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For some reason maple9.03 or 9.5 doesn't figure this one out. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The angular parts of any cent ral-force problem are spherical harmonics." }}}{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 53 "Plm:=proc(theta,l::nonnegint,m::integer) local x,y,f; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x:=cos(theta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "if m>0 then f:=subs(y=x,diff(P(l,y),y$m));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "else f:=subs(y=x,P(l,y)); fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(-1)^m*sin(theta)^m*f; end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Plm(theta,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#:\"\"\")-%$sinG6#%&thetaG\"\"#F&-%$cosGF *F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "For the spherical harmon ics we don't need the Plm's with negative argument." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Y:=proc(theta,phi,l::nonnegint,m::integer ) local m1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "m1:=abs(m); if m1>l \+ then RETURN(\"|m\} has to be <= l for Y_lm\"); fi;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "exp(I*m*phi)*Plm(theta,l,m1)*(-1)^m*sqrt((2*l+1)*(l -m1)!/(4*Pi*(l+m1)!)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\" \"#F&*(,&*&#\"\"&F'F&*$)-%$cosG6#%&thetaG\"\"$F&F&F&*&#F3F'F&F/F&!\"\" F&\"\"(F%%#PiG#F6F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " Y(theta,phi,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#7F& *,-%$expG6#*&%$phiGF&^#F&F&F&-%$sinG6#%&thetaGF&,&*&#\"#:\"\"#F&*$)-%$ cosGF1F7F&F&F&#\"\"$F7!\"\"F&\"#@#F&F7%#PiG#F>F7F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "No:=(l,m)->int(int(evalc(Y(theta,phi,l,m )*conjugate(Y(theta,phi,l,m))),phi=0..2*Pi)*sin(theta),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NoGj+6$%\"lG%\"mG6\"6$%)operatorG %&arrowGF)-%$intG6$*&-F.6$-%&evalcG6#*&-%\"YG6&%&thetaG%$phiG9$9%\"\" \"-%*conjugateG6#F7F>/F;;\"\"!,$*&\"\"#F>%#PiGF>F>F>-%$sinG6#F:F>/F:;F DFHF)F)F)6$FDFD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1,0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "No(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Suppose I need to calculate th e electrostatic potential for a wavefunction in a superposition of hyd rogenic orbitals." }}{PARA 0 "" 0 "" {TEXT -1 52 "What is the angular \+ dependence of the product terms?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "IP:=(f,g)->int(int(evalc(g*conjugate(f)),phi=0..2*Pi) *sin(theta),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IPGj+6 $%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%$intG6$*&-F.6$-%&evalcG6#*&9%\" \"\"-%*conjugateG6#9$F8/%$phiG;\"\"!,$*&\"\"#F8%#PiGF8F8F8-%$sinG6#%&t hetaGF8/FH;F@FDF)F)F)6$F@F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IP(Y(theta,phi,1,0),Y(theta,phi,3,0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The requi red matrix element involves as the operator the position vector " } {TEXT 19 9 "[x, y, z]" }{TEXT -1 14 " expressed as " }{TEXT 19 57 " r* [sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)]" }}{PARA 0 "" 0 "" {TEXT -1 82 "The radial part of the matrix element is defined fir st. We need to set the charge:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Z:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "RME:=(n,l,np,lp)->int(r^3*R( n,l,r)*R(np,lp,r),r=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$RMEGj+6&%\"nG%\"lG%#npG%#lpG6\"6$%)operatorG%&arrowGF+-%$intG6$*()% \"rG\"\"$\"\"\"-%\"RG6%9$9%F4F6-F86%9&9'F4F6/F4;\"\"!%)infinityGF+F+F+ 6&FBFBFBFB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "RME(2,1,1,0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$G\"\"\"\"\"$V#!\"\"\"\"'#F& \"\"#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Now the part where the polar and azimuthal angles theta and phi are integrated:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "AME:=(l,m,lp,mp)->abs(IP(Y(theta,p hi,l,m),sin(theta)*cos(phi)*Y(theta,phi,lp,mp)))^2+abs(IP(Y(theta,phi, l,m),sin(theta)*sin(phi)*Y(theta,phi,lp,mp)))^2+abs(IP(Y(theta,phi,l,m ),cos(theta)*Y(theta,phi,lp,mp)))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$AMEGj+6&%\"lG%\"mG%#lpG%#mpG6\"6$%)operatorG%&arrowGF+,(*$)-%$abs G6#-%#IPG6$-%\"YG6&%&thetaG%$phiG9$9%*(-%$sinG6#F;\"\"\"-%$cosG6#F \+ " 0 "" {MPLTEXT 1 0 13 "AME(1,0,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The phase-spac e factor " }{TEXT 19 7 "omega^3" }{TEXT -1 23 " is defined now, where \+ " }{TEXT 19 10 "hbar*omega" }{TEXT -1 114 " is the photon energy calcu lated from the difference of electronic energy levels. We choose atomi c units in which " }{TEXT 19 10 "hbar=m=e=1" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ED:=(n,np)->-Z^2/2*(1/n^2-1/ np^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EDGj+6$%\"nG%#npG6\"6$%)o peratorG%&arrowGF),$*&#\"\"\"\"\"#F0*&)%\"ZGF1F0,&*&F0F0*$)9$F1F0!\"\" F0*&F0F0*$)9%F1F0F:F:F0F0F:F)F)F)6$\"\"!F@" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "c:=137.04: alpha:=1/c:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "Acof:=(n,l,m,np,lp,mp)->evalf(RME(n,l,np,lp)^2*AME( l,m,lp,mp)*ED(n,np)^3*4/3*alpha/c^2,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AcofGj+6(%\"nG%\"lG%\"mG%#npG%#lpG%#mpG6\"6$%)operatorG%&arro wGF--%&evalfG6$,$*&#\"\"%\"\"$\"\"\"*,-%$RMEG6&9$9%9'9(\"\"#-%$AMEG6&F ?9&FA9)F9-%#EDG6$F>F@F8%&alphaGF9%\"cG!\"#F9F9F7F-F-F-6(\"\"!FOFO\")K1 eb\"'I)*e\")!=TV$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Acof(2 ,1,0,1,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%;:!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "This is the A coefficient in atomic unit s. To obtain the lifetime, we take the inverse, and convert to nanosec onds. The atomic time unit is approximately 2.4E-17 s?" }}{PARA 0 "" 0 "" {TEXT -1 15 "In nanoseconds:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "atu:=2.418884E-17; # the atomic time unit to 7 digits (it is known more accurately)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a tuG$\"(%))=C!#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(1/ Acof(2,1,0,1,0,0)*atu*1E9,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$f \"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The special case of max imal angular momentum states which can only decay to the nearest neigh bour:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "C1:=n->evalf(1/Aco f(n,n-1,n-1,n-1,n-2,n-2)*atu*1E9,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#C1Gj+6#%\"nG6\"6$%)operatorG%&arrowGF(-%&evalfG6$**\"\"\"F0-%%Aco fG6(9$,&F4F0F0!\"\"F5F5,&F4F0\"\"#F6F7F6%$atuGF0$F0\"\"*F0\"\"%F(F(F(6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "C1(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%'f\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(C1(n),n=2..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 65$\"%'f\"!\"$$\"%Y:!\"#$\"%XsF($\"%\\B!\"\"$\"%ugF-$\"%\\8\"\"!$\"%$o #F2$\"%3\\F2$\"%6%)F2$\"%o8\"\"\"$\"%J@F;$\"%+KF;$\"%kYF;$\"%=mF;$\"%y \"*F;$\"%Z7\"\"#$\"%m;FH$\"%*=#FH$\"%PGFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 243 "These numbers look right when compared to an analytic ev aluation of the integrals. In particular, they can be compared with th e closed-form expression (31) in M Seidl and P O Lipas: Eur. J. Phys. \+ 17, 25 (1996) for the lifetimes in nanoseconds:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "C1a:=n->eva lf(0.0622*3*n^4*(n-1)^2/(2*n-1)*(1+1/4/n/(n-1))^(2*n),4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$C1aGj+6#%\"nG6\"6$%)operatorG%&arrowGF(-%&ev alfG6$*.$\"$A'!\"%\"\"\"\"\"$F39$\"\"%,&F5F3F3!\"\"\"\"#,&*&F9F3F5F3F3 F3F8F8),&F3F3*&#F3F6F3*&F3F3*&F5F3F7F3F8F3F3,$*&F9F3F5F3F3F3F6F(F(F(6# \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(C1a(n),n=2..20 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "65$\"%%f\"!\"$$\"%W:!\"#$\"%UsF($ \"%ZB!\"\"$\"%qgF-$\"%[8\"\"!$\"%#o#F2$\"%0\\F2$\"%1%)F2$\"%o8\"\"\"$ \"%J@F;$\"%+KF;$\"%lYF;$\"%:mF;$\"%y\"*F;$\"%Y7\"\"#$\"%m;FH$\"%*=#FH$ \"%QGFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "For other initial sta tes (non-maximal angular momentum), the situation is more complicated \+ due to the presence of various decay channels. We begin by providing e xplicit expressions that sum over the allowed " }{TEXT 19 1 "m" } {TEXT -1 37 "-sublevels. The selection rules are: " }{TEXT 19 5 "+/- 1 " }{TEXT -1 4 " in " }{TEXT 19 1 "l" }{TEXT -1 10 " (L), and " }{TEXT 19 7 "0, +/-1" }{TEXT -1 4 " in " }{TEXT 19 1 "m" }{TEXT -1 32 ". For \+ the initial state we pick " }{TEXT 19 5 "n,l,m" }{TEXT -1 22 ". We nee d to know how " }{TEXT 19 4 "Acof" }{TEXT -1 32 " responds for non-all owed cases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Acof(2,1,0,2,2,0);" }}{PARA 8 "" 1 "" {TEXT -1 50 "Error, (in A) numeric exception: division by zero\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Thus, we need to protect the procedure from ill egal calls." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "AcofSp:=(n, l,m,np)->if l+1 < np then add(Acof(n,l,m,np,l+1,mp),mp=max(m-1,-l-1).. min(m+1,l+1)) else 0 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AcofSpG j+6&%\"nG%\"lG%\"mG%#npG6\"6$%)operatorG%&arrowGF+@%2,&9%\"\"\"F3F39'- %$addG6$-%%AcofG6(9$F29&F4F1%#mpG/F=;-%$maxG6$,&F " 0 " " {MPLTEXT 1 0 16 "AcofSp(2,1,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "AcofSp(4,1,1,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&PT)!#;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "AcofSp(4,1,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Why would the last one be forbidden? 4p1->2d final state does not exist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Now look at lowering " }{TEXT 19 2 " l" }{TEXT -1 17 " (L) by one unit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "AcofSm:=(n,l,m,np)->if l-1 < np and l-1> =0 then add(Acof(n,l,m,np,l-1,mp),mp=max(m-1,-l+1)..min(m+1,l-1)) else 0 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AcofSmGj+6&%\"nG%\"lG%\"m G%#npG6\"6$%)operatorG%&arrowGF+@%32,&9%\"\"\"F4!\"\"9'1\"\"!F2-%$addG 6$-%%AcofG6(9$F39&F6F2%#mpG/FA;-%$maxG6$,&F@F4F4F5,&F4F4F3F5-%$minG6$, &F@F4F4F4F2F8F+F+F+6&F8F8F8F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "AcofSm(4,2,1,2),AcofSm(4,2,1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%%*\\!#8$\"&Mq\"!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 " This shows the phase-space factor effect. The Einstein coefficient pro vides a decay rate: the decay from " }{TEXT 19 3 "n=4" }{TEXT -1 14 " \+ is faster to " }{TEXT 19 3 "n=2" }{TEXT -1 15 " than it is to " } {TEXT 19 3 "n=3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Now let us demonstrate that the decay rat e for the case when " }{TEXT 19 1 "l" }{TEXT -1 40 " decreases beats t he one for increasing " }{TEXT 19 1 "l" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "AcofSm(4,1,0,3),AcofSp(4,1,0,3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%Au!#9$\"%8%)!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "There is almost an order-of-magnitude differen ce. The answer is independent of the initial magnetic quantum number ( apart from inaccuracies)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "AcofSm(4,1,-1,3),AcofSp(4,1,-1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%Au!#9$\"&PT)!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 11 "Exe rcise 1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "Calculate the transition rates to all possible final states for | 3s>, |3p>, |3d>, and for |4s>, |4p>, |4d>, |4f> and compare them to a \+ table (e.g., Radzig+Smirnov). Then calculate the total transition rate s and lifetimes for these states." }}{PARA 0 "" 0 "" {TEXT -1 89 "Why \+ is it that |3s> has a finite lifetime within this framework, but |2s>, |4s> does not?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "70 3 0" 79 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }