{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 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 "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 "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 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 16 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 35 "The H-atom in parabolic \+ coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "Our aim here is to demonstrate what some hydrogen atom eigenfu nctions look like when expressed in parabolic coordinates. A few point s are demonstrated by this:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "1) for a pure Coulomb potential the separation in spherical polar coordinates is not the only way to proceed; eigenf unctions can be found which are not eigenfunctions of the angular mome ntum operator " }{TEXT 19 3 "L^2" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 158 "2) for a pure Coulomb potential some additional symmetri es are present beyond spherical symmetry; these symmetries (conservati on law) are responsible for the " }{TEXT 258 1 "n" }{TEXT -1 1 "-" } {TEXT 257 1 "l" }{TEXT -1 212 " degeneracy of the spectrum; they make \+ the separation possible in another set of coordinates; the conservatio n law behind this symmetry is the conservation of the Laplace-Runge-Le nz vector (as an operator in QM);" }}{PARA 0 "" 0 "" {TEXT -1 126 "3) \+ the usual angular momentum eigenstates can be expressed as linear comb inations of the eigenstates in parabolic coordinates;" }}{PARA 0 "" 0 "" {TEXT -1 152 "4) the eigenstates in parabolic coordinates are usefu l when additional interactions are turned on, such as a constant elect ric field along the (chosen) " }{TEXT 259 1 "z" }{TEXT -1 6 "-axis." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "To save time on computational steps we do not re-derive the energy spectrum ( although it can be done!). Our purpose is to demonstrate what the prob ability (charge) density looks like for the new eigenstates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The new eigenst ates are classified by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 1 "m" }{TEXT -1 26 " - quantum number for the " }{TEXT 19 3 "L_z" }{TEXT -1 25 " eigenvalues (as before);" }}{PARA 0 "" 0 "" {TEXT 19 2 "n1" }{TEXT -1 72 " - quantum number for the first 'radial' function in the xi-coordinate, " }{TEXT 19 6 "xi=r+z" }{TEXT -1 1 "; " }}{PARA 0 "" 0 "" {TEXT 19 2 "n2" }{TEXT -1 74 " - quantum number fo r the second 'radial' function in the eta-coordinate, " }{TEXT 19 7 "e ta=r-z" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The principal quantum number n (which determines th e eigenenergies) is related to the three above as" }}{PARA 0 "" 0 "" {TEXT 19 19 "n = n1 + n2 + m + 1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "For details consult the b ook by HA Bethe and EE Salpeter: " }{TEXT 256 48 "Quantum Mechanics of One- and Two-Electron Atoms" }{TEXT -1 133 "; chapter 6. Just keep in mind that Bethe and Salpeter do not follow the usual convention for a ssociated Laguerre polynomials, their " }{XPPEDIT 18 0 "L[n+m]^m;" "6# )&%\"LG6#,&%\"nG\"\"\"%\"mGF)F*" }{TEXT -1 11 " is really " }{XPPEDIT 18 0 "L[n]^m;" "6#)&%\"LG6#%\"nG%\"mG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=14:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots) :" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b een redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We choose a va lue for the magnetic quantum number." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "SE:=(E,Zp,f)->diff(s*diff (f,s),s)+(E/2*s-m^2/4/s+Zp)*f=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #SEGj+6%%\"EG%#ZpG%\"fG6\"6$%)operatorG%&arrowGF*/,&-%%diffG6$*&%\"sG \"\"\"-F16$9&F4F5F4F5*&,(*&#F5\"\"#F5*&9$F5F4F5F5F5*(\"\"%!\"\"%\"mGF= F4FBFB9%F5F5F8F5F5\"\"!F*F*F*6%FEFEFE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SE(E,Z1,f(s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( -%%diffG6$-%\"fG6#%\"sGF+\"\"\"*&F+F,-F&6$F(-%\"$G6$F+\"\"#F,F,*&,&*(F 3!\"\"%\"EGF,F+F,F,%#Z1GF,F,F(F,F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Here s stands for either of the two parabolic coordinate s, " }{TEXT 19 2 "xi" }{TEXT -1 4 " or " }{TEXT 19 3 "eta" }{TEXT -1 78 ". For the pure Coulomb problem the two differential equations for \+ the factors " }{TEXT 19 6 "f1(xi)" }{TEXT -1 5 " and " }{TEXT 19 7 "f2 (eta)" }{TEXT -1 73 " are identical. This will change when we add an e lectric field along the " }{TEXT 261 1 "z" }{TEXT -1 6 "-axis." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Let us il lustrate the meaning of the parabolic coordinates by drawing lines of \+ constant " }{TEXT 19 2 "xi" }{TEXT -1 5 " and " }{TEXT 19 3 "eta" } {TEXT -1 10 " over the " }{TEXT 263 1 "x" }{TEXT -1 1 "-" }{TEXT 262 1 "z" }{TEXT -1 10 " plane at " }{TEXT 264 1 "y" }{TEXT -1 48 "=0. Thi s choice of plane really illustrates the " }{TEXT 19 3 "rho" }{TEXT -1 1 "-" }{TEXT 265 1 "z" }{TEXT -1 13 " plane where " }{TEXT 19 3 "rh o" }{TEXT -1 77 " is the cylindrical coordinate measuring the distance from the rotation axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eq1:=sqrt(x^2+z^2)+z=c1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/, &*$,&*$)%\"xG\"\"#\"\"\"F-*$)%\"zGF,F-F-#F-F,F-F0F-%#c1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eq2:=sqrt(x^2+z^2)-z=c2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&*$,&*$)%\"xG\"\"#\"\"\"F-*$)%\"zGF ,F-F-#F-F,F-F0!\"\"%#c2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1 to 10 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "PL1[i]: =implicitplot(lhs(eq1)=i*0.5,z=-5..5,x=-5..5,color=magenta,numpoints=5 000); PL2[i]:=implicitplot(lhs(eq2)=i*0.5,z=-5..5,x=-5..5,color=green, numpoints=5000); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dis play([seq(PL1[i],i=1..10),seq(PL2[i],i=1..10)],scaling=constrained);" }}{PARA 13 "" 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\\F-7$$!3+(>UwLkOQ\"Fc[nF57$Fgc`pFba`p7$F`c`p7$$!3p&*)\\x&f%Hu\"Fc[n$ \"3IT'=(>>=A[F-7$7$$!3%>EUwLkOQ\"Fc[nF^`jmF\\d`p7$7$Fcd`pFchx7$$!3!HBm .K^*[5Fc[n$\"3s4GSEvi$*[F-7$7$Fegn$\"3jy7N=V2H\\F-Fgd`p7$7$$\"3Wj))3Sv 3f\\FdjbmF(Fdc`p7$7$Fegn$\"3ux7N=V2H\\F-7$$!3e4,,!R]h\\$Fggn$\"3iA))oo *QX'\\F-7$7$$\"3cyy,Sv3f\\FdjbmFi]^lFhe`pFfajm-%+AXESLABELSG6%%\"zG%\" xG-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 21 "This plot shows that:" }}{PARA 0 "" 0 "" {TEXT -1 86 "1) the coordinates are orthogonal; (in a scale-constrained plot the line s of constant " }{TEXT 19 6 "xi/eta" }{TEXT -1 27 " intersect at right angles)" }}{PARA 0 "" 0 "" {TEXT -1 88 "2) when the coordinates are v aried from 0 to infinity they define a mesh that spans the " }{TEXT 269 1 "x" }{TEXT -1 1 "-" }{TEXT 268 1 "z" }{TEXT -1 38 " plane; we us e that plane at positive " }{TEXT 19 1 "x" }{TEXT -1 18 " and rotate i t by " }{TEXT 19 3 "phi" }{TEXT -1 11 " about the " }{TEXT 266 1 "z" } {TEXT -1 12 "-axis. Then " }{TEXT 267 1 "x" }{TEXT -1 42 " plays the r ole of cylindrical coordinate " }{TEXT 19 3 "rho" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 69 "3) The nucleus will be placed at the orig in of the coordinate system;" }}{PARA 0 "" 0 "" {TEXT -1 155 "4) In ad dition to axial symmetry, the coordinate system is reflective of the l eft-right symmetry, but not of the spherical symmetry in the Coulomb p roblem." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Define the energy in t erms of the principal quantum number and fix " }{TEXT 260 1 "Z" } {TEXT -1 59 " - the nuclear charge. We also introduce a scale paramete r " }{TEXT 19 7 "epsilon" }{TEXT -1 51 " for the coordinates which cha nges with the energy." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "En :=n->-Z^2/2/n^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnGj+6#%\"nG6\" 6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&%\"ZGF09$!\"#F/!\"\"F(F(F(6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Z:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "epsilon:=n->sqrt(-2*En(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonGj+6#%\"nG6\"6$%)operatorG%&arrowGF(-%%sqrtG6 #,$*&\"\"#\"\"\"-%#EnG6#9$F2!\"\"F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Now choose the principal quantum number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "En(n), epsilon(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"\"\"#=#\"\"\"\"\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 "Let us take the following at titude: we know the answers in sperical polar coordinates. Let us pick a certain principal quantum number n, we know the energy, let us then see which functions of eta and xi combined lead to allowed states (al l for fixed m). The problem is now an eigenvalue problem in Z1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "simplify(SE(En(n),Z1,exp(-ep silon(n)*s/2)*s^(m/2)*f(s)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*& #\"\"\"\"\"'F'*&-%$expG6#,$*&F(!\"\"%\"sGF'F/F',,-%\"fG6#F0F/*&F(F'-%% diffG6$F2F0F'F'*(\"\"#F'F0F'F6F'F/*(F(F'F0F'-F76$F2-%\"$G6$F0F:F'F'*(F (F'F2F'%#Z1GF'F'F'F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SE_1:=simplify(%/(s^(m/2)*exp(-s*epsilon(n)/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%SE_1G/,,*&#\"\"\"\"\"'F)-%\"fG6#%\"sGF)!\"\"-%% diffG6$F+F.F)*&#F)\"\"$F)*&F.F)F0F)F)F/*&F.F)-F16$F+-%\"$G6$F.\"\"#F)F )*&F+F)%#Z1GF)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "This eq uation can be made identical with the differential equation satisfied \+ by the associated Laguerre polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now pick a value for the " }{TEXT 19 2 "n1" }{TEXT -1 32 " quantum number (starting at 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n1:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We need the \+ associated Laguerre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "LagA:=(n,k,x)->expand(((n+k)!)*add((-x)^m/(n-m)!/(k+m )!/m!,m=0..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LagAGj+6%%\"nG% \"kG%\"xG6\"6$%)operatorG%&arrowGF*-%'expandG6#*&-%*factorialG6#,&9$\" \"\"9%F7F7-%$addG6$**),$9&!\"\"%\"mGF7-F36#,&F6F7FAF@F@-F36#,&F8F7FAF7 F@-F36#FAF@/FA;\"\"!F6F7F*F*F*6%FLFLFL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "LagA(2,k,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&# \"\"$\"\"#\"\"\"%\"kGF(F(*&#F(F'F(*$)F)F'F(F(F(F(F(*&F'F(%\"xGF(!\"\"* &F/F(F)F(F0*&F+F(*$)F/F'F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Maple offers these as generalized Laguerre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(LaguerreL(2,k,x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"$\"\"#\"\"\"%\"kGF(F(*&#F(F'F (*$)F)F'F(F(F(F(F(*&F'F(%\"xGF(!\"\"*&F/F(F)F(F0*&F+F(*$)F/F'F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f||n1:=LagA(n1,m,epsilon(n )*s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f0G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Z1:='Z1': simplify(subs(f(s)=f||n1, SE_1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,#\"\"\"\"\"'!\"\"-%%diff G6$F&%\"sGF&*&#F&\"\"$F&*&F,F&F)F&F&F(*&F,F&-F*6$F&-%\"$G6$F,\"\"#F&F& %#Z1GF&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify(%) ; # weird: why do we need a repeat of simplify?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"\"\"\"\"'!\"\"%#Z1GF&\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "Z1:=solve(%,Z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z1G#\"\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " Z2:=Z-Z1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z2G#\"\"&\"\"'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Can we find a solution for the sec ond function with that value?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 19 7 "Z2 = Z1" }{TEXT -1 33 " \+ then we are done (the functions " }{TEXT 19 2 "f1" }{TEXT -1 5 " and \+ " }{TEXT 19 2 "f2" }{TEXT -1 23 " will be identical and " }{TEXT 19 5 "n1=n2" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "If not, we have to find a solution with quantum numb er " }{TEXT 19 2 "n2" }{TEXT -1 11 " such that " }{TEXT 19 9 "Z2=Z - Z 1" }{TEXT -1 14 " is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 28 "Given \+ that the equation for " }{TEXT 19 7 "f2(eta)" }{TEXT -1 20 " is the sa me as for " }{TEXT 19 6 "f1(xi)" }{TEXT -1 74 " let us proceed by find ing the possible solutions for different values of " }{TEXT 19 2 "n1" }{TEXT -1 44 ". At the end we will combine allowed states " }{TEXT 19 14 "f1(xi)*f2(eta)" }{TEXT -1 11 " such that " }{TEXT 19 7 "Z=Z1+Z2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Now step through the next possible solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n1:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f||n1 :=LagA(n1,m,epsilon(n)*s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G,& \"\"\"F&*&\"\"$!\"\"%\"sGF&F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 " We need to reset " }{TEXT 19 2 "Z1" }{TEXT -1 54 " to the unassigned s ymbol so that we can solve for it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Z1:='Z1':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(subs(f(s)=f||n1,SE_1));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,0#\"\"\"\"\"'!\"\"*&\"#=F(%\"sGF&F&-%%diffG6$,&F&F&*&\"\"$F(F+F&F( F+F&*&#F&F1F&*&F+F&F,F&F&F(*&F+F&-F-6$F/-%\"$G6$F+\"\"#F&F&%#Z1GF&*(F1 F(F " 0 "" {MPLTEXT 1 0 16 "Z1:=so lve(%,Z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z1G#\"\"\"\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "This is not the one we need, but o bviously we can combine Z1=1/2 with Z2=1/2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 11 "Exercise 1:" }}{PARA 0 "" 0 " " {TEXT -1 10 "For given " }{TEXT 19 3 "Z=1" }{TEXT -1 33 " find all t he possible values of " }{TEXT 19 8 "\{n1, Z1\}" }{TEXT -1 5 " for " } {TEXT 19 12 "n1=0,1,2,..." }{TEXT -1 6 " with " }{TEXT 19 6 "Z1 < Z" } {TEXT -1 41 ". These can be used to form combinations " }{TEXT 19 5 "n 1/n2" }{TEXT -1 11 " such that " }{TEXT 19 7 "Z1+Z2=Z" }{TEXT -1 62 ". Then investigate what happens for hydrogenlike atoms, e.g., " }{TEXT 19 3 "Z=2" }{TEXT -1 32 ". Are there more allowed states?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f2:= LagA(2,m,epsilon(n)*s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G,(\" \"\"F&*&#\"\"#\"\"$F&%\"sGF&!\"\"*&#F&\"#=F&*$)F+F)F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Z1:='Z1':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(subs(f(s)=f2,SE_1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4#\"\"\"\"\"'!\"\"*&\"\"*F(%\"sGF&F&*&\"$3\"F(F+\"\"# F(-%%diffG6$,(F&F&*&#F.\"\"$F&F+F&F(*&#F&\"#=F&*$)F+F.F&F&F&F+F&*&#F&F 5F&*&F+F&F/F&F&F(*&F+F&-F06$F2-%\"$G6$F+F.F&F&%#Z1GF&**F.F&F5F(FDF&F+F &F(*(F8F(FDF&F+F.F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(%,Z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Thus, we have 3 candidates for " } {TEXT 286 1 "m" }{TEXT -1 5 "=0, " }{TEXT 287 1 "n" }{TEXT -1 3 "=3: " }}{PARA 0 "" 0 "" {TEXT -1 43 "\{n1=0 , n2=2\}, \{n1=1 , n2=1\}, \{n 1=2 , n2=0\}" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Three other states are possible at this level in the usual repr esentation:" }}{PARA 0 "" 0 "" {TEXT -1 13 "3s, 3p0, 3d0." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 69 "Now let us plot the density of the asymmetric state (2, 0) [or (0,2)]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "psi:=eval (exp(-epsilon(n)*s/2)*s^(m/2)*f2,s=xi)*eval(exp(-epsilon(n)*s/2)*s^(m/ 2)*f0,s=eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG*(-%$expG6#,$* &\"\"'!\"\"%#xiG\"\"\"F,F.,(F.F.*&#\"\"#\"\"$F.F-F.F,*&#F.\"#=F.*$)F-F 2F.F.F.F.-F'6#,$*&F+F,%$etaGF.F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* &#\"\"\"\"#=F&*&-%$expG6#,&*&\"\"'!\"\"%#xiGF&F/*&F.F/%$etaGF&F/F&,(F' F&*&\"#7F&F0F&F/*$)F0\"\"#F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "psi_1:=simplify(eval(psi,\{eta=r-z,xi=r+z\}));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_1G,$*&#\"\"\"\"#=F(*&-%$expG6#, $*&\"\"$!\"\"%\"rGF(F1F(,.F)F(*&\"#7F(F2F(F1*&F5F(%\"zGF(F1*$)F2\"\"#F (F(*(F:F(F2F(F7F(F(*$)F7F:F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "psi_1:=eval(psi_1,r=sqrt(x^2+y^2+z^2));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&psi_1G,$*&#\"\"\"\"#=F(*&-%$expG6#,$*&\"\"$! \"\",(*$)%\"xG\"\"#F(F(*$)%\"yGF6F(F(*$)%\"zGF6F(F(#F(F6F1F(,0F)F(*&\" #7F(F2F=F1*&F@F(F " 0 "" {MPLTEXT 1 0 23 "psi_1:=eval(psi_1,y=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_1G,$*&#\"\"\"\"#=F(*&-%$expG6#, $*&\"\"$!\"\",&*$)%\"xG\"\"#F(F(*$)%\"zGF6F(F(#F(F6F1F(,.F)F(*&\"#7F(F 2F:F1*&F=F(F9F(F1F3F(*&F6F(F8F(F(*(F6F(F2F:F9F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rho_1:=(psi_1)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rho_1G,$*&#\"\"\"\"$C$F(*&)-%$expG6#,$*&\"\"$!\" \",&*$)%\"xG\"\"#F(F(*$)%\"zGF7F(F(#F(F7F2F7F(),.\"#=F(*&\"#7F(F3F;F2* &F@F(F:F(F2F4F(*&F7F(F9F(F(*(F7F(F3F;F:F(F(F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "There are various ways to represent the probabi lity (or charge) density:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 "We could simply graph the density over " }{TEXT 285 1 "x" }{TEXT -1 5 " and " }{TEXT 284 1 "z" }{TEXT -1 5 " (at " } {TEXT 283 1 "y" }{TEXT -1 4 "=0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 119 "However, we are more interested in a gra ph that displays the probability to find the electron a certain distan ce along " }{TEXT 278 1 "z" }{TEXT -1 72 " (that's the coordinate that did not get transformed, i.e., cylindrical " }{TEXT 279 1 "z" }{TEXT -1 13 " = cartesian " }{TEXT 280 1 "z" }{TEXT -1 18 "), and a distance " }{TEXT 282 1 "x" }{TEXT -1 132 " away from the rotation axis. For t he latter we include the Jacobian element in cylindrical coordinates, \+ i.e., we show the distance " }{TEXT 281 1 "x" }{TEXT -1 1 "=" }{TEXT 19 3 "rho" }{TEXT -1 19 " times the density." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot3d(abs(x)*rho_1,x=-25..25,z=-25..25,axes=box ed,shading=zhue,style=patchcontour,numpoints=5000);" }}}{EXCHG {PARA 13 "" 1 "" {GLPLOT3D 562 368 368 {PLOTDATA 3 "6'-%%GRIDG6%;$!#D\"\"!$ \"#DF)F&X,%)anythingG6\"6\"[gl'!%\"!!#[__l\"co\"co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133F234ABADDC9BC293F1F6B672204DAC73F19658767CA77993F1461D33B285EBC3F10 3EDE96C010993F09B9DBCDB9C0933F043D64ED78AA47-%'COLOURG6#%%ZHUEG-%&STYL EG6#%-PATCHCONTOURG-%+AXESLABELSG6%%\"xG%\"zGQ!F/-%*AXESSTYLEG6#%$BOXG " 1 4 0 1 10 0 2 4 1 2 2 1.000000 40.000000 4.000000 0 0 "Curve 1" }}} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Now repeat for a symmetric state: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "psi:=eval(exp(-epsilon( n)*s/2)*s^(m/2)*f1,s=xi)*eval(exp(-epsilon(n)*s/2)*s^(m/2)*f1,s=eta); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG**-%$expG6#,$*&\"\"'!\"\"%# xiG\"\"\"F,F.,&F.F.*&\"\"$F,F-F.F,F.-F'6#,$*&F+F,%$etaGF.F,F.,&F.F.*&F 1F,F6F.F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(psi );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"*F&*(,&\"\"$!\"\" %#xiGF&F&,&F*F+%$etaGF&F&-%$expG6#,&*&\"\"'F+F,F&F+*&F4F+F.F&F+F&F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "psi_1:=simplify(eval(ps i,\{eta=r-z,xi=r+z\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_1G,$ *&#\"\"\"\"\"*F(*(,(\"\"$!\"\"%\"rGF(%\"zGF(F(,(F,F-F.F(F/F-F(-%$expG6 #,$*&F,F-F.F(F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ps i_1:=eval(psi_1,r=sqrt(x^2+y^2+z^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_1G,$*&#\"\"\"\"\"*F(*(,(\"\"$!\"\"*$,(*$)%\"xG\"\"#F(F(*$)% \"yGF3F(F(*$)%\"zGF3F(F(#F(F3F(F9F(F(,(F,F-F.F(F9F-F(-%$expG6#,$*&F,F- F/F:F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psi_1:=eval (psi_1,y=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_1G,$*&#\"\"\"\" \"*F(*(,(\"\"$!\"\"*$,&*$)%\"xG\"\"#F(F(*$)%\"zGF3F(F(#F(F3F(F6F(F(,(F ,F-F.F(F6F-F(-%$expG6#,$*&F,F-F/F7F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "rho_1:=(psi_1)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&rho_1G,$*&#\"\"\"\"#\")F(*(),(\"\"$!\"\"*$,&*$)%\"xG\"\"#F(F(*$)% \"zGF4F(F(#F(F4F(F7F(F4F(),(F-F.F/F(F7F.F4F()-%$expG6#,$*&F-F.F0F8F.F4 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot3d(abs(x)*rho _1,x=-25..25,z=-25..25,axes=boxed,shading=zhue,style=patchcontour,nump oints=5000);" }}{PARA 13 "" 1 "" {GLPLOT3D 562 368 368 {PLOTDATA 3 "6' -%%GRIDG6%;$!#D\"\"!$\"#DF)F&X,%)anythingG6\"6\"[gl'!%\"!!#[__l\"co\"c o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energy remains unaffected by an electric field along the " } {TEXT 274 1 "z" }{TEXT -1 85 "-axis. The contributions to the added po tential energy expectation value at positive " }{TEXT 273 1 "z" } {TEXT -1 37 " is cancelled by those from negative " }{TEXT 272 1 "z" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 75 "Construct surface plots of the density for other s tates. Given a choice of " }{TEXT 276 1 "n" }{TEXT -1 5 " and " } {TEXT 277 1 "m" }{TEXT -1 19 ", find the allowed " }{TEXT 19 5 "n1/n2 " }{TEXT -1 31 " pairs and graph the densities." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }