{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "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 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 259 19 "Perturbation Theory" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "In this w orksheet we provide two examples of perturbation calculations:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 417 "1) to pr ovide a motivation for perturbative calculations we consider the first relativistic correction to the kinetic energy in the hydrogen-like gr ound state. It represents an example where a natural smallness paramet er is present (1/c^2), and where we really are only interested in the \+ first-order correction to the energy as the higher-order contributions do compete with corrections to the energy operator itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "2) to show in \+ a model problem how one can calculate higher-order corrections, and th at there are examples where the series expansion is only semi-converge nt we calculate the energy corrections to high, but finite order for a harmonic oscillator perturbed by a power-law (quartic) anharmonic pot ential. The Dalgarno-Lewis method is used to carry out the calculation ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 38 "A first-order perturbation calculation" }}{PARA 0 "" 0 "" {TEXT -1 204 "Suppose we are interested in a calculation to account fo r the correction to the Schroedinger kinetic energy due to special rel ativity. We just consider the hydrogenic ground state which is non-deg enerate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume(c>0,m>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "taylor (sqrt(m^2*c^4+p^2*c^2),p=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"p G*&%#m|irG\"\"\")%#c|irG\"\"#F'\"\"!,$*&F'F'F&!\"\"#F'F*F*,$*&F'F'*&)F &\"\"$F'F(F'F.#F.\"\")\"\"%-%\"OG6#F'\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The rest energy in atomic units is given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(m=1,c=137,m*c^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&p(=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "%*27.12*_eV;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%$_eVG$\")G:!4 &!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "This is slightly off fr om the accepted electron mass of 512 keV due to our approximate value \+ of the fine structure constant as 1/137." }}{PARA 0 "" 0 "" {TEXT -1 247 "We know from the virial theorem that the kinetic energy in the hy drogen atom ground state has the same magnitude as the binding energy \+ (13.6 eV). We recognize that the inverse powers of c will generate con verging series. Note that in atomic units " }{TEXT 256 1 "m" }{TEXT -1 3 "=1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 280 "To first order we need to calculate the expectation valu e of the fourth derivative for the H0-eigenstate, i.e., the hydrogenic 1s-state. The angular momentum is zero, and therefore only a radial d erivative is required when considering the momentum operator as it act s on s-states." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "psi:=2*r* exp(-r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&%\"rG\"\"\"-%$e xpG6#,$F'!\"\"F(\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "in t(psi^2,r=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "It is most convenient to perform \+ the calculation in momentum space, where the momentum operator is a mu ltiplicative operator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p hi:=simplify(sqrt(2/Pi)*int(psi*sin(k*r),r=0..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&*&-%%sqrtG6#\"\"#\"\"\"%\"kGF,F,*&-F )6#%#PiGF,),&F,F,*$)F-F+F,F,F+F,!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "int(phi^2,k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "If we c hoose units in which h_bar equals unity, we do not distinguish between p and k (momentum and wavenumber)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(phi^2*k^2/2,k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "W e recognize that the non-relativistic kinetic energy is calculated cor rectly. Let us now obtain the " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(phi^2*(-k^4/(8*c^2)),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%#c|irG\"\"#F%!\"\"#!\"&\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(subs(c=137,%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+v*e*HL!#9" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 157 "This is a very small correction when compared to the n on-relativistic answer of 0.5. Now we repeat the calculation for arbit rary Z (hydrogen-like situation)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(Z>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "psi:=2*Z^(3/2)*r*exp(-Z*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$psiG,$*()%#Z|irG#\"\"$\"\"#\"\"\"%\"rGF,-%$expG6#,$*&F(F,F-F,!\"\"F, F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int(psi^2,r=0..infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "phi:=simplify(sqrt(2/Pi)*int(psi*sin(k*r),r=0. .infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&*(-%%sqrtG6 #\"\"#\"\"\")%#Z|irG#\"\"&F+F,%\"kGF,F,*&-F)6#%#PiGF,),&*$)F.F+F,F,*$) F1F+F,F,F+F,!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "N o:=int(phi^2,k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NoG \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "T_nr:=int(phi^2*k ^2/2,k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%T_nrG,$*$)% #Z|irG\"\"#\"\"\"#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T _r1:=int(phi^2*(-k^4/(8*c^2)),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%T_r1G,$*&*$)%#Z|irG\"\"%\"\"\"F+*$)%#c|irG\"\"#F+!