{VERSION 2 3 "SGI IRIS UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Define a package named QMf unctions which contains the common functions" }}{PARA 0 "" 0 "" {TEXT -1 29 "needed for quantum mechanics." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "QMfunctions := table():" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Define the Legendre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "QMfunctions[Legendre] := proc(n::nonnegint, x::a nything)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " if n=0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " RETURN(1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " elif n=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " RETURN(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " expand( ((2*n-1)*x*Legendre(n-1,x) - (n-1)*Legendre (n-2,x))/n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "First derivative of the Legendre polynomi als:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "QMfunctions[dLegend re] := proc(n::nonnegint, x::anything)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " if \+ n = 0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " RETURN(0);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " elif n = 1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " RETURN(1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " expand( n*Legendre( n-1,x) + x * dLegendre(n-1,x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "e nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Define the associated Lege ndre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "QMfunc tions[aLegendre] := proc(n::nonnegint, m::nonnegint, x::anything)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " if m = 0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " RETURN(Legendre(n,x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " elif m = 1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " RETURN(sq rt(1-x^2)*dLegendre(n,x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 " factor( 2*(m-1)*x/sqrt(1-x^2) *aLegendre(n,m-1,x)+((m-1)*(m-2)-n*(n+1))*aLegendre(n,m-2,x) );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Define the spherical harmonics:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "QMfunctions[SphericalHarmonic] := proc(n::nonneg int, m::integer, theta::anything, phi::anything)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " if m < 0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 " RETURN( \+ sqrt( (2*n+1)*(n+m)!/4/Pi/(n-m)! ) * exp(I*m*phi) * aLegendre(n,-m,cos (theta)) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 " (-1)^m * sqrt( (2*n+1)*(n-m)!/4/Pi/(n+m)! ) \+ * exp(I*m*phi) * aLegendre(n,m,cos(theta));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Define th e Spherical Bessel functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "QMfunctions[SphericalBessel] := proc(n::nonnegint, x::anything) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 13 " if n=0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " RETURN( sin(x)/x );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " elif n=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " RETURN( sin( x)/x^2 - cos(x)/x );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 " simplify( (2*n-1)*SphericalBessel (n-1,x)/x - SphericalBessel(n-2,x) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Define Spherical Neu mann functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "QMfuncti ons[SphericalNeumann] := proc(n::nonnegint, x::anything) option remem ber;\n if n=0 then\n RETURN( -cos(x)/x );\n elif n=1 then\n RE TURN( -cos(x)/x^2 - sin(x)/x );\n fi;\n simplify( (2*n-1)*SphericalN eumann(n-1,x)/x - SphericalNeumann(n-2,x) );\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Define Laguerre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "QMfunctions[Laguerre] := proc(n::nonnegi nt, x::anything) option remember;\n if n=0 then\n RETURN(1);\n e lif n=1 then\n RETURN(1-x);\n fi;\n expand( ((2*n-1-x)*Laguerre(n -1,x) - (n-1)^2*Laguerre(n-2,x)));\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Define Associated Laguerre polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "QMfunctions[aLaguerre] := proc(n::nonnegi nt, m::nonnegint, x::anything)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " \+ option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " if n=0 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " RETURN(m!);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 " elif n=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " RETURN((m+1)!*(1+m-x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 " expand(((2*n-1-x+m)*( n+m)*aLaguerre(n-1,m,x) - (n+m)*(n-1+m)^2*aLaguerre(n-2,m,x))/n);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Define Hermite polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "QMfunctions[Hermitep] := proc(n::nonnegint, x::anythi ng)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " if n=0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " RETURN(1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " elif n=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " RETURN(2*x); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " expand( 2*x*Hermitep(n-1,x) - 2*(n-1)*Hermitep(n-2, x) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Define the radial wave functions for the hydrogen at om:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "QMfunctions[Hydrogen ] := proc(n::nonnegint, k::nonnegint, a::anything, r::anything)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 " factor( sqrt( (2/n/a)^3*(n-k-1)!/2/n/((n+k)!) ^3 )*exp(-r/n/a)*(2*r/n/a)^k * aLaguerre(n-k-1,2*k+1,2*r/n/a));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Include the package QMfunctions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with(QMfunctions):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Save the functions into a library that can be called lat er on. Use back-quotes to specify the directory on your" }}{PARA 0 " " 0 "" {TEXT -1 111 "machine where you wish to save the package QMfunc tions. Then you can use this later by starting a session with" }} {PARA 0 "" 0 "" {TEXT -1 70 "> ibname := `/usr2/people/griff/courses/P hys313/maplepacks`, libname; " }}{PARA 0 "" 0 "" {TEXT -1 20 "> with(Q Mfunctions):" }}{PARA 0 "" 0 "" {TEXT -1 117 "Everything will then be \+ ready for use. If you use this worksheet, each time you wish to do a \+ calculation you'll need" }}{PARA 0 "" 0 "" {TEXT -1 84 "to specificall y load each function by clicking on its definition and hitting return. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "save( QMfunctions, `/us r2/people/griff/courses/Phys313/maplepacks/QMfunctions.m`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "Play with the functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Legendre(1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Legendre(2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"##\"\"$F&#!\"\"F&\"\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Legendre(3,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"$#\"\"&\"\"#F%#!\"$F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Legendre(3,cos(theta));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%$cosG6#%&thetaG\"\"$#\"\"&\"\"#F %#!\"$F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Legendre(5,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\"&#\"#j\"\")*$F%\"\"$# !#N\"\"%F%#\"#:F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "aLegen dre(3,2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"xG\"\"\",&F%F&! \"\"F&F&,&F%F&F&F&F&!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " dLegendre(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"##\"#: F&#!\"$F&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dLegendr e(3,cos(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%$cosG6#%&the taG\"\"##\"#:F)#!\"$F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dLegendre(6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\" &#\"$$p\"\")*$F%\"\"$#!$:$\"\"%F%#\"$0\"F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "aLegendre(1,0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "aLegendre(4,3,co s(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**-%$cosG6#%&thetaG\" \"\",&F%F)!\"\"F)\"\"#,&F%F)F)F)F,,$*&F*F)F-F)F+#F+F,\"$0\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor(aLegendre(4,3,x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"xG\"\"\",&F%F&!\"\"F&\"\"#,&F% F&F&F&F),$*&F'F&F*F&F(#F(F)\"$0\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "SphericalHarmonic(3,-1,theta,phi);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*,\"#@#\"\"\"\"\"#%#PiG#!\"\"F(-%$expG6#,$*&%\"IGF' %$phiGF'F+F',&F'F'*$-%$cosG6#%&thetaGF(F+F&,&F4#\"#:F(#!\"$F(F'F'#F'\" #7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "SphericalBessel(2,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(-%$sinG6#%\"xG!\"$*&-%$cosGF (\"\"\"F)F.\"\"$*&F&F.F)\"\"#F.F.F)F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "SphericalNeumann(21,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,N*&%\"xG\"#:-%$sinG6#F'\"\"\"!-+ZA02=*&F'\"\"%-%$c osGF+F,\"9vV))=)f.O)GXuY*&F'\"#6F)F,!2D'e!ykUPa&*&F'\"#8F)F,\"0vGI_=cP \"*&F'\"\"*F)F,\"5+v*G7*&F'\"#5F0F,!3+!Q*[O#))*p))*&F'\"\"'F0F,! 8++fnC$RR%Qd?\"*&F'\"\"(F)F,!7+]Zz&RS!HSN9*&F'\"\")F0F,\"6+v%z&RS!HSN9 *&F)F,F'F,\";D1W.'))zod/28J\"*&F'\"#9F0F,!.+5uc6U&*&F'\"\"&F)F,\"8Dc!R \\1HB/\"[O)*&F)F,F'\"\"$!:Dc;:#f:c\\ifD?*&F'\"#7F0F,\"1Dc*)*z10)H*&F0F ,F'\"\"#!:+DJ;abdat>mR'*&F'\"#;F0F,\"+v&*e>]*&F'\"#@F)F,F,*&F'\"#F)F,!&ll#F0FI*&F'\"#?F0F,\"$J#F,F'! #A!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Laguerre(6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%\"xG!%?V\"$?(\"\"\"*$F$\"\"#\"%+a *$F$\"\"$!%+C*$F$\"\"%\"$]%*$F$\"\"&!#O*$F$\"\"'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "aLaguerre(2,3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"%+7\"\"\"%\"xG!$+'*$F&\"\"#\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Hermitep(9,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"*\"$7&*$F%\"\"(!%;#**$F%\"\"&\"&%Q[*$F%\" \"$!&S1)F%\"&S-$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl ots);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7S%(animateG%*animate3dG%-cha ngecoordsG%,complexplotG%.complexplot3dG%*conformalG%,contourplotG%.co ntourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,densityplotG%(d isplayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+gradplot3dG% -implicitplotG%/implicitplot3dG%(inequalG%-listcontplotG%/listcontplot 3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%(logplotG%+ matrixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3dG%*polarplotG%, polygonplotG%.polygonplot3dG%.polyhedraplotG%'replotG%*rootlocusG%,sem ilogplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixplotG% +sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([SphericalBessel(3,x), Spheric alBessel(4,x)],x=0..30);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7 ^s7$$\"+]i9Rl!#5$\"+nzO+E!#77$$\"1+++XV)RQ*!#;$\"1=R_\"zmD\\(!#=7$$\"1 +++WA)GA\"!#:$\"1L)4H6#z,;!#<7$$\"1+++TS\"Ga\"F8$\"1F1)p]E#fIF;7$$\"1+ ++Qeui=F8$\"1p*fpF17$$\"1+++WTAUPF8$\"1smj. 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