{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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em phasis" -1 276 "Times" 1 12 96 52 84 1 0 1 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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Fourier transforms of Hermite fun ctions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., \+ Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "The Schrodinger wave equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 65 "The tim e-independent Schrodinger equation for the wave function " }{XPPEDIT 18 0 "y = Psi(x);" "6#/%\"yG-%$PsiG6#%\"xG" }{TEXT -1 21 " in one dime nsion is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-h^2/(8 *Pi^2*m)" "6#,$*&%\"hG\"\"#*(\"\")\"\"\"*$%#PiGF&F)%\"mGF)!\"\"F-" } {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+1/2;" "6#,&*(%\"dG\"\"#% \"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F(F(F&F,F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "m*omega^2*x^2*y = E*y;" "6#/**%\"mG\"\"\"*$%&omegaG\"\" #F&%\"xGF)%\"yGF&*&%\"EGF&F+F&" }{TEXT -1 14 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 273 1 "h" }{TEXT -1 26 " is Planck's constant and " }{TEXT 272 1 "E" } {TEXT -1 21 " is the total energy." }}{PARA 0 "" 0 "" {TEXT -1 58 "For time-independent states, the energies are quantized by" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E=(n+1/2)*h*nu" "6#/%\"EG*(,&%\" nG\"\"\"*&F(F(\"\"#!\"\"F(F(%\"hGF(%#nuGF(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "n = 0, 1, 2, 3,` . . . `" "6'/%\"nG\"\"!\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "nu=omega/(2*Pi)" "6#/%#nuG*&%&omegaG\"\"\"*&\"\"#F'%#Pi GF'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "It is conven ient to write this equation using the scaled dimensionless variable " }{XPPEDIT 18 0 "u = ``(2*Pi/sqrt(h/(m*nu)))*x;" "6#/%\"uG*&-%!G6#*(\" \"#\"\"\"%#PiGF+-%%sqrtG6#*&%\"hGF+*&%\"mGF+%#nuGF+!\"\"F5F+%\"xGF+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "x \+ = u*sqrt(h/(2*Pi*m*omega));" "6#/%\"xG*&%\"uG\"\"\"-%%sqrtG6#*&%\"hGF' **\"\"#F'%#PiGF'%\"mGF'%&omegaGF'!\"\"F'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = ``(2*Pi*m*omega/h);" "6#/*(%\"dG\"\"#% \"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"-%!G6#*,F&F(%#PiGF(%\"mGF(%&omegaGF(% \"hGF," }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*u^2)" "6#*(%\"dG\"\"# %\"yG\"\"\"*&F$F'*$%\"uGF%F'!\"\"" }{TEXT -1 18 ", so (i) becomes " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-h*omega/(4*Pi)" "6# ,$*(%\"hG\"\"\"%&omegaGF&*&\"\"%F&%#PiGF&!\"\"F+" }{TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*u^2) + h*omega/(4*Pi)" "6#,&*(%\"dG\"\"#%\"yG \"\"\"*&F%F(*$%\"uGF&F(!\"\"F(*(%\"hGF(%&omegaGF(*&\"\"%F(%#PiGF(F,F( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^2*y=E*y" "6#/*&%\"uG\"\"#%\"yG\"\" \"*&%\"EGF(F'F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*u^2) = (u^2 -4*Pi*E/(h*omega))*y" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"uGF&F(!\" \"*&,&*$F+F&F(**\"\"%F(%#PiGF(%\"EGF(*&%\"hGF(%&omegaGF(F,F,F(F'F(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Now rename the variable \+ " }{TEXT 274 1 "u" }{TEXT -1 4 " as " }{TEXT 275 1 "x" }{TEXT -1 69 ", and note that quantized values for the dimensionless scaled energy " }{XPPEDIT 18 0 "4*Pi*E/(h*omega) = (2*E)/(h*nu)" "6#/**\"\"%\"\"\"%#Pi GF&%\"EGF&*&%\"hGF&%&omegaGF&!\"\"*(\"\"#F&F(F&*&F*F&%#nuGF&F," } {TEXT -1 7 " are " }{XPPEDIT 18 0 "E[n] = 2*n+1" "6#/&%\"EG6#%\"nG,& *&\"\"#\"\"\"F'F+F+F+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "n = 0, 1, 2 , 3,` . . . `" "6'/%\"nG\"\"!\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 48 "This gives the family of differenti al equations " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2 *y/(d*x^2) = (x^2-2*n-1)*y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF &F(!\"\"*&,(*$F+F&F(*&F&F(%\"nGF(F,F(F,F(F'F(" }{TEXT -1 3 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "n = 0, 1, 2, 3,` . . . `" "6'/%\"nG\"\"!\"\" \"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Hermite polynomials" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 19 "Hermite polynomials" }{TEXT -1 1 " " } {XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 33 " satis fy the recurrence relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "H[n+1](x) = 2*x *H[n](x) - 2*n*H[n-1](x)" "6#/-&%\"HG6# ,&%\"nG\"\"\"F*F*6#%\"xG,&*(\"\"#F*F,F*-&F&6#F)6#F,F*F**(F/F*F)F*-&F&6 #,&F)F*F*!\"\"6#F,F*F9" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 0 "H[0](x)=1" "6#/-&%\"HG6#\"\"!6#%\"xG\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "H[1](x)=2*x" "6#/-&%\"HG6#\"\"\"6#% \"xG*&\"\"#F(F*F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "They form an " }{TEXT 261 17 "orthogonal \+ family" }{TEXT -1 16 " on the interval" }{XPPEDIT 18 0 "``(-infinity,i nfinity);" "6#-%!G6$,$%)infinityG!\"\"F'" }{TEXT -1 35 " with respect \+ to the inner product " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`.`*g(x) = Int(exp(-x^2)*f(x)*g(x),x = -infinity .. infinit y);" "6#/*(-%\"fG6#%\"xG\"\"\"%\".GF)-%\"gG6#F(F)-%$IntG6$*(-%$expG6#, $*$F(\"\"#!\"\"F)-F&6#F(F)-F,6#F(F)/F(;,$%)infinityGF8F@" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "exp(-x^2)" "6#-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 19 " is the associated " } {TEXT 261 15 "weight function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "They are available in Map le in the package " }{TEXT 0 9 "orthopoly" }{TEXT -1 76 ". To avoid co nfusion with the Heaviside function let's use the Maple symbol " } {TEXT 276 7 "Hm(n,X)" }{TEXT -1 28 " for the Hermite polynomial " } {XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "alias(Hm=orthopoly[H]):\nfor ct from 1 to 8 do 'H'[ct](x)=Hm(ct,x) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\"\"6#%\"xG,$* &\"\"#F(F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\"#6#%\" xG,&F(!\"\"*&\"\"%\"\"\")F*F(F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -&%\"HG6#\"\"$6#%\"xG,&*&\"\")\"\"\")F*F(F.F.*&\"#7F.F*F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\"%6#%\"xG,(\"#7\"\"\"*&\" #;F-)F*F(F-F-*&\"#[F-)F*\"\"#F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-&%\"HG6#\"\"&6#%\"xG,(*&\"#K\"\"\")F*F(F.F.*&\"$g\"F.)F*\"\"$F.!\" \"*&\"$?\"F.F*F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\"'6 #%\"xG,*\"$?\"!\"\"*&\"#k\"\"\")F*F(F0F0*&\"$![F0)F*\"\"%F0F-*&\"$?(F0 )F*\"\"#F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\"(6#%\"xG ,**&\"$G\"\"\"\")F*F(F.F.*&\"%W8F.)F*\"\"&F.!\"\"*&\"%gLF.)F*\"\"$F.F. *&\"%!o\"F.F*F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"HG6#\"\")6# %\"xG,,\"%!o\"\"\"\"*&\"$c#F-)F*F(F-F-*&\"%%e$F-)F*\"\"'F-!\"\"*&\"&SM \"F-)F*\"\"%F-F-*&F7F-)F*\"\"#F-F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Int(exp(-x^2)*'Hm(5,x)'*'Hm(8,x)',x=-infinity..infinity);\nvalue (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%$expG6#,$*$)%\"xG \"\"#\"\"\"!\"\"F/-%#HmG6$\"\"&F-F/-F26$\"\")F-F//F-;,$%)infinityGF0F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Solutions for the Schrodinger equation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "We start by finding the solution of" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = (x^2-2*n-1)*y;" "6#/*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&,(*$F+F&F(*&F&F(%\"nGF(F,F (F,F(F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "for the cas e " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }{TEXT -1 30 ", with the \+ initial conditions " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 16 " and y '(0) = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "de := diff(y(x),x$2)=(x^2-3 )*y(x);\nic := y(0)=0,D(y)(0)=1;\ndsolve(\{de,ic\},y(x)):\ng := unappl y(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"y G6#%\"xG-%\"$G6$F,\"\"#*&,&*$)F,F0\"\"\"F5\"\"$!\"\"F5F)F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(*&-%$expG6#,$*&#\"\"\"\"\"#F3*$)9$F4F3F3!\"\"F3F7F3F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(g(x),x=-5...5);" }}{PARA 13 "" 1 "" {GLPLOT2D 566 241 241 {PLOTDATA 2 "6%-%'CURVESG6$7do7$$!\"&\"\"!$!37N$RgeEL'=!#A7$$!3YLLLe%G ?y%!#<$!3]U,WYNFv^F-7$$!3OmmT&esBf%F1$!3!)G(\\.u+!47!#@7$$!3ALL$3s%3zV F1$!3O!)*R\"y\"H6+$F97$$!3_LL$e/$QkTF1$!3vw$oc/!4TrF97$$!3ommT5=q]RF1$ !3n_(fDeUAh\"!#?7$$!3ILL3_>f_PF1$!3-E6mwua%G$FI7$$!3K++vo1YZNF1$!3AdKA e^(\\c'FI7$$!3;LL3-OJNLF1$!3#H0M\")))>4G\"!#>7$$!3p***\\P*o%Q7$F1$!3o] 7sAr1vBFY7$$!3Kmmm\"RFj!HF1$!3Kn3+Vv`dUFY7$$!33LL$e4OZr#F1$!3LTEu&)*GR \"oFY7$$!3u*****\\n\\!*\\#F1$!3D#))=r0<15\"!#=7$$!3%)*****\\ixCG#F1$!3 avIHZ@2(o\"Fho7$$!3#******\\KqP2#F1$!3P>.1h=)\\T#Fho7$$!39LL3-TC%)=F1$ !3s<_ZW=)G>$Fho7$$!3[mmm\"4z)e;F1$!3))\\9MJFO!>%Fho7$$!3Mmmmm`'zY\"F1$ !3%oPhOF9^B-\" F1$!3=$Rf%z%*HigFho7$$!3S**\\(=n;R&**Fho$!3atIm#fx^1'Fho7$$!3_k\"zW#>K %o*Fho$!3/.ES)G+#fgFho7$$!3tIL3xrs9%*Fho$!3&4s!z[K9WgFho7$$!31i;H#oPb( ))Fho$!39#RJ$yz*f)fFho7$$!3S$***\\(=[jL)Fho$!3)*RD0+gQ*)eFho7$$!3q%*** *\\Pw%4tFho$!3kk7i+L(ef&Fho7$$!3)f***\\iXg#G'Fho$!3y?O=Eo`XE&\\XFho7$$!3ndmmT&Q(RTFho$!3/4B\"Ro)y*z$Fho7$$!3I hm\"HdGe:$Fho$!3aIoOS'GD+$Fho7$$!3%\\mmTg=><#Fho$!3+Vd313H@@Fho7$$!3g* **\\7yQ16\"Fho$!3l_?O%*)4Q5\"Fho7$$!3vDMLLe*e$\\FI$!3#\\_Z5Fho7$$\"3!=nm\"zRQb@Fho$\"3Q%f='e[*e5#Fh o7$$\"3mOLL$e,]6$Fho$\"3_^=$H7\"[nHFho7$$\"3_,+](=>Y2%Fho$\"3qgcUkJ.]P Fho7$$\"36QLe*[K56&Fho$\"3e+_;+GA&[%Fho7$$\"3summ\"zXu9'Fho$\"3[:3#HR* **)3&Fho7$$\"3#yLLe9i\"=sFho$\"3kH)*z59uibFho7$$\"3#4+++]y))G)Fho$\"3a 3M$Hs-!zeFho7$$\"3V,+DJ?i7))Fho$\"3#GOH,JHn(fFho7$$\"3%>++DcljL*Fho$\" 3CprpqQ.QgFho7$$\"3=-]7GtB)f*Fho$\"3Q:)3AJ*QbgFho7$$\"3W-+v$44,')*Fho$ \"3eq@wsT6kgFho7$$\"3Q+v$f3)>75F1$\"3CPz)H#ySkgFho7$$\"3H++]i_QQ5F1$\" 3w(QhLP([cgFho7$$\"3O+v=U,1*3\"F1$\"3r3]S8tr=gFho7$$\"3U+](=-N(R6F1$\" 3e$H*z/k#H&fFho7$$\"3[+Dc,*4/>\"F1$\"3Qb*)))=Z9heFho7$$\"3b++D\"y%3T7F 1$\"3ot$Fho7$$\"3?mm\"zpe*z?F1$\"31jGi%oc7R#Fho7$$\"3;,++D \\'QH#F1$\"3)p'3$3t\")=l\"Fho7$$\"3%HL$e9S8&\\#F1$\"3;#H*Rd**o46Fho7$$ \"3s++D1#=bq#F1$\"3M)4=BEuD'pFY7$$\"3\"HLL$3s?6HF1$\"3*)exW^'zX?%FY7$$ \"3a***\\7`Wl7$F1$\"3=DaIV<;dBFY7$$\"3enmmm*RRL$F1$\"3=E:7:2F'G\"FY7$$ \"3%zmmTvJga$F1$\"3EKJC%3tcf'FI7$$\"3]MLe9tOcPF1$\"3o#eIP'pdTKFI7$$\"3 1,++]Qk\\RF1$\"3bbq`&*3c=;FI7$$\"3![LL3dg6<%F1$\"3aYN/I^\\`pF97$$\"3%y mmmw(GpVF1$\"3(oGTt+Eb7$F97$$\"3C++D\"oK0e%F1$\"3G?? " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In general, if " } {TEXT 277 1 "n" }{TEXT -1 40 " is a non-ngative integer, the equation \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = ( x^2-2*n-1)*y" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&,(*$F +F&F(*&F&F(%\"nGF(F,F(F,F(F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "has the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y(x) = ``" "6#/-%\"yG6#%\"xG%!G" }{XPPEDIT 18 0 "h[n]( x) = 1/sqrt(2^n*n!);" "6#/-&%\"hG6#%\"nG6#%\"xG*&\"\"\"F,-%%sqrtG6#*&) \"\"#F(F,-%*factorialG6#F(F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "H[n ](x)*exp(-x^2/2)" "6#*&-&%\"HG6#%\"nG6#%\"xG\"\"\"-%$expG6#,$*&F*\"\"# F1!\"\"F2F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 8 " is the " }{TEXT 278 1 "n" }{TEXT -1 4 " th " }{TEXT 261 18 "Hermite polynomi al" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 73 "alias(Hm=orthopoly[H]):\nh := (n,x) -> 1/sqr t(2^n*n!)*Hm(n,x)*exp(-x^2/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" hGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*(-%%sqrtG6#*&)\"\"#9$\"\"\" -%*factorialG6#F4F5!\"\"-%#HmG6$F49%F5-%$expG6#,$*&#F5F3F5*$)F=F3F5F5F 9F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The factor " } {XPPEDIT 18 0 "1/sqrt(2^n*n!)" "6#*&\"\"\"F$-%%sqrtG6#*&)\"\"#%\"nGF$- %*factorialG6#F+F$!\"\"" }{TEXT -1 19 " is chosen so that " }{XPPEDIT 18 0 "Int(h[n](x)^2,x=-infinity..infinity) = sqrt(Pi)" "6#/-%$IntG6$*$ -&%\"hG6#%\"nG6#%\"xG\"\"#/F.;,$%)infinityG!\"\"F3-%%sqrtG6#%#PiG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(h(n,x)^2,x=-infinity..infinity);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&#\"\"\"\"*gXz&=F)*&), ,*&\"$7&F))%\"xG\"\"*F)F)*&\"%;#*F))F1\"\"(F)!\"\"*&\"&%Q[F))F1\"\"&F) F)*&\"&S1)F))F1\"\"$F)F7*&\"&S-$F)F1F)F)\"\"#F))-%$expG6#,$*&FBF7F1FBF 7FBF)F)F)/F1;,$%)infinityGF7FL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%# PiG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " } {XPPEDIT 18 0 "y(x)=1/sqrt(2^3*3!)" "6#/-%\"yG6#%\"xG*&\"\"\"F)-%%sqrt G6#*&\"\"#\"\"$-%*factorialG6#F/F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "H[3](x)*exp(-x^2/2);" "6#*&-&%\"HG6#\"\"$6#%\"xG\"\"\"-%$expG6#,$*& F*\"\"#F1!\"\"F2F+" }{TEXT -1 15 ", for the case " }{XPPEDIT 18 0 "n = 3" "6#/%\"nG\"\"$" }{TEXT -1 16 ", we have . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "y(x)=h(3,x); \ndiff(y(x),x$2)=normal(diff(h(3,x),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&#\"\"\"\"#7F+*(\"\"$#F+\"\"#,&*&\"\") F+)F'F.F+F+*&F,F+F'F+!\"\"F+-%$expG6#,$*&F0F6F'F0F6F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,(**\"\" (\"\"\"\"\"$#F2F.F*F2-%$expG6#,$*&F.!\"\"F*F.F:F2F2*&#\"# \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 62 "On the other hand, the right side of the different ial equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y /(d*x^2)=(x^2-7)*y(x)" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\" \"*&,&*$F+F&F(\"\"(F,F(-F'6#F+F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " becomes . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y(x)=h(3,x):\n(x^2-7)*y(x)=(x^2-7)* h(3,x);\n``=expand(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*$ )%\"xG\"\"#\"\"\"F*\"\"(!\"\"F*-%\"yG6#F(F*,$*&#F*\"#7F***F%F*\"\"$#F* F),&*&\"\")F*)F(F5F*F**&F3F*F(F*F,F*-%$expG6#,$*&F)F,F(F)F,F*F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(**\"\"(\"\"\"\"\"$#F(\"\"#%\"xGF (-%$expG6#,$*&F+!\"\"F,F+F2F(F(*&#\"# " 0 "" {MPLTEXT 1 0 21 "plot(h(3,x),x=-5..5);" }}{PARA 13 " " 1 "" {GLPLOT2D 581 242 242 {PLOTDATA 2 "6%-%'CURVESG6$7_q7$$!\"&\"\" !$!35$ey$)*GAc]!#@7$$!3YLLLe%G?y%!#<$!3Iz:sSi\"pF\"!#?7$$!3OmmT&esBf%F 1$!3m\"p7$$!3ommT5=q]RF1$!3=JtDoXVEEFD7$$!3ILL3_>f_PF1$!3%f;\"*Q Y:>x%FD7$$!3K++vo1YZNF1$!3-o$f?+xES)FD7$$!3;LL3-OJNLF1$!3.`%)>zB^B9!#= 7$$!3p***\\P*o%Q7$F1$!3M)\\bxKk[E#FY7$$!3Kmmm\"RFj!HF1$!3Yitk3D9:MFY7$ $!33LL$e4OZr#F1$!3xhfgj-Q=YFY7$$!3u*****\\n\\!*\\#F1$!3/*fGF:d1.'FY7$$ !3%)*****\\ixCG#F1$!3?587[xtEsFY7$$!35+++vR7y@F1$!3H!=t3Ba9h(FY7$$!3#* *****\\KqP2#F1$!3eksWGe[4yFY7$$!3I;a8Z&z+0#F1$!3&p'G%oV]R#yFY7$$!36L3F p()QE?F1$!3%3_[S*R8EyFY7$$!3$*\\iS\"*zp-?F1$!3^i>%**44c\"yFY7$$!3am;a8 s+z>F1$!3#*pAZqV(>z(FY7$$!3&**\\7ylD;$>F1$!3u^WbIi\"Rq(FY7$$!39LL3-TC% )=F1$!3S[3GI:PfvFY7$$!3#)**\\(ofh:x\"F1$!3;XT8&**R$ypFY7$$!3[mmm\"4z)e ;F1$!3)[t39Xjt0'FY7$$!3Smm;HAUj:F1$!3UGvk#*y.A]FY7$$!3Mmmmm`'zY\"F1$!3 WifH,^azPFY7$$!3%)*\\(oaeW79F1$!3kZ;\"F1$\"3?qQsT=oyXFD7$$!3OL$3x'*)fZ6F1$\"3k%[VI ,n`D\"FY7$$!33+v=#za%)4\"F1$\"3a))*4%p0iN?FY7$$!3!ommmh5$\\5F1$\"35o6b BPR(y#FY7$$!3tIL3xrs9%*FY$\"3'=Z#e5`i#G%FY7$$!3S$***\\(=[jL)FY$\"3#Qjx T[WZZ&FY7$$!3q%****\\Pw%4tFY$\"3$[r4<'p-SiFY7$$!3)f***\\iXg#G'FY$\"3a( QvUi]Ae'FY7$$!3qGe*)4ju9gFY$\"3![u=$R(zrf'FY7$$!3Uh;Hd!))ou&FY$\"3))*f m7&3q!e'FY7$$!37%\\(o/)H!zaFY$\"3Zb&))y*QuKlFY7$$!3$oK$3_:<6_FY$\"3Eln #\\b(R`kFY7$$!3C#*\\(o/ban%FY$\"3hi2/rGp,iFY7$$!3ndmmT&Q(RTFY$\"37l&)e g=]HeFY7$$!3Ihm\"HdGe:$FY$\"3')fXt&oU_&[FY7$$!3%\\mmTg=><#FY$\"3C`TJ() pjeNFY7$$!3FK$3Fpy7k\"FY$\"3;cPDjgPaFFY7$$!3g***\\7yQ16\"FY$\"35j]kjF8 '*=FY7$$!3Epm\"zp))**z&FD$\"3Au#f:1]1+\"FY7$$!3vDMLLe*e$\\F4$\"3k6*HqO z*[&)F47$$\"3bsmTgxE=]FD$!3s[^0MIQm')FD7$$\"3womT5D,`5FY$!3O^U/;FY$!3ChtQr\"pfp#FY7$$\"3!=nm\"zRQb@FY$!3QY6.8*[X`$FY7 $$\"3mOLL$e,]6$FY$!31qyB#oTt![FY7$$\"3_,+](=>Y2%FY$!3I=_iU8LwdFY7$$\"3 !)p;aQe#Gf%FY$!3#4S:6*=,_hFY7$$\"36QLe*[K56&FY$!3ghk(=F=dT'FY7$$\"3EsT 5:e8q`FY$!3mI1IyGE/lFY7$$\"3T1]iS\"R#HcFY$!3C9Ra++Dcl jL*FY$!3s5(y1VU2Q%FY7$$\"3H++]i_QQ5F1$!3kw&zY1@&\\HFY7$$\"3O+v=U,1*3\" F1$!3mWwY=4)==#FY7$$\"3U+](=-N(R6F1$!3c)pCpNr;Q\"FY7$$\"3[+Dc,*4/>\"F1 $!3@l?mOZFTv$FY7$$\"3KmmT5FEn:F1$\" 3!*o)*4Jq\"z1&FY7$$\"3iKLL$Qx$o;F1$\"3Sz9'FY7$$\"3cmm;z)Qjx\"F1$ \"3y`Tx\")Q))4qFY7$$\"3Y+++v.I%)=F1$\"33\\i_cvdfvFY7$$\"3!p;Hd&\\@L>F1 $\"3nx_vmDx2xFY7$$\"3ML$ek`H@)>F1$\"3[P@1NH&ez(FY7$$\"3M;H#o#oe1?F1$\" 3K$G8^xD#=yFY7$$\"3y*\\(=#o_Hce(FY7$$\"3;,++D\\'QH#F1$\"3Cjxf`