{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 "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 "R 3 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "c ITP Bonn 1995 \+ filename: schroe.ms" }}{PARA 0 "" 0 "" {TEXT -1 103 "Autor: Komma \+ Datum: 27.1.94" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Die Schr\366dingergleichung ist die Bewegungsgleich ung der nicht relativistischen Quantenphysik. Gibt es Analogien zu den Bewegungsgleichungen der klassischen Physik?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Schr\366dingergleichung f\374r den potentialfreien Fall (V = 0):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "sgl:=I*h*diff(psi(x, t),t)=-h^2/(2*m)*diff(psi(x,t),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$sglG/*(^#\"\"\"F(%\"hGF(-%%diffG6$-%$psiG6$%\"xG%\"tGF1F(,$*&*&)F )\"\"#F(-F+6$F--%\"$G6$F0F6F(F(%\"mG!\"\"#F=F6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "Eine \+ m\366glichst allgemein formulierte L\366sung der SGL besteht aus einer reellwertigen Amplitude (Funktion von Ort und Zeit) und der Wirkungsf unktion des Systems als Phase (Notation: h statt h-quer):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=A(x,t)*exp(I/h*S(x,t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG*&-%\"AG6$%\"xG%\"tG\"\"\"-%$exp G6#*&*&^#F+F+-%\"SGF(F+F+%\"hG!\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Was passiert, wenn wir diese Wellenfunktion in die SGL ei nsetzen?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "psi:=(x,t)->u; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGR6$%\"xG%\"tG6\"6$%)operato rG%&arrowGF)%\"uGF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "sgl;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*(^#\"\"\"F&%\"hGF&,&*&-%%diffG6$-%\"AG6$%\"xG%\"tGF1F&-%$expG6 #*&*&F%F&-%\"SGF/F&F&F'!\"\"F&F&*&**F%F&F-F&-F+6$F7F1F&F2F&F&F'F9F&F&, $*&*&)F'\"\"#F&,**&-F+6$F--%\"$G6$F0FBF&F2F&F&*&**^#FBF&-F+6$F-F0F&-F+ 6$F7F0F&F2F&F&F'F9F&*&**F%F&F-F&-F+6$F7FGF&F2F&F&F'F9F&*&*(F-F&)FOFBF& F2F&F&*$FAF&F9F9F&F&%\"mGF9#F9FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Umformen" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gl:=sgl/u;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#glG/*&*(^#\"\"\"F)%\"hGF),&*&-%%diffG6$-%\"AG6$% \"xG%\"tGF4F)-%$expG6#*&*&F(F)-%\"SGF2F)F)F*!\"\"F)F)*&**F(F)F0F)-F.6$ F:F4F)F5F)F)F*F " 0 "" {MPLTEXT 1 0 17 "gl:=simplify(g l);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#glG/*&*&^#\"\"\"F),&*&-%%dif fG6$-%\"AG6$%\"xG%\"tGF3F)%\"hGF)F)*(F(F)F/F)-F-6$-%\"SGF1F3F)F)F)F)F/ !\"\",$*&,**&-F-6$F/-%\"$G6$F2\"\"#F))F4FDF)F)**^#FDF)-F-6$F/F2F)-F-6$ F8F2F)F4F)F)**F(F)F/F)-F-6$F8FAF)F4F)F)*&F/F))FJFDF)F:F)*&F/F)%\"mGF)F :#F:FD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gl:=expand(gl);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#glG/,&*&*(^#\"\"\"F*-%%diffG6$-% \"AG6$%\"xG%\"tGF2F*%\"hGF*F*F.!\"\"F*-F,6$-%\"SGF0F2F4,**&*&-F,6$F.-% \"$G6$F1\"\"#F*)F3FAF*F**&F.F*%\"mGF*F4#F4FA*&**^#F4F*-F,6$F.F1F*-F,6$ F7F1F*F3F*F**&F.F*FDF*F4F**&*(^#FEF*-F,6$F7F>F*F3F*F*FDF4F**&*&#F*FAF* )FKFAF*F*FDF4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "W enn diese Gleichung erf\374llt sein soll, so mu\337 sie f\374r den Rea lteil und den Imagin\344rteil erf\374llt sein" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Realteil:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "-evalc(R e(lhs(gl)))=-evalc(Re(rhs(gl)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%diffG6$-%\"SG6$%\"xG%\"tGF+,&*&*&-F%6$-%\"AGF)-%\"$G6$F*\"\"#\"\"\") %\"hGF6F7F7*&F1F7%\"mGF7!\"\"#F7F6*&#F7F6F7*&*$)-F%6$F'F*F6F7F7F;F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "Der Gradient der Wi rkungsfuktion ist der Impuls. Also steht hier die Hamilton-Jacobi-Glei chung, allerdings mit einem Zusatz, der die Dimension eines Potentials hat und mit h^2 geht. Bohm nennt ihn das Quantenpotential (das f\374r h -> 0 verschwindet)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Imagin\344rteil:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "igl:=evalc(Im(lhs(gl)))=evalc(Im(rh s(gl)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iglG/*&*&-%%diffG6$-%\" AG6$%\"xG%\"tGF/\"\"\"%\"hGF0F0F+!\"\",&*&*(-F)6$F+F.F0-F)6$-%\"SGF-F. F0F1F0F0*&F+F0%\"mGF0F2F2*&#F0\"\"#F0*&*&-F)6$F:-%\"$G6$F.F@F0F1F0F0F= F2F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "igl:=igl*A(x,t)/h ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iglG/-%%diffG6$-%\"AG6$%\"xG% \"tGF-*&*&F)\"\"\",&*&*(-F'6$F)F,F0-F'6$-%\"SGF+F,F0%\"hGF0F0*&F)F0%\" mGF0!\"\"F=*&#F0\"\"#F0*&*&-F'6$F8-%\"$G6$F,F@F0F:F0F0F " 0 "" {MPLTEXT 1 0 17 "igl:=expand(igl);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iglG/-%%diffG6$-%\"AG6$%\"xG%\"tGF- ,&*&*&-F'6$F)F,\"\"\"-F'6$-%\"SGF+F,F3F3%\"mG!\"\"F9*&#F3\"\"#F3*&*&F) F3-F'6$F6-%\"$G6$F,F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Das sieht na ch einer Kontinuit\344tsgleichung aus, wenn man n\344mlich die Amplitu de mit der Wurzel einer Teilchendichte identifiziert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subs(A(x,t)=sqrt(rho(x,t)),igl);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$*$-%%sqrtG6#-%$rhoG6$%\"xG %\"tG\"\"\"F/,&*&*&-F%6$F'F.F0-F%6$-%\"SGF-F.F0F0%\"mG!\"\"F;*&#F0\"\" #F0*&*&F(F0-F%6$F8-%\"$G6$F.F>F0F0F:F;F0F;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "eval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&-%% diffG6$-%$rhoG6$%\"xG%\"tGF-\"\"\"*$-%%sqrtG6#F)F.!\"\"#F.\"\"#,&*&*&- F'6$F)F,F.-F'6$-%\"SGF+F,F.F.*&%\"mGF.-F16#F)F.F3#F3F5*&#F.F5F.*&*&-F1 6#F)F.-F'6$F=-%\"$G6$F,F5F.F.F@F3F.F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "%*2*sqrt(rho(x,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%diffG6$-%$rhoG6$%\"xG%\"tGF+,$*&-%%sqrtG6#F'\"\"\",&*&*&-F%6$F'F *F1-F%6$-%\"SGF)F*F1F1*&%\"mGF1-F/6#F'F1!\"\"#F?\"\"#*&#F1FAF1*&*&F.F1 -F%6$F9-%\"$G6$F*FAF1F1F " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%dif fG6$-%$rhoG6$%\"xG%\"tGF+,&*&*&-F%6$F'F*\"\"\"-F%6$-%\"SGF)F*F1F1%\"mG !\"\"F7*&*&F'F1-F%6$F4-%\"$G6$F*\"\"#F1F1F6F7F7" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Und di eser Zusammenhang f\374hrt bekanntlich auf die statistische Interpreta tion der Quantenmechanik. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe. uni-tuebingen.de" }}}}{MARK "1 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }