{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 257 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 24 0 0 0 0 0 0 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 " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 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 "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 36 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "T imes" 1 36 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 "Normal" -1 260 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 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 272 55 "Az 1-dimenzi\363s Scr \366dinger egyenlet numerikus megold\341sai" }{TEXT 269 1 "\000" } {TEXT 271 0 "" }}{PARA 259 "" 0 "" {TEXT 270 13 "Bartha Ferenc" }} {PARA 260 "" 0 "" {TEXT -1 17 "2002. \341prilis 15." }}{PARA 0 "" 0 " " {TEXT -1 32 "A Schr\366dinger-egyenlet: " }{XPPEDIT 18 0 "i* diff(Psi(x,t),t)=H*Psi(x,t):" "6#/*&%\"iG\"\"\"-%%diffG6$-%$PsiG6$%\"x G%\"tGF.F&*&%\"HGF&-F+6$F-F.F&" }{TEXT -1 11 " ahol " }{XPPEDIT 18 0 "H*Psi=-diff(Psi,x$2)+V(x)*Psi;" "6#/*&%\"HG\"\"\"%$PsiGF&,&-%%di ffG6$F'-%\"$G6$%\"xG\"\"#!\"\"*&-%\"VG6#F/F&F'F&F&" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "Stacion\341rius k\366t\366tt \341llapotai ra: " }{XPPEDIT 18 0 "H*phi[n](x)=E[n]*phi[n]" "6#/*&%\"HG\"\"\"- &%$phiG6#%\"nG6#%\"xGF&*&&%\"EG6#F+F&&F)6#F+F&" }{TEXT -1 26 " \+ \351s a hat\341rfelt\351ltel: " }{XPPEDIT 18 0 "phi[n](1)=0,phi[n](-1) =0:" "6$/-&%$phiG6#%\"nG6#\"\"\"\"\"!/-&F&6#F(6#,$F*!\"\"F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}} {SECT 1 {PARA 257 "" 0 "" {TEXT -1 4 "Nume" }{TEXT 264 0 "" }{TEXT -1 5 "rov: " }{TEXT 265 0 "" }{TEXT 266 64 "Numerov integr\341l\341s balr \363l a fordul\363pontig, majd jobbr\363l ugyanoda" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 351 "Numerov := proc(Rm::Vector,Rp::Vector,delta ::evaln,Nx,Mp::evaln,Nn::evaln)\nlocal i,j,n;\nn:=0:j:=1:\nwhile not(( Rp[j]<1) or (j=Nx)) do\n Rp[j+1]:=evalf(U(j)-1.0/Rp[j]):j:=j+1:\nend \+ do:\n if(j=Nx) then j:=ceil(Nx/2) end if:\nfor i from Nx by -1 to j d o\n Rm[i-1]:=evalf(U(i)-1.0/Rm[i]):\n if(Rm[i-1]<0) then n:=n+1 end \+ if:\nend do:\ndelta:=Rp[j-1]-1.0/Rm[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Mp:=j:Nn:=n:end proc:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 9 "TridSolve" }{TEXT -1 2 ": " }{TEXT 261 3 "Az " }{TEXT 259 4 "Av=d" }{TEXT 262 21 " egyenlet megold\341sa, " }{TEXT 260 1 "A" }{TEXT 263 13 " triangul\341ris" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 418 "Tri dSolve := proc(La::Vector,Da::Vector,Ua::Vector,d::Vector,v::Vector,n) \nlocal beta,gam,j:\ngam:= Vector(n):beta:= Da[1]; \nif beta=0 then \+ ERROR(`ez igy nem lesz jo`) end if;\nv[1]:= d[1]/beta:\011\nfor j from 2 to n do\n gam[j]:= Ua[j-1]/beta:beta := Da[j]-La[j]*gam[j];\n if \+ beta=0 then ERROR(`no-no!!