{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 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 "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 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 }{PSTYLE "Normal" -1 258 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 16 "c ITP Bonn 1995" }} {PARA 258 "" 0 "" {TEXT -1 13 "Autor: Komma " }}{PARA 258 "" 0 "" {TEXT -1 15 "Datum: 31.12.93" }}{PARA 258 "" 0 "" {TEXT -1 78 "Index: \+ H-Orbitale, zugeordnete Legendre-Polynome, verallgem. Laguerre-Polynom e" }}{PARA 258 "" 0 "" {TEXT -1 65 "Thema: Darstellung der Wahrscheinl ichkeitsdichten des H-Elektrons" }}{PARA 258 "" 0 "" {TEXT -1 84 "Syno psis: 3d-Darstellung und Polarplot der Kugelfunktionen, Plot der Radia lfunktion," }}{PARA 258 "" 0 "" {TEXT -1 94 " Wahrschei nlichkeitsdichte als Funktion von r und theta (3d- und contour-Plot). " }}{PARA 258 "" 0 "" {TEXT -1 76 " Demonstration zur \+ Verwendung von Prozeduren und Funktionen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Modifizieret durch O. Marti, Unive rsit\344t Ulm, am 22. 5. 2005, nun kompatibel mit Maple V 9.5" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 10 "H-Orb itals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 94 "With(orthopoly ) macht das package bekannt, in dem die Legendre-Polynome P(l,x) zu fi nden sind." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(plots): #with(orthopoly);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 53 "Man kann P(l,x) aber auch leicht selbst definieren: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "P:=(l,x)- >if l<>0 then 1/(2^l*l!)*diff((x^2-1)^l,x$l)\n else 1 \n fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGj+6$%\"lG%\"x G6\"6$%)operatorG%&arrowGF)@%09$\"\"!*()\"\"#F/!\"\"-%*factorialG6#F/F 4-%%diffG6$),&*$)9%F3\"\"\"F@F@F4F/-%\"$G6$F?F/F@F@F)F)F)6$F0F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "P(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"$\"\"\"F(**F'F(\"\"#!\"\",&*$)F&F*F(F(F(F +F(F&F(F(" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 79 "Dann schreibt man \+ eine kleine Prozedur f\374r die zugeordneten Legendre-Polynome:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Y:=proc(l,m,x);\nif m=0 then P(l,x) \n else (1-x^2)^(m/2)*diff(P(l,x),x$m)\nfi;\nend;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"YGj+6%%\"lG%\"mG%\"xG6\"F*F*@%/9%\"\"!-%\"PG 6$9$9&*&),&\"\"\"F7*$)F3\"\"#F7!\"\",$*&#F7F:F7F-F7F7F7-%%diffG6$F/-% \"$G6$F3F-F7F*F*F*6%F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 5 "Tes t:" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Y(3,0,u);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"uG\"\"$\"\"\"F(**F'F(\"\"#!\" \",&*$)F&F*F(F(F(F+F(F&F(F(" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 218 "F\374r die Plots sollte man x durch cos(theta)^2 ersetzen. Diese Kons truktion ist einfacher, als von vornherein mit theta zu arbeiten (wege n der Ableitungen). Man ben\366tigt aber eine dummy-Variable (ebenfall s wegen diff).\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ply:=dummy->sub s(dummy=cos(theta)^2,Y(l,m,dummy));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$plyGj+6#%&dummyG6\"6$%)operatorG%&arrowGF(-%%subsG6$/9$*$)-%$cosG6 #%&thetaG\"\"#\"\"\"-%\"YG6%%\"lG%\"mGF0F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "l:=2:\nm:=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "ply(poo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&#\"\"$\"\"#\"\"\"*$)-%$cosG6#%&thetaG\"\"%F(F(F(#F(F'!\"\"" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 11 "Normierung\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "l:='l':\nm:='m': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Ny2:=(2*l+1)*(l-m)!/(l+m)!; #Normierung^2*4*Pi" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Ny2G*(,&*&\"\"#\"\"\"%\"lGF)F)F)F)F )-%*factorialG6#,&F*F)%\"mG!\"\"F)-F,6#,&F*F)F/F)F0" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Y(l,m,x);" }}{PARA 258 "" 0 "" {TEXT -1 62 "Ko ntrolle der \"spherical harmonics\" nach Vorgabe von l und m:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "l:=2:\nm:=0:\nY(l,m,x);\nply(o);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"$\"\"\"\"\"#!\"\"%\"xGF'F&#F& F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"$\"\"#\"\"\"*$)-%$cos G6#%&thetaG\"\"%F(F(F(#F(F'!\"\"" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 64 "Nun kann man sich das Quadrat der Kugelfunktion zeigen lassen: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "l:=3:m:=0:\nlatex(ply(o));\n sphereplot(Ny2*ply(o)^2,phi=Pi/2..2*Pi,theta=0..Pi,axes=boxed,scaling= constrained,grid=[15,100],style=PATCHNOGRID);" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\left( \\cos \\left( \\theta \\right) \\right) ^\{6\}+ 3/2\\, \\left( \\left( " }}{PARA 6 "" 1 "" {TEXT -1 63 "\\cos \\left( \\theta \\right) \\right) ^\{4\}-1 \\right) \\left( \\cos" }}{PARA 6 "" 1 "" {TEXT -1 36 " \\left( \\theta \\right) \\right) ^\{2\}" }} {PARA 13 "" 1 "" {GLPLOT3D 496 496 496 {PLOTDATA 3 "6'-%%MESHG6#X.%)an 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2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 23 "Nun die Radialfunktion:" }}{PARA 258 "" 0 "" {TEXT -1 85 "Man b en\366tigt die verallgemeinerten Laguerre-Polynome die im orthopoly-pa ckage stehen." }}{PARA 258 "" 0 "" {TEXT -1 39 "Wer dieses package nic ht hat, schreibt:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "#u=2*r/(a*n)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a:='a': # (Bohrscher Radius =1, wird ggf. weiter unten benoeti gt)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "L:=(j,k,u)->if j<>0 \+ then 1/j!*exp(u)/u^k*diff(u^(j+k)*exp(-u),u$j)\n el se 1 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGj+6%%\"jG%\"kG%\"uG6 \"6$%)operatorG%&arrowGF*@%09$\"\"!**-%*factorialG6#F0!\"\"-%$expG6#9& \"\"\")F:9%F6-%%diffG6$*&)F:,&F0F;F=F;F;-F86#,$F:F6F;-%\"$G6$F:F0F;F;F *F*F*6%F1F1F1" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 19 "Radialfunktion (u):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Ru:=(n,l,u)->u^l*exp(-u/2 )*L(n-l-1,2*l+1,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RuGj+6%%\"nG %\"lG%\"uG6\"6$%)operatorG%&arrowGF**()9&9%\"\"\"-%$expG6#,$*&#F2\"\"# F2F0F2!\"\"F2-%\"LG6%,(9$F2F1F:F2F:,&*&F9F2F1F2F2F2F2F0F2F*F*F*6%\"\"! 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print(p1);\np2:=polarplot([Ny2*ply(o)^2,theta+Pi/2,theta=0..2*Pi],\n \+ title=txt,scaling=constrained);\nprint(p2);\np3:=plot(Nr2* (r*plr(o))^2,r=0..30);\nprint(p3);\np4:=plot3d([r*cos(theta),r*sin(the ta),Nr2*Ny2*(r*plr(o)*ply(o))^2],\n r=0..30,theta=Pi/2..2*Pi ,title=txt,axes=boxed);\nprint(p4);\np5:=plot3d([r*cos(theta),r*sin(th eta),Nr2*Ny2*(r*plr(o)*ply(o))^2], r=0..30,theta=0 ..2*Pi,axes=boxed,orientation=[0,0],\n style=patchcontour,ti tle=txt,scaling=constrained,\n shading=z);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0myhydrogenplotsGj+6%%\"nG%\"lG%\"mG6#%$tx tG6\"F,C->%\"aG\"\"\">8$(((%%nlm=G9$9%9&>%#p1G-%+sphereplotG6)*&%$Ny2G F0)-%$plyG6#%\"oG\"\"#F0/%$phiG;,$*&FF!\"\"%#PiGF0F0,$*&FFF0FMF0F0/%&t hetaG;\"\"!FM/%%axesG%&boxedG/%&titleGF2/%(scalingG%,constrainedG/%%gr idG7$\"#:\"$+\"-%&printG6#F;>%#p2G-%*polarplotG6%7%F?,&FQF0*&FFFLFMF0F 0/FQ;FSFNFWFY-F\\o6#F_o>%#p3G-%%plotG6$*(%$Nr2GF0)%\"rGFFF0)-%$plrGFDF FF0/Fbp;FS\"#I-F\\o6#F[p>%#p4G-%'plot3dG6'7%*&FbpF0-%$cosG6#FQF0*&FbpF 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Mit anames(); erf\344hrt man die Namen der Plots (al lerdings auch alle anderen zugewiesenen Namen der jeweiligen session - - anames(); also m\366glichst zu Beginn der session verwenden)." }}}} {MARK "0 9 0" 92 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }