*(FFF2FIF 2FVF2F7**F8F2FhnF2FHF2F1F2F7**FFF2FhnF2FHF2F3F2F2FcoF7F2FCF7FfoF2F7**F 8F2-F;6#,$*&*&F@F2,(FQF2*(FFF2F1F2F3F2F2*&FjnF2FVF2F2F2F2FCF7F7F2FWF2F ^sF2F2**F8F2FfrF2-F[qFhoF2FjpF2F2**F8F2-F;6#,$*&*&F@F2,&FQF2*&FFF2FVF2 F2F2F2FCF7F7F2F^uF2-F[qF[tF2F2**F8F2-F;6#,$*&*&F@F2,(FQF2*(F8F2F1F2F3F 2F2*&FUF2FVF2F2F2F2FCF7F7F2FdrF2FguF2F2**F8F2F`uF2FgoF2FjsF2F7**F8F2F] rF2FWF2F`sF2F7**F8F2-F;6#,$*&*&F@F2,(FQF2*(FSF2F1F2F3F2F7*&FUF2FVF2F2F 2F2FCF7F7F2FguF2FaqF2F2**F8F2FetF2FdrF2FjpF2F2**F8F2-F;6#,$*&*&F@F2,(F QF2*(F8F2F1F2F3F2F7*&FUF2FVF2F2F2F2FCF7F7F2FgoF2F`sF2F7**F8F2FKF2FdrF2 F^uF2F2**F8F2F^wF2F^uF2FaqF2F2**F8F2FdvF2FjsF2F`sF2F2-F;6#,$*&*&F@F2), &F3FFF1F7F8F2F2FCF7F7F2-F;6#,$*&*&F@F2),&F1F2*&FFF2F3F2F2F8F2F2FCF7F7F 2-F;6#,$*&*&F@F2),&F1F2*&F8F2F3F2F2F8F2F2FCF7F7F2-F;6#,$*&*&F@F2),&F3F 8F1F7F8F2F2FCF7F7F2**F8F2FiuF2FWF2FjsF2F7**F8F2FbsF2FjpF2FguF2F2F2F@F2 F2F2*$)FC#F2FFF2F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "asumpsi:=safe3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "Man bekommt \"station\344re\" Zust\344nde f\374r x0=n*Pi/k, wo bei n auch \"ein geeigneter Bruch\" sein darf. K kann man zweckm\344 \337iger Weise 1/a w\344hlen, weil Impuls und Ortsunsch\344rfe umgekeh rt proportional sind, ist dann die Ortsunsch\344rfe gerade zur \"volle n Impulsunsch\344rfe\" gew\344hlt." }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Drei Para meters\344tze zur Auswahl" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#a:=2;k:=1/2;x0:=3*a; T:=2*x0/k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#a:=2;k:=1/2;x0:=1.1*Pi/k;T:=2*x0/k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=5;k:=1/a;x0:=2*Pi/k;T:=2*x0/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x0G,$%#PiG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$%#PiG\"$+ \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate(asumpsi,x=-x0-1..x0+1,t=0.. 2*T,frames=20,color=red);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "F\374r Abbildungen" }}{PARA 0 "" 0 "" {TEXT -1 11 "t:=14/20*T: " }}{PARA 0 "" 0 "" {TEXT -1 38 "plot(asumpsi,x=-x0-1..x0+1,color=red) ;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 10 "p2 ryd:=\": " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 222 "Das Paket zerflie\337t also, aber es bildet sich durch Interfe renz wieder. Zun\344chst an drei Stellen (Anfang=Ende), dann an zwei S tellen, zwischenzeitlich an vielen Stellen und dann wieder \"in der Mi tte\", d.h. im Schwerpunkt ." }}{PARA 0 "" 0 "" {TEXT -1 50 "So kann m an sich die Sache auch in Ruhe anschauen:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "plot3d(asumpsi,x=-x0-1..x0+1,t=0..4*T,style=wirefra me,axes=boxed);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "Wenn man das Amplitundenquadrat in Polarkoordinaten darstellt, wird es ans chaulicher, wie das Elektron in seinem SPS vergeht und entsteht. F\374 r einen ersten \334berblick k\366nnen Sie wieder den abgespeicherten P lot verwenden, also bei read `pak3.m` weitermachen. Aber dann werden S ie sicher die Parameter \344ndern wollen." }}{PARA 0 "" 0 "" {TEXT -1 38 "Elektronenpudding in Polarkoordinaten:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "a:=5;k:=1/a;x0:=2*Pi/k;T:=2*x0/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"k G#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G,$%#PiG\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$%#PiG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "paket3:=animate([asumpsi,Pi*x/x0,x=-x0..x0],t=0..4*T ,frames=80,color=red,coords=polar,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "paket3;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "save paket3,`pak3.m`;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#read `pak3.m`:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#paket3;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "f \374r Abbildungen" }}{PARA 0 "" 0 "" {TEXT -1 9 "t:=4.1*T:" }}{PARA 0 "" 0 "" {TEXT -1 77 "plot([asumpsi,Pi*x/x0,x=-x0..x0],color=red,coords =polar,scaling=constrained);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 9 "p9ryd:=\":" }}{PARA 2 "" 0 "" {TEXT -1 1 " \n" }}{PARA 0 "" 0 "" {TEXT -1 72 "save p1ryd,p2ryd,p3ryd,p4ryd, p5ryd ,p6ryd,p7ryd,p8ryd,p9ryd,`abb1ryd.m`:" }}{PARA 0 "" 0 "" {TEXT -1 71 " k\374rzer: save p.(1..9).ryd, filename; (nur *Namens*listen, keine arr ays)" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 137 "Nun transformieren wir zur\374ck auf das Laborsystem (+- ist nur f\374r den Umlaufsinn ma\337gebend, frame s kann durchaus auf 200 gesetzt werden)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 "Man beachte im Folgenden die vers chiedenen Charakteristiken und die h\366chst realistische Abstrahlung, d.h. den Verlust der vom Schwerpunkt zu weit abliegenden Partialwelle n mit \"zu hohen\" und \"zu niedrigen\" Frequenzen." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Ggf. zuerst wieder zu r ead`pak4.m`" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=5;k:=1/a;x0:=2*Pi/k;T:=2*x0/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"kG#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G,$%#PiG\"#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$%#PiG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 118 "paket4:=animate([asumpsi,Pi*x/x0+Pi*k*t/x0,x=-x0.. x0],t=0..4*T,frames=200,color=red,coords=polar,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "paket4;" }}{PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "save paket4,`paket4.m`;" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#read ` pak4.m`:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#paket4;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "f\374r Abbildungen" }}{PARA 0 "" 0 "" {TEXT -1 9 "t:=4.1*T:" }}{PARA 0 "" 0 "" {TEXT -1 87 "plot([asump si,Pi*x/x0+Pi*k*t/x0,x=-x0..x0],color=red,coords=polar,scaling=constra ined);" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 10 "p13ryd:=\":" }}{PARA 0 "" 0 "" {TEXT -1 46 "save p10ryd,p11ryd,p12 ryd,p13ryd, `abb2ryd.m`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Aber das ist alles noch nicht ansc haulich genug ... ?" }}{PARA 0 "" 0 "" {TEXT -1 168 "Wir tragen also d as Absolutquadrat der \"Wellenfunktion eines Elektrons auf einer Kreis bahn\" \374ber der Ebene der Kreisbahn auf. Das geht am besten in Zyli nderkoordinaten." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot3d([r,Pi*x/x0+Pi*k*t/x0,asumpsi],r=1..2,x=-x0..x0,coords=cy lindrical,scaling=constrained,grid=[10,60]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Wir wollen das Paket wieder laufen lassen , aber animate funktioniert nicht mit coords=cylindrical" }}{PARA 0 " " 0 "" {TEXT -1 8 "#t:='t':" }}{PARA 0 "" 0 "" {TEXT -1 116 "#animate3 d([r,Pi*x/x0+Pi*k*t/x0,asumpsi],r=1..2,x=-x0..x0,t=0..4*T,frames=2,coo rds=cylindrical,scaling=constrained);" }}{PARA 0 "" 0 "" {TEXT -1 131 "Also m\374ssen wir uns eine Plotsequence selbst schreiben und bauen b ei dieser Gelegenheit noch eine k\374nstliche radiale Unsch\344rfe ein ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Mit \+ read`pak5.m` (\374bern\344chste Input-region) kommen Sie schneller wei ter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pls:='pls':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t:='t':N:=80:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "pls :=seq(plot3d([r,Pi*x/x0+Pi*k*t/x0,exp(-2*(r-1.5)^2)*asumpsi],r=1..3,x= -x0..x0,coords=cylindrical,grid=[10,30])," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t=seq(i*4*T/N,i=0..N)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "paket5:=display([pls],insequence=true,orientation=[40 ,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "paket5;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "save paket5,`paket5.m`;" }} {PARA 0 "" 0 "" {TEXT -1 61 "Auf die Berechnung der (41) frames m\374s sen Sie aber warten ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " #read `pak5.m`:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#paket5; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Es kommt ein stroboskopische r Effekt herein (je nach grid), aber so benimmt sich ein Elektron in e inem hoch angeregten Zustand, und dazu gibt es auch Messungen!" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "F\374r Abbildungen" }}{PARA 0 "" 0 "" {TEXT -1 20 "for i from 0 to 4 do" }}{PARA 0 "" 0 "" {TEXT -1 9 "t:=i*T/2:" }}{PARA 0 "" 0 "" {TEXT -1 118 "zustand[i]:=plot3d([r,Pi*x/x0+Pi*k*t/x0,exp(-2*(r-1.5)^2 )*asumpsi],r=0.5..3,x=-x0..x0,coords=cylindrical,grid=[20,30]," }} {PARA 0 "" 0 "" {TEXT -1 46 "color=black,style=hidden,orientation=[45, 27]):" }}{PARA 0 "" 0 "" {TEXT -1 3 "od:" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 73 "save zustand[0],zustand[1],zusta nd[2],zustand[3],zustand[4], `abb3ryd.m`:" }}{PARA 2 "" 0 "" {TEXT -1 33 "Error, save can only save names\n\n" }}{PARA 0 "" 0 "" {TEXT -1 61 "es geht mit save nur name.(bereich), filename (keine arrays)!" }} {PARA 0 "" 0 "" {TEXT -1 21 "for i from 0 to 4 do " }}{PARA 0 "" 0 "" {TEXT -1 19 "z.i:=zustand[i]:od:" }}{PARA 0 "" 0 "" {TEXT -1 33 "save \+ z0,z1,z2,z3,z4, `abb3ryd.m`:" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 4 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }