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2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Time s" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 213 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 214 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "## Consider the Yu kawa potential and try to find the ground state. As we saw, the minima l energy is " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Emin:=(beta0,beta)->(1/2/beta^2-4*b eta0/beta/(beta+2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I&beta0G6 \"I%betaGF%F%6$I)operatorGF%I&arrowGF%F%,&*$9%!\"##\"\"\"\"\"#*(9$F/F, !\"\",&F,F/F0F/F-!\"%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# where beta is determined from minimazing of Emin which gives" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "betaeq:=beta->(beta+2)^3-8* beta*beta0*(1+3/2*beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I%betaG6 \"F%6$I)operatorGF%I&arrowGF%F%,&*$,&9$\"\"\"\"\"#F-\"\"$F-*(F,F-I&bet a0GF%F-,&F-F-F,#F/F.F-!\")F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# for example consider beta0=10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "res:=[evalf(solve(subs(beta0=10,betaeq(beta)),beta)) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%^$$\"+jz#f9\"!\"($!\"#!#5^$$!+=y nMpF)$!+;;5kM!#<^$$\"+=Yr15F)$\"+;;5kMF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "[Emin(10,res[1]),Emin(10,res[2]),Emin(10,res[3])];" } }{PARA 11 "" 1 "" {XPPMATH 20 "7%^$$\"+'4Z)R7!#9$\"\"\"!#D^$$\"+*>8I[$ !\")$\"#m!#;^$$!+OW^qSF-$!#cF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "# the third one is negative indicating a bound state with E=- 40.70514436 (at least from variationl method)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# Let's compare with numerical solution of the S chroedinger equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Eq1 :=(beta0,E)->-1/2*(diff(psi(u),u$2)+2/u*diff(psi(u),u))-beta0*exp(-u)/ u*psi(u)-E*psi(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I&beta0G6\"I\" EGF%F%6$I)operatorGF%I&arrowGF%F%,*-I%diffG%*protectedG6$-I$psiGF%6#I \"uGF%-I\"$GF-6$F2\"\"##!\"\"F6*&F2F8-F,6$F/F2\"\"\"F8**9$F<-I$expG6$F -I(_syslibGF%6#,$F2F8F " 0 "" {MPLTEXT 1 0 73 "dsolve(\{Eq1(10,-0.1),psi(0)=1,D(psi)(0)=0 \},numeric,output=listprocedure);" }}{PARA 206 "" 1 "" {TEXT 216 0 "" }{HYPERLNK 216 "Error, (in dsolve/numeric/checksing) ode system has a \+ removable singularity at u=0. Initial data is restricted to \{psi(u) = -.100000000000000*diff(psi(u),u)\}" 4 "http://www.maplesoft.com/suppo rt/help/errors/view.aspx?path=Error,%20(in%20dsolve%2Fnumeric%2Fchecks ing)%20ode%20system%20has%20a%20removable%20singularity%20at%20u%3D0.% 20Initial%20data%20is%20restricted%20to%20%7Bpsi(u)%20%3D%20-.10000000 0000000*diff(psi(u),u)%7D" "" }{TEXT 216 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# OK, we need to redefine the differential equati on" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "exp(beta0*u)*subs(psi (u)=exp(-beta0*u)*chi(u),Eq1(beta0,E));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I$expG6$%*protectedGI(_syslibG6\"6#*&I&beta0GF(\"\"\"I\"uGF(F,F ,,*-I%diffGF&6$-F06$*&-F$6#,$F*!\"\"F,-I$chiGF(6#F-F,F-F-#F8\"\"#*&F-F 8F2F,F8*,F+F,-F$6#,$F-F8F,F-F8F5F,F9F,F8*(I\"EGF(F,F5F,F9F,F8F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&I&beta0G6\"\"\"#-I$chiGF%6#I\"uGF%\"\"\"#!\"\"F&*& F$F+-I%diffG%*protectedG6$F'F*F+F+-F06$F/F*F,*(F*F-F$F+F'F+F+*&F*F-F/F +F-**F$F+-I$expG6$F1I(_syslibGF%F)F-F*F-F'F+F-*&I\"EGF%F+F'F+F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "Eq2:=(beta0,E)->-1/2*beta0^ 2*chi(u)+beta0*diff(chi(u),u)-1/2*diff(chi(u),`$`(u,2))+1/u*beta0*chi( u)-1/u*diff(chi(u),u)-beta0/exp(u)/u*chi(u)-E*chi(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I&beta0G6\"I\"EGF%F%6$I)operatorGF%I&arrowGF%F%,0 *&9$\"\"#-I$chiGF%6#I\"uGF%\"\"\"#!\"\"F-*&F,F2-I%diffG%*protectedG6$F .F1F2F2-F76$F.-I\"$GF86$F1F-F3*(F1F4F,F2F.F2F2*&F1F4F6F2F4**F,F2-I$exp G6$F8I(_syslibGF%F0F4F1F4F.F2F4*&9%F2F.F2F4F%F%F%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "# OK now it's working" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "sol1:=dsolve(\{Eq2(10,-40.705803),chi(0)=1,D(c hi)(0)=0\},numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "chi1s:=subs(sol1,chi(u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"uG6\"6)I%_resGF%I%_datGF%I*_solnprocGF%I&_xoutGF%I 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" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "#We find E=-40.705803 show ing that the variationl procedure gave a quite accurate result. 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