\" \"#!\"&\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs(Z=20,c =137,evalf([1*c^2,T_nr,T_r1])); # Calcium" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"&p(=$\"+++++?!\"($!+gV$zK&!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "subs(Z=79,c=137,evalf([1*c^2,T_nr,T_r1])); # Gold" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"&p(=$\"+++]?J!\"'$!+=<-(H \"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs(Z=92,c=137,eva lf([1*c^2,T_nr,T_r1])); # Uranium" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 %\"&p(=$\"++++KU!\"'$!+n\"fbQ#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "It is evident that:" }}{PARA 0 "" 0 "" {TEXT -1 64 "a) the kinetic energy becomes comparable to the rest energy for " }{TEXT 257 1 "Z" } {TEXT -1 73 " approaching 100 (heavy-ion collisions allow one to explo re this regime)." }}{PARA 0 "" 0 "" {TEXT -1 78 "b) the relativistic k inetic energy becomes appreciable for moderate values of " }{TEXT 258 1 "Z" }{TEXT -1 50 ". The Schroedinger equation is no longer adequate. " }}{PARA 0 "" 0 "" {TEXT -1 68 "c) there are two expansions that do c ome up in the present context: " }}{PARA 0 "" 0 "" {TEXT -1 76 "(i) th e expansion of the relativistic kinetic energy as a power series in p; " }}{PARA 0 "" 0 "" {TEXT -1 146 "(ii) the energy correction from the \+ first term in this expansion beyond the non-relativistic expression ha s been evaluated only in first-order PT." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 979 "It is important to distinguish th e two sources of expansions. The expansion of the Hamiltonian itself h as as its origin the model-like character of the approach; a proper ap proach to special relativity would start from a different wave equatio n, and would be fully compliant with the modern world after Einstein. \+ This approach for spin-1/2 particles such as electrons was introduced \+ by Dirac. It leads not only to an equation where the relativistic kine tic energy expression is quantized, but one finds that all the magneti c interactions resulting from the particle's spin and orbital angular \+ momenta is automatically included. One has to include perturbatively, \+ however corrections to the Hamiltonian that arise from interactions wi th the nuclear magnetic moment (hyperfine structure), and further deta ils such as nuclear finite size effects (the nucleus is not a point pa rticle and one should take into account at short distances the extent \+ of the nuclear charge distribution)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 49 "The Dalgarno-Lewis method for \+ perturbation theory" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 144 "In 1955 A. Dalgarno and J.T. Lewis published a method \+ that allows to calculate the perturbation series to high orders for no n-degenerate states." }}{PARA 0 "" 0 "" {TEXT -1 380 "The method is ex plained in the classic textbook by L.I. Schiff (QM, McGraw-Hill, 3rd e d., 1968, chapter 33). The method is based on the conversion of an eig envalue problem into a series of inhomogeneous differential equations. These equations determine successively the corrections to the eigenfu nctions, and the energy corrections are obtained by a simple expectati on value. For " }{XPPEDIT 18 0 "H = H0+lambda*W;" "6#/%\"HG,&%#H0G\"\" \"*&%'lambdaGF'%\"WGF'F'" }{TEXT -1 20 " the equations read:" }}{PARA 0 "" 0 "" {TEXT -1 20 "(H0-E[0]) psi[0] = 0" }}{PARA 0 "" 0 "" {TEXT -1 34 "(H0-E[0]) psi[1] = (E[1]-W) psi[0]" }}{PARA 0 "" 0 "" {TEXT -1 48 "(H0-E[0]) psi[2] = (E[1]-W) psi[1] + E[2] psi[0]" }}{PARA 0 "" 0 " " {TEXT -1 62 "(H0-E[0]) psi[3] = (E[1]-W) psi[2] + E[2] psi[1] + E[3] psi[0]" }}{PARA 0 "" 0 "" {TEXT -1 3 "..." }}{PARA 0 "" 0 "" {TEXT -1 5 "Here " }{XPPEDIT 18 0 "E = E[0]+lambda*E[1]+lambda^2*E[2]+lambda ^3*E[3]+O(lambda^4);" "6#/%\"EG,,&F$6#\"\"!\"\"\"*&%'lambdaGF)&F$6#F)F )F)*&F+\"\"#&F$6#F/F)F)*&F+\"\"$&F$6#F3F)F)-%\"OG6#*$F+\"\"%F)" } {TEXT -1 175 ", and a similar expansion holds for the wavefunction. Th e energy corrections are calculated from the expectation value E[i] = \+ . psi[0] has to be normalized." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 738 "The equations to be solv ed for psi[i] are inhomogeneous, as they are driven by a known right-h and side, namely the wavefunction correction obtained in the previous \+ step. The method is started with the known solution psi[0], E[0], and \+ E[1] is calculated directly. At every stage one has to calculate a new wavefunction from solving a distinct inhomogeneous problem, although \+ the operator on the LHS remains the same. The energy correction is the n obtained by a simple matrix element. This is in contrast with the us ual approach where the second-order energy already involves an infinit e sum. Note that the differential equations are of a structure where t hey can be solved by a Green's function that involves a sum over all H 0 eigenstates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "In this worksheet we consider cases of power-law potenti als that allow exact solutions to the Dalgarno-Lewis equations. The H0 problem is assumed to be the simple harmonic oscillator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "W:=x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG*$) %\"xG\"\"%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "u00:=ex p(-x^2/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "No:=1/sqrt(int(u00^2, x=-infinity..infinity)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "u0:=No* u00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u00G-%$expG6#,$*$)%\"xG\"\" #\"\"\"#!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G*&-%$expG6#,$ *$)%\"xG\"\"#\"\"\"#!\"\"F-F.*$)%#PiG#F.\"\"%F.F0" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "E0:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#E0G#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E1:=in t(u0*W*u0,x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#E1G#\"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Now that we \+ have the first-order energy correction we can proceed with the calcula tion of the first-order wavefunction calculation:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "SE1:=-1/2*diff(f(x),x$2)+(x^2-1)/2*f(x)=u0*( E1-W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SE1G/,&-%%diffG6$-%\"fG6# %\"xG-%\"$G6$F-\"\"##!\"\"F1*(#\"\"\"F1F6,&*$)F-F1F6F6F6F3F6F*F6F6*&*& -%$expG6#,$F8F2F6,&#\"\"$\"\"%F6*$)F-FCF6F3F6F6*$)%#PiG#F6FCF6F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "We can solve SE1 by a matrix meth od. The left-hand side will be common to all subsequent equations. Inh omogeneous differential equations in matrix representations become mat rix inversion problems. We use a harmonic oscillator basis for the mat rix representation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with (orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG %\"TG%\"UG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xi:=n->exp(-x ^2/2)*H(n,x)/(Pi^(1/4)*sqrt(n!*2^n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xiGR6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&-%$expG6#,$*$)%\"xG\"\" #\"\"\"#!\"\"F5F6-%\"HG6$9$F4F6F6*&)%#PiG#F6\"\"%F6-%%sqrtG6#*&-%*fact orialG6#F " 0 "" {MPLTEXT 1 0 66 "int(xi(4)^2,x=-infinity..infinity); # an example of normalizatio n." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(f(x)=xi(n),lhs(SE1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%%diffG6$*&*&-%$expG6#,$*$)%\"xG\"\"#\"\"\"#!\"\"F0F 1-%\"HG6$%\"nGF/F1F1*&)%#PiG#F1\"\"%F1-%%sqrtG6#*&-%*factorialG6#F7F1) F0F7F1F1F3-%\"$G6$F/F0F2*&**#F1F0F1,&F-F1F1F3F1F)F1F4F1F1*&)F:#F1F6#F@F1F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xi(m)*simpl ify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*()-%$expG6#,$*$)%\"xG \"\"#\"\"\"#!\"\"F.F.F/-%\"HG6$%\"mGF-F/,&*&F-F/-%%diffG6$-F36$%\"nGF- F-F/F.-F96$F;-%\"$G6$F-F.F1F/F/*(-%%sqrtG6#%#PiGF/-FE6#*&-%*factorialG 6#F5F/)F.F5F/F/-FE6#*&-%&GAMMAG6#,&F=F/F/F/F/)F.F=F/F/F1#F/F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ME:=unapply(Int(expand(%),x= -infinity..infinity),m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MEGR6 $%\"mG%\"nG6\"6$%)operatorG%&arrowGF)-%$IntG6$,&*&**)-%$expG6#,$*$)%\" xG\"\"#\"\"\"#!\"\"F;F;F<-%\"HG6$9$F:FF<*&#FF/F:;,$%)infinityGF>FfoF)F)F)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 236 "Now we show that the structure of the matrix is v ery simple indeed: only diagonal matrix elements occur, and the struct ure is that =n. This diagonal matrix is trivial to invert ! Check it for yourself using matrix inversion." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "value(ME(1,2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(ME(2,2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(ME (2,4));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(ME(2,6));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(ME(0,0));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "value(ME(4,4));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value(ME(6,6)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(ME(1, 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 8 "" 1 "" {TEXT -1 54 " Error, (in GAMMA) numeric exception: division by zero\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rhs(SE1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%$ex pG6#,$*$)%\"xG\"\"#\"\"\"#!\"\"F,F-,&#\"\"$\"\"%F-*$)F+F3F-F/F-F-*$)%# PiG#F-F3F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "RH1:=unappl y(Int(expand(xi(m)*rhs(SE1)),x=-infinity..infinity),m);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$RH1GR6#%\"mG6\"6$%)operatorG%&arrowGF(-%$IntG 6$,&*&*&)-%$expG6#,$*$)%\"xG\"\"#\"\"\"#!\"\"F:F:F;-%\"HG6$9$F9F;F;*&- %%sqrtG6#%#PiGF;-FD6#*&-%*factorialG6#FAF;)F:FAF;F;F=#\"\"$\"\"%*&*(F2 F;F>F;)F9FPF;F;*&-FD6#FFF;-FD6#FIF;F=F=/F9;,$%)infinityGF=FfnF(F(F(" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "The solution is given as a supe rposition of the basis functions, where the coefficients follow from t he matrix inversion mentioned above in combination with the right-hand -side calculation." }}{PARA 0 "" 0 "" {TEXT -1 257 "The first two calc ulations state that we do not need to admix the psi[0] state (c0=0), w hich is good since we can't invert the corresponding matrix coefficien t; also all states with odd symmetry do not contribute (the integrals \+ vanish for symmetry reasons):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c0:=simplify(value(RH1(0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#c0G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c1:=simplify (value(RH1(1))/1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Now we could get an infinite seque nce of non-vanishing right-hand-side calculations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2:=simplify(value(RH1(2))/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,$*$-%%sqrtG6#\"\"#\"\"\"#!\"$\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:=simplify(value(RH1(4))/4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c4G,$*$-%%sqrtG6#\"\"'\"\"\"# !\"\"\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=simplify( value(RH1(6))/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "In fact, all higher coefficients v anish!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=simplify(valu e(RH1(8))/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c8G\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "The solution to the equations SE1 which determines the first-order wave-function correction is then giv en exactly as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sol:=c2*x i(2)+c4*xi(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,&*&*&-%$expG 6#,$*$)%\"xG\"\"#\"\"\"#!\"\"F/F0,&F,\"\"%F/F2F0F0*$)%#PiG#F0F4F0F2#! \"$\"\")*&#F0\"#kF0*&*&F(F0,(*$)F.F4F0\"#;*&\"#[F0F-F0F2\"#7F0F0F0*$)F 7#F0F4F0F2F0F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We now verify t hat it does solve SE1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s ubs(f(x)=sol,SE1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpl ify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&*&-%$expG6#,$*$)%\"xG \"\"#\"\"\"#!\"\"F.F/,&!\"$F/*&\"\"%F/)F-F5F/F/F/F/*$)%#PiG#F/F5F/F1#F 1F5F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalb(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "So we found an exact solution! Now we calculate the second-order e nergy correction using psi[1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E2:=int(u0*sol*(W),x=-infinity..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#E2G#!#@\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(u0*sol*(W-E1),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#@\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "That is the exact answer! The two above calculations agree sin ce u0 is orthogonal to sol, i.e., to psi[1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Now do the same for the next eq uation: (H0-W0)*psi[2]=(E[1]-W)*psi[1]+E[2]*psi[0]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Note that it is only the \+ right-hand side that changes!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "E1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "RH2:=unapply(Int(expand(xi(m)*((E1- W)*sol+E2*u0)),x=-infinity..infinity),m):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "value(RH2(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2:=simplify(value (RH2(2))/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:=simplify(value( RH2(4))/4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=simplify(value(R H2(6))/6);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=simplify(value(RH 2(8))/8);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(RH2(10));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,$*$-%%sqrtG6#\"\"#\"\"\"#\"#v\" #;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c4G,$*$-%%sqrtG6#\"\"'\"\"\"# \"\"*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G,$*$-%%sqrtG6#\"\" &\"\"\"#\"#<\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c8G,$*$-%%sqrtG 6#\"#q\"\"\"#\"\"$\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "All higher coefficients vanish. N ote that we produced two more coefficients than before: this is due to the x^4 interaction, which can couple two adjacent columns or rows in the matrix that represents this interaction in the HO basis (generate the matrix representation and observe). There are four terms." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sol2:=c2*xi(2)+c4*xi(4)+c6*x i(6)+c8*xi(8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "SE2:=lhs( SE1)=((E1-W)*sol+E2*u0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(f( x)=sol2,SE2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%): evalb (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Let us check the orthogonality of the successive sol ution corrections:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(s ol2*u0,x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(sol2*sol,x=-infini ty..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$z#\"#K" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 249 "Note that orthogonality to ground state is preserved. The solution psi[1] is not orthogonal to psi[2]. \+ The orthogonality to the ground state means that the energy correction calculation can be made with W or with W displaced by some energy con stant:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E3:=int(u0*W*sol2 ,x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G#\"$ L$\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(u0*sol2*(W-E1 ),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$L$\"# ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now the fourth order:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "RH3:=unapply(Int(expand(xi(m )*((E1-W)*sol2+E2*sol+E3*u0)),x=-infinity..infinity),m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(RH3(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2: =simplify(value(RH3(2))/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:= simplify(value(RH3(4))/4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=s implify(value(RH3(6))/6);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=si mplify(value(RH3(8))/8);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c10:=si mplify(value(RH3(10))/10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c12:= simplify(value(RH3(12))/12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G ,$*$-%%sqrtG6#\"\"#\"\"\"#!%F:\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#c4G,$*$-%%sqrtG6#\"\"'\"\"\"#!%(=%\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G,$*$-%%sqrtG6#\"\"&\"\"\"#!%h<\"#k" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#c8G,$*$-%%sqrtG6#\"#q\"\"\"#!$6\"\"#K" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c10G,$*$-%%sqrtG6#\"\"(\"\"\"#!$v$ \"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c12G,$*$-%%sqrtG6#\"$J#\" \"\"#!#:\"$c#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "value(RH3( 14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "All subsequent coefficients vanish. There are four t erms in the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "so l3:=c2*xi(2)+c4*xi(4)+c6*xi(6)+c8*xi(8)+c10*xi(10)+c12*xi(12):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "SE3:=lhs(SE1)=((E1-W)*sol2+E 2*sol+E3*u0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(f(x)=sol3,SE3 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%): evalb(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "We continue to find exact solutions to the Dalgarno-Lewi s equations, and we preserve the orthogonality to the ground state:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(sol3*u0,x=-infinity..i nfinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(sol2*sol3,x=-infinity..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!'4zE\"$c#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 49 "Orthogonality to ground state is again preserved." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E4:=int(u0*W*sol3,x=-infinit y..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E4G#!&&)3$\"$G\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(u0*sol3*(W-E1),x=-inf inity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!&&)3$\"$G\"" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "This is the fourth order. Now wa nt the fifth! Our series expansion for the ground-state energy so far \+ reads as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "E0+lambda*E1+l ambda^2*E2+lambda^3*E3+lambda^4*E4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,,#\"\"\"\"\"#F%*&#\"\"$\"\"%F%%'lambdaGF%F%*&#\"#@\"\")F%*$)F+F&F%F% !\"\"*&#\"$L$\"#;F%)F+F)F%F%*&#\"&&)3$\"$G\"F%*$)F+F*F%F%F2" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "It is somewhat worrisome that the \+ coefficients are growing..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "RH4:=unapply(Int(expand(xi(m)*((E1-W)*sol3+E2*sol2+E3*sol+E4*u0)), x=-infinity..infinity),m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(RH4(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2:=simplify(value(RH4(2))/2 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:=simplify(value(RH4(4))/4) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=simplify(value(RH4(6))/6); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=simplify(value(RH4(8))/8); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c10:=simplify(value(RH4(10))/10 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c12:=simplify(value(RH4(12))/ 12);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c14:=simplify(value(RH4(14) )/14);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c16:=simplify(value(RH4(1 6))/16);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c18:=simplify(value(RH4 (18))/18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,$*$-%%sqrtG6#\"\" #\"\"\"#\"'Td;\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c4G,$*$-%%sq rtG6#\"\"'\"\"\"#\"&f\\$\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G ,$*$-%%sqrtG6#\"\"&\"\"\"#\"'8(f\"\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c8G,$*$-%%sqrtG6#\"#q\"\"\"#\"&$>i\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c10G,$*$-%%sqrtG6#\"\"(\"\"\"#\"&v()*\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c12G,$*$-%%sqrtG6#\"$J#\"\"\"#\"$b$\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c14G,$*$-%%sqrtG6#\"$e)\"\"\"#\"% b6\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c16G,$*$-%%sqrtG6#\"%I9 \"\"\"#\"$:$\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c18G\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sol4:=c2*xi(2)+c4*xi(4)+c6* xi(6)+c8*xi(8)+c10*xi(10)+c12*xi(12)+c14*xi(14)+c16*xi(16):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SE4:=lhs(SE1)=((E1-W)*sol3+E 2*sol2+E3*sol+E4*u0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(f(x)= sol4,SE4):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%): evalb(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E5:=int(u0*W*sol4,x=-infinity..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E5G#\"'Jn\"*\"$c#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(u0*sol4*(W-E1),x=-infinity..infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"'Jn\"*\"$c#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Do one more, the sixth:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 100 "RH5:=unapply(Int(expand(xi(m)*((E1-W)*sol4+E2 *sol3+E3*sol2+E4*sol+E5*u0)),x=-infinity..infinity),m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(RH5(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2: =simplify(value(RH5(2))/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:= simplify(value(RH5(4))/4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=s implify(value(RH5(6))/6);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=si mplify(value(RH5(8))/8);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c10:=si mplify(value(RH5(10))/10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c12:= simplify(value(RH5(12))/12);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c14 :=simplify(value(RH5(14))/14);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c 16:=simplify(value(RH5(16))/16);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c18:=simplify(value(RH5(18))/18);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c20:=simplify(value(RH5(20))/20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c22:=simplify(value(RH5(22))/22);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c24:=simplify(value(RH5(24))/24);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,$*$-%%sqrtG6#\"\"#\"\"\"#!(.F_&\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c4G,$*$-%%sqrtG6#\"\"'\"\"\"#!)hSz5\"%C5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G,$*$-%%sqrtG6#\"\"&\"\"\"#!( FBd(\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c8G,$*$-%%sqrtG6#\"#q \"\"\"#!&`,$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c10G,$*$-%%sqr tG6#\"\"(\"\"\"#!':@a\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c12G,$ *$-%%sqrtG6#\"$J#\"\"\"#!(b5;$\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$c14G,$*$-%%sqrtG6#\"$e)\"\"\"#!'&>:'\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c16G,$*$-%%sqrtG6#\"%I9\"\"\"#!&X.$\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c18G,$*$-%%sqrtG6#\"&b@\"\"\"\"#!&:H\"\"% #>)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c20G,$*$-%%sqrtG6#\"&*=Y\"\" \"#!$X*\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c22G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c24G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "sol5:=c2*xi(2)+c4*xi(4)+c6*xi(6)+c8*xi(8)+c10*x i(10)+c12*xi(12)+c14*xi(14)+c16*xi(16)+c18*xi(18)+c20*xi(20):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "SE5:=lhs(SE1)=((E1-W)*sol4+E 2*sol3+E3*sol2+E4*sol+E5*u0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "su bs(f(x)=sol5,SE5):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%): \+ evalb(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E6:=int(u0*W*sol5,x=-infinity..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E6G#!),%=b'\"%C5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "And, one more:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "RH6:=unapply(Int(xi(m)*((E1-W)*sol5+E2*sol4+E3*so l3+E4*sol2+E5*sol+E6*u0),x=-infinity..infinity),m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(RH6(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c2: =simplify(value(RH6(2))/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c4:= simplify(value(RH6(4))/4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c6:=s implify(value(RH6(6))/6);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c8:=si mplify(value(RH6(8))/8);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c10:=si mplify(value(RH6(10))/10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c12:= simplify(value(RH6(12))/12);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c14 :=simplify(value(RH6(14))/14);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c 16:=simplify(value(RH6(16))/16);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c18:=simplify(value(RH6(18))/18);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c20:=simplify(value(RH6(20))/20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c22:=simplify(value(RH6(22))/22);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c24:=simplify(value(RH6(24))/24);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c26:=simplify(value(RH6(26))/26);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#c2G,$*$-%%sqrtG6#\"\"#\"\"\"#\"*(>.IV\"%[?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c4G,$*$-%%sqrtG6#\"\"'\"\"\"#\"*Z3Q P#\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c6G,$*$-%%sqrtG6#\"\"&\" \"\"#\"*(fjyx\"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c8G,$*$-%%sqr tG6#\"#q\"\"\"#\"+z]fC>\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c1 0G,$*$-%%sqrtG6#\"\"(\"\"\"#\"+b\\$)QF\"%#>)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c12G,$*$-%%sqrtG6#\"$J#\"\"\"#\")XUoT\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c14G,$*$-%%sqrtG6#\"$e)\"\"\"#\"*0#Hm=\"& %Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c16G,$*$-%%sqrtG6#\"%I9\"\" \"#\")0@0f\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c18G,$*$-%%sqrt G6#\"&b@\"\"\"\"#\")vg87\"&oF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c 20G,$*$-%%sqrtG6#\"&*=Y\"\"\"#\"'vGK\"%#>)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c22G,$*$-%%sqrtG6#\"'ej<\"\"\"#\"'&yp\"\"&Ob'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c24G,$*$-%%sqrtG6#\"'Rgn\"\"\"#\"&& R5\"'s58" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c26G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "sol6:=c2*xi(2)+c4*xi(4)+c6*xi(6)+c 8*xi(8)+c10*xi(10)+c12*xi(12)+c14*xi(14)+c16*xi(16)+c18*xi(18)+c20*xi( 20)+c22*xi(22)+c24*xi(24):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "SE6:=lhs(SE1)=((E1-W)*sol5+E2*sol4+E3*sol3+E4*sol2+E5*sol+E6*u0): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(f(x)=sol6,SE6):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%): evalb(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "E7:=int(u0*sol6*(W-W1),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E7G#\"+tYHBF\"%[?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "This worrisome trend of growing coefficients has continued:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E1,E2,E3,E4,E5,E6,E7;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)#\"\"$\"\"%#!#@\"\")#\"$L$\"#;#!&&)3$ \"$G\"#\"'Jn\"*\"$c##!),%=b'\"%C5#\"+tYHBF\"%[?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "lambda:='lambda':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "EO2:=E0+E1*lambda+E2*lambda^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EO2G,(#\"\"\"\"\"#F'*&#\"\"$\"\"%F'%'lambdaGF'F'*&# \"#@\"\")F'*$)F-F(F'F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EO3:=EO2+E3*lambda^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EO3G,* #\"\"\"\"\"#F'*&#\"\"$\"\"%F'%'lambdaGF'F'*&#\"#@\"\")F'*$)F-F(F'F'!\" \"*&#\"$L$\"#;F')F-F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EO4:=EO3+E4*lambda^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EO4G,, #\"\"\"\"\"#F'*&#\"\"$\"\"%F'%'lambdaGF'F'*&#\"#@\"\")F'*$)F-F(F'F'!\" \"*&#\"$L$\"#;F')F-F+F'F'*&#\"&&)3$\"$G\"F'*$)F-F,F'F'F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EO5:=EO4+E5*lambda^5;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$EO5G,.#\"\"\"\"\"#F'*&#\"\"$\"\"%F'%'lambdaGF 'F'*&#\"#@\"\")F'*$)F-F(F'F'!\"\"*&#\"$L$\"#;F')F-F+F'F'*&#\"&&)3$\"$G \"F'*$)F-F,F'F'F4*&#\"'Jn\"*\"$c#F')F-\"\"&F'F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "EO6:=EO5+E6*lambda^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EO6G,0#\"\"\"\"\"#F'*&#\"\"$\"\"%F'%'lambdaGF'F'*&# \"#@\"\")F'*$)F-F(F'F'!\"\"*&#\"$L$\"#;F')F-F+F'F'*&#\"&&)3$\"$G\"F'*$ )F-F,F'F'F4*&#\"'Jn\"*\"$c#F')F-\"\"&F'F'*&#\"),%=b'\"%C5F'*$)F-\"\"'F 'F'F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EO7:=EO6+E7*lambda ^7;" }}{PARA 11 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$Fax$\"3?#3\"=VL69JF\\in7$Ffx$\"3'RX*\\y-a+PF\\in7$F[y$\"3CXxGq&y/K%F \\in7$F`y$\"3ITQOEefQ^F\\in7$Fa`n$\"3=\\]5?YQXbF\\in7$Fey$\"3]=aT]1[zf F\\in7$Fi`n$\"3Y,h$[B*4ukF\\in7$Fjy$\"3(fET$3VR.qF\\in7$Fgfm$\"3WY)fr: CVa(F\\in7$F_z$\"39B\">&)\\!p?\")F\\in7$F_gm$\"3gU39Thw\"y)F\\in7$F*$ \"3'p7bxme%)[*F\\in-Fgz6&FizFjzF(Fjz-%+AXESLABELSG6$Q'lambda6\"Q!6\"-% %VIEWG6$;F($\"\"&!\"\";$!\"#F)$\"\"#F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Let us \+ find the exact eigenenergy at lambda=0.1 and compare the various appro ximations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "E_num:=0.5591 45;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E_numG$\"'X\"f&!\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "SE:=-1/2*diff(u(x),x$2)+(1/2 *x^2+1/10*x^4-E_num)*u(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEG /,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"##!\"\"F1*&,(*$)F-F1\"\"\"#F8F 1*&#F8\"#5F8)F-\"\"%F8F8$\"'X\"f&!\"'F3F8F*F8F8\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "solSE:=dsolve(\{SE,u(0)=1,D(u)(0)=0\},u(x ),numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ux:=subs(solSE,u(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot('ux(x)',x=0..4,view=[0..4,-0.2..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 694 186 186 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$\"\"! F)$\"\"\"F)7$$\"+L3VfV!#6$\"+V&y$*)**!#57$$\"+m;')=()F/$\"+.Gdd**F27$$ \"+7z>^7F2$\"+Wtz7**F27$$\"+e'40j\"F2$\"+W>I_)*F27$$\"+<6m$[#F2$\"++7? g'*F27$$\"+R*F27$$\"+eF>(>%F2$\"+)p&4c!*F27$$\"+\">K'*) \\F2$\"+&p#G*o)F27$$\"+Dt:5eF2$\"+p`Mg#)F27$$\"+\"fX(emF2$\"+,[bsxF27$ $\"+DCh/vF2$\"+*e]4D(F27$$\"+L/pu$)F2$\"+]]\"zo'F27$$\"+;c0T\"*F2$\"++ zEyhF27$$\"+I,Q+5!\"*$\"+\"*[k)f&F27$$\"+]*3q3\"F\\p$\"+\"el+-&F27$$\" +q=\\q6F\\p$\"+U%HXZ%F27$$\"+fBIY7F\\p$\"+?\">e*RF27$$\"+j$[kL\"F\\p$ \"+l&zUX$F27$$\"+`Q\"GT\"F\\p$\"+=m'Q-$F27$$\"+s]k,:F\\p$\"+ZnUgDF27$$ \"+`dF!e\"F\\p$\"+J$>j=#F27$$\"+sgam;F\\p$\"+d0k;=F27$$\"+F\\p$\"+[^*3&**F/7$$\"+Uc -)*>F\\p$\"+.$>#zyF/7$$\"+f`@'3#F\\p$\"+15+\"4'F/7$$\"+nZ)H;#F\\p$\"+P 6X1[F/7$$\"+Ky*eC#F\\p$\"+1j!3n$F/7$$\"+S^bJBF\\p$\"+#RXht#F/7$$\"+0TN :CF\\p$\"+3v*3-#F/7$$\"+7RV'\\#F\\p$\"+;^0&[\"F/7$$\"+:#fke#F\\p$\"+i7 lO5F/7$$\"+`4NnEF\\p$\"+?u7!R(!#77$$\"+],s`FF\\p$\"+P2Ar]Fcv7$$\"+zM)> $GF\\p$\"+rcmkNFcv7$$\"+qfa\"Fcv7$$\"+$)G[kJF\\p$\"+=[i$\\*!#87$$ \"+7yh]KF\\p$\"+*>`K*))Fbx7$$\"+()fdLLF\\p$\"+GzRL5Fcv7$$\"+-FT=MF\\p$ \"+Q'=HW\"Fcv7$$\"+Epa-NF\\p$\"+6(H$[AFcv7$$\"+Sv&)zNF\\p$\"+%3M^d$Fcv 7$$\"+GUYoOF\\p$\"+oJdRjFcv7$$\"+2^rZPF\\p$\"+M,a)3\"F/7$$\"+sI@KQF\\p $\"+r^M))>F/7$$\"+S2lsQF\\p$\"+L0jvEF/7$$\"+2%)38RF\\p$\"+YztEOF/7$$\" +/UacRF\\p$\"+8'fa0&F/7$$\"\"%F)$\"+R)Ga5(F/-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F[\\l;$!\"#Fe\\lF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Pt:=[[1,E0+0.1*E1],[2,subs( lambda=0.1,EO2)],[3,subs(lambda=0.1,EO3)],[4,subs(lambda=0.1,EO4)],[5, subs(lambda=0.1,EO5)],[6,subs(lambda=0.1,EO6)],[7,subs(lambda=0.1,EO7) ]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PtG7)7$\"\"\"$\"++++]d!#57$ \"\"#$\"+++]([&F*7$\"\"$$\"++]i&p&F*7$\"\"%$\"+QfLaaF*7$\"\"&$\"+&)RV7 eF*7$\"\"'$\"+]egs^F*7$\"\"($\"+x&RB]'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([Pt,E_num],1..8);" }}{PARA 13 "" 1 "" {GLPLOT2D 551 170 170 {PLOTDATA 2 "6&-%'CURVESG6$7)7$$\"\"\"\"\"!$\"3c********** **\\d!#=7$$\"\"#F*$\"3g**********\\([&F-7$$\"\"$F*$\"3s********\\i&p&F -7$$\"\"%F*$\"31+++QfLaaF-7$$\"\"&F*$\"3`*****\\)RV7eF-7$$\"\"'F*$\"3] +++]egs^F-7$$\"\"(F*$\"3J+++x&RB]'F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F S-F$6$7S7$F($\"3/+++++X\"f&F-7$$\"3smm;z+e_6!#R`G\"FgnF X7$$\"3hmmT&pSYV\"FgnFX7$$\"3imm\"z'=$\\e\"FgnFX7$$\"3]L$3Ft3Xt\"FgnFX 7$$\"3cm;aLc=t=FgnFX7$$\"3J+](=`xn,#FgnFX7$$\"3#omT&y/Gl@FgnFX7$$\"3y* *\\PurI8BFgnFX7$$\"3wLL$e#3dlCFgnFX7$$\"3ymm\"Ht%o*f#FgnFX7$$\"3K++]F_ m]FFgnFX7$$\"32++]icE-HFgnFX7$$\"3;++]s2O[IFgnFX7$$\"3um;aG\"H5=$FgnFX 7$$\"3^LL$ej%yQLFgnFX7$$\"3mLLLVUUsMFgnFX7$$\"35+](o()yyi$FgnFX7$$\"3G LLLoD[lPFgnFX7$$\"3P+](oibk\"RFgnFX7$$\"3a+]i!o<-1%FgnFX7$$\"3qLL3-$=- @%FgnFX7$$\"33M$3xplzM%FgnFX7$$\"3/nm\"H([a'\\%FgnFX7$$\"3Km;ayo(3l%Fg nFX7$$\"3k+]7VLA&y%FgnFX7$$\"3'pm;a?@.$\\FgnFX7$$\"3)******\\\\@-3&Fgn FX7$$\"3Q++v$opoA&FgnFX7$$\"3c+](oMf(o`FgnFX7$$\"3#)***\\ii.j_&FgnFX7$ $\"3%GLL$oT'ym&FgnFX7$$\"3'3++DE5!>eFgnFX7$$\"3Mm;a)3rf&fFgnFX7$$\"3*4 ++vW0d5'FgnFX7$$\"3;L$3-\"QfYiFgnFX7$$\"3C+]PWF'QR'FgnFX7$$\"3[LL$e/Xy `'FgnFX7$$\"3m**\\(=<\"e)o'FgnFX7$$\"3%ymmm(zvLoFgnFX7$$\"3-nm\"zAAA)p FgnFX7$$\"3LM$3-7d%HrFgnFX7$$\"3#4++]p]ZE(FgnFX7$$\"3$QL$e*R7)>uFgnFX7 $$\"3'pmmmV,&evFgnFX7$$\"3<+](o(GP1xFgnFX7$$\"3g+]78Z!z%yFgnFX7$$\"\") F*FX-FM6&FOFSFPFS-%+AXESLABELSG6$Q!6\"Fjw-%%VIEWG6$;F(Fcw%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "Clearly this is an exam ple where perturbation theory does not work! For the x^4 potential eve n a coupling of lambda=0.1 leads to a useless asymptotic series. For s maller lambda is should be useful, however." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "E_num:=0.514085;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&E_numG$\"'&39&!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " SE:=-1/2*diff(u(x),x$2)+(1/2*x^2+1/50*x^4-E_num)*u(x)=0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#SEG/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"## !\"\"F1*&,(*$)F-F1\"\"\"#F8F1*&#F8\"#]F8)F-\"\"%F8F8$\"'&39&!\"'F3F8F* F8F8\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "solSE:=dsolve( \{SE,u(0)=1,D(u)(0)=0\},u(x),numeric,output=listprocedure):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ux:=subs(solSE,u(x)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot('ux(x)',x=0..4,view=[0. .4,-0.2..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 694 172 172 {PLOTDATA 2 "6 %-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"+m;')=()!#6$\"+FN*4'**!#57$$\"+ e'40j\"F2$\"+rQAk)*F27$$\"+<6m$[#F2$\"+CHo(o*F27$$\"+(>%F2$\"+0g\"G8*F27$$\"+\">K'*)\\F2$\"+m)Rgz)F27$$\"+Dt:5eF 2$\"+wnC-%)F27$$\"+\"fX(emF2$\"+DSHazF27$$\"+DCh/vF2$\"+HT!\\Z(F27$$\" +L/pu$)F2$\"+=?kcpF27$$\"+;c0T\"*F2$\"+NfV'['F27$$\"+I,Q+5!\"*$\"+)zk) \\fF27$$\"+]*3q3\"Fbo$\"+s(3;T&F27$$\"+q=\\q6Fbo$\"+k\"*y+\\F27$$\"+fB IY7Fbo$\"+s2/\\WF27$$\"+j$[kL\"Fbo$\"+eqyKRF27$$\"+`Q\"GT\"Fbo$\"+#)QJ jIF27$$\"+`dF!e\"Fbo$\"++&3**o#F27$$\"+sgam;F bo$\"+rM?8BF27$$\"+F27$$\"+e/TM=Fbo$\"+@!\\;o\"F27$$ \"+cK78>Fbo$\"+DPEJ9F27$$\"+Uc-)*>Fbo$\"+2mg#>\"F27$$\"+f`@'3#Fbo$\"+l kGu(*F/7$$\"+nZ)H;#Fbo$\"+!f&[a\")F/7$$\"+Ky*eC#Fbo$\"+E:r[mF/7$$\"+S^ bJBFbo$\"+CTQM`F/7$$\"+0TN:CFbo$\"+9d\\gUF/7$$\"+7RV'\\#Fbo$\"+o\")R(R $F/7$$\"+:#fke#Fbo$\"+c@6:EF/7$$\"+`4NnEFbo$\"+D]$y/#F/7$$\"+],s`FFbo$ \"+Q*[=c\"F/7$$\"+zM)>$GFbo$\"+AjD67F/7$$\"+qfa(oc\"Fhv7$$\"+Epa-NFbo$\"+;,Z_7Fhv7$$\"+Sv&)zNFbo$\"+z7&**4 \"Fhv7$$\"+GUYoOFbo$\"+8u!\\2\"Fhv7$$\"+2^rZPFbo$\"+-:J%>\"Fhv7$$\"+sI @KQFbo$\"+VRi(\\\"Fhv7$$\"+2%)38RFbo$\"+7ua8?Fhv7$$\"\"%F)$\"+'Rp\\#HF hv-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG 6$;F(Ffz;$!\"#F`[lF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "Pt:=[[1,E0+1/50*E1],[2,subs(lambda=1/50,EO2)],[3,subs(lambda=1/50,EO3 )],[4,subs(lambda=1/50,EO4)],[5,subs(lambda=1/50,EO5)],[6,subs(lambda= 1/50,EO6)],[7,subs(lambda=1/50,EO7)]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PtG7)7$\"\"\"#\"$.\"\"$+#7$\"\"##\"&z-\"\"&++#7$\"\"$#\"(L#G5 \"(+++#7$\"\"%#\")jCD#)\"*+++g\"7$\"\"&#\",J#[r7T\",+++++)7$\"\"'#\".* zFTOD#)\"/++++++;7$\"\"(#\"0tX2O\"RD#)\"1+++++++;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "plot([Pt,E_num],1..8);" }}{PARA 13 "" 1 "" {GLPLOT2D 551 170 170 {PLOTDATA 2 "6&-%'CURVESG6$7)7$$\"\"\"\"\"!$\"39 ++++++]^!#=7$$\"\"#F*$\"3=+++++]R^F-7$$\"\"$F*$\"3w********\\;T^F-7$$ \"\"%F*$\"39++]P*y29&F-7$$\"\"&F*$\"3<+]()GN*39&F-7$$\"\"'F*$\"3QvV()z D&39&F-7$$\"\"(F*$\"3V\"3m/gp39&F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*FS- F$6$7S7$F($\"39+++++&39&F-7$$\"3smm;z+e_6!#R`G\"FgnFX7$ $\"3hmmT&pSYV\"FgnFX7$$\"3imm\"z'=$\\e\"FgnFX7$$\"3]L$3Ft3Xt\"FgnFX7$$ \"3cm;aLc=t=FgnFX7$$\"3J+](=`xn,#FgnFX7$$\"3#omT&y/Gl@FgnFX7$$\"3y**\\ PurI8BFgnFX7$$\"3wLL$e#3dlCFgnFX7$$\"3ymm\"Ht%o*f#FgnFX7$$\"3K++]F_m]F FgnFX7$$\"32++]icE-HFgnFX7$$\"3;++]s2O[IFgnFX7$$\"3um;aG\"H5=$FgnFX7$$ \"3^LL$ej%yQLFgnFX7$$\"3mLLLVUUsMFgnFX7$$\"35+](o()yyi$FgnFX7$$\"3GLLL oD[lPFgnFX7$$\"3P+](oibk\"RFgnFX7$$\"3a+]i!o<-1%FgnFX7$$\"3qLL3-$=-@%F gnFX7$$\"33M$3xplzM%FgnFX7$$\"3/nm\"H([a'\\%FgnFX7$$\"3Km;ayo(3l%FgnFX 7$$\"3k+]7VLA&y%FgnFX7$$\"3'pm;a?@.$\\FgnFX7$$\"3)******\\\\@-3&FgnFX7 $$\"3Q++v$opoA&FgnFX7$$\"3c+](oMf(o`FgnFX7$$\"3#)***\\ii.j_&FgnFX7$$\" 3%GLL$oT'ym&FgnFX7$$\"3'3++DE5!>eFgnFX7$$\"3Mm;a)3rf&fFgnFX7$$\"3*4++v W0d5'FgnFX7$$\"3;L$3-\"QfYiFgnFX7$$\"3C+]PWF'QR'FgnFX7$$\"3[LL$e/Xy`'F gnFX7$$\"3m**\\(=<\"e)o'FgnFX7$$\"3%ymmm(zvLoFgnFX7$$\"3-nm\"zAAA)pFgn FX7$$\"3LM$3-7d%HrFgnFX7$$\"3#4++]p]ZE(FgnFX7$$\"3$QL$e*R7)>uFgnFX7$$ \"3'pmmmV,&evFgnFX7$$\"3<+](o(GP1xFgnFX7$$\"3g+]78Z!z%yFgnFX7$$\"\")F* FX-FM6&FOFSFPFS-%+AXESLABELSG6$Q!6\"Fjw-%%VIEWG6$;F(Fcw%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "Eventually the expansion \+ does diverge! The message is: for small coupling perturbation theory ( even though it can result in a semi-convergent series) can be extremel y useful. This is particularly true for problems where many terms of t he expansion can be calculated symbolically or by numerical techniques ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Exe rcise 1:" }}{PARA 0 "" 0 "" {TEXT -1 142 "Pick a different power-law p otential (possibly a linear combination). 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