`) end if;\n v[j]:=(d[j]-La[j]*v[j-1])/beta :\nend do:\nfor j from n-1 by -1 to 1 do\n v[j]:=v[j]-gam[j+1]*v[j+1] ;\nend do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Potenci\341l megad\341sa" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Most " }{MPLTEXT 1 0 19 "V:=(x) -> (10*x )^2:" }{TEXT -1 6 "-1 " 0 " " {MPLTEXT 1 0 46 "plot([seq([x[i],u[i]],i=1..Nx)],title=`V(x)`);" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "K\366t\366tt \341llapotok keres \351se" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "A saj\341t\341llapotokna k rendre 0,1,2,3.... bels\365 z\351r\363helye van. Most " }{MPLTEXT 1 0 13 "NodesMax:=3:\n" }{TEXT -1 55 "maxim\341lis z\351r\363helyig oldj uk meg a saj\341t\351rt\351k-probl\351m\341t " }{MPLTEXT 1 0 15 "toler :=10^(-7):" }{TEXT -1 12 "pontoss\341ggal" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 139 "T:=(j)->(u[j]-En)*prefact:\nU:=(j)->(2.0+10.0*T(j) )/(1.0-T(j)):\nprefact:=dx^2/12:Rp[1]:=infinity:Rm[Nx]:=infinity:\nnma x:=50:En:=Emin:Nn:=0:\n" }{TEXT -1 21 "A kereset \341llapotban " } {TEXT 267 1 "m" }{TEXT -1 22 " bels\365 z\351r\363hely legyen" } {MPLTEXT 1 0 396 " for m from 0 to NodesMax do\nn:=0:delta:=1:flag:=fa lse:\n while ((abs(delta)>toler) and nm) then En:=En-abs(Emin)/2 \+ end if:\n if (Nn=m) then\n EOld:=ENew:ENew:=En:\n if (flag) \+ then\n En:=(-delta*EOld+ENew*Delta)/(Delta-delta):\n else\n \+ En:=En+abs(En)/10:\n end if:\n Delta:=delta:flag:=true:\n \+ end if:" }{TEXT -1 2 " \n" }{MPLTEXT 1 0 120 " Numerov(Rm,Rp,delta,Nx ,Mp,Nn):\nprint(`L\351p\351s`=n,'delta'=delta,'E'=En,`Fordul\363pont`= Mp,`Nodes`=Nn,`versus`=m);\n end do:\n" }{TEXT -1 67 "Ha megvan a kere sett megold\341s, akkor el\365\341llitjuk a hull\341mf\374ggv\351nyt: " }{MPLTEXT 1 0 202 " f[Mp]:=1.0:\n for j from Mp by -1 to 2 do:f[j-1] :=f[j]/Rp[j-1]:end do:\n for j from Mp+1 to Nx do:f[j]:=f[j-1]/Rm[j]:e nd do:\n for j from 1 to Nx do:phi[m][j]:=f[j]/(1-T(j)):end do:\n E[m] :=En:Node[m]:=Nn:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Grafika: a megtal\341lt saj\341t \341llapotok" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "ux:=max(uma x,1):\nplv:=[seq([x[i],(u[i]-umin)/ux],i=1..Nx)]:\npln:=seq([seq([x[i] ,phi[m][i]],i=1..Nx)],m=0..NodesMax):\nplot([plv,pln]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Evoluci\363 Crank-Nicholson l\351p\351sek kel " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Beoszt\341s a t-tengelyen: " }{MPLTEXT 1 0 10 "dt:=0.005:" }{TEXT -1 14 "L\351p\351sek sz\341ma \+ " }{MPLTEXT 1 0 8 "Nt:=20: " }{TEXT -1 26 "Anim\341ci\363s frame-ek sz \341ma: " }{MPLTEXT 1 0 7 "Nf:=10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Ng:=ceil(Nt/Nf);M:=0:\nLa:=Vector(Nx):Da:=Vector(Nx):Ua:=Vector(Nx ):\nd:=Vector(Nx):v:=Vector(Nx):" }}{PARA 0 "" 0 "" {TEXT -1 15 "Kezd \365felt\351tel: " }{MPLTEXT 1 0 221 "t:=0:\nfor j from 1 to Nx do\n \+ tmp:=0:\n for m from 0 to NodesMax do:tmp:=tmp+phi[m][j]:end do:\n U 0[j]:=tmp:\nend do:\nUT[0]:=seq(U0[j],j=1..Nx):\nPT[0]:=seq(U0[j],j=1. .Nx):\nplot([plv,[seq([x[j],abs(UT[0][j])^2],j=1..Nx)]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Konstansok \351s a tridiagon\341lis m\341 trix el\365k\351szit\351se" }{MPLTEXT 1 0 161 "\na:=I*dt/2/dx^2:b:=I* dt/2:a0:=1+2*a:b0:=1-2*a:\nfor j from 1 to Nx do:La[j]:=-a:Ua[j]:=-a:e nd do:\nLa[Nx]:=0:Da[1]:=1:Da[Nx]:=1:Ua[1]:=0:d[1]:=U0[1]:d[Nx]:=U0[Nx ]:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "\311s mehet a ciklus ...." }{MPLTEXT 1 0 234 "\nfor n from 1 to Nt do:t:=t+dt:\n for j fro m 2 to Nx-1 do\n bj:=b*V(x[j]):Da[j]:=a0+bj:\n d[j]:=a*U0[j+1]+( b0-bj)*U0[j]+a*U0[j-1]:\n end do:\n TridSolve(La,Da,Ua,d,v,Nx):print ('t'=t);\n for j from 2 to Nx-1 do:U0[j]:=v[j]:end do:\n" }{TEXT -1 64 " Aktu\341lis megold\341st alkalmasint menteni kell az anim\341 ci\363hoz:" }{MPLTEXT 1 0 43 " \n if((n mod Ng)=0 or n=Nt) then:M:=M +1:\n" }{TEXT -1 50 " Az 'analitikus' hull\341mf\374ggv\351nyt is \+ legy\341rtjuk" }{MPLTEXT 1 0 226 "\n for l from 1 to Nx do:tmp:=0: \n for m from 0 to NodesMax do\n tmp:=tmp+phi[m][l]*exp(-I*E [m]*t):\n end do:\n pt[l]:=tmp:\n end do:\n PT[M]:=seq(p t[j],j=1..Nx):UT[M]:=seq(U0[j],j=1..Nx):\n end if:\nend do:" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Anim\341ci\363k: " }{TEXT 268 39 "(az indul\341shoz a k\351pre kell kattintani)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "soln:=display(seq(listplot([seq([x[j],abs(UT[k] [j])^2],j=1..Nx)]) ,k=0..M),insequence=true):\nsola:=display(seq( listplot([seq([x[j],abs(PT[k][j])^2],j=1..Nx)]) ,k=0..M),insequen ce=true):\ndisplay([soln,sola],title=`density`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 504 "solnR:=dis play(seq(listplot([seq([x[j],Re(UT[k][j])],j=1..Nx)]) ,k=0..M),in sequence=true):\nsolnI:=display(seq(listplot([seq([x[j],Im(UT[k][j])], j=1..Nx)]) ,k=0..M),insequence=true):\nsolaR:=display(seq(listplo t([seq([x[j],Re(PT[k][j])],j=1..Nx)]) ,k=0..M),insequence=true): \nsolaI:=display(seq(listplot([seq([x[j],Im(PT[k][j])],j=1..Nx)]) \+ ,k=0..M),insequence=true):\ndisplay([solnR,solaR],title=`hull\341mf \374ggv\351ny, val\363s r\351sz`);\ndisplay([solnI,solaI],title=`hull \341mf\374ggv\351ny, k\351pzetes r\351sz`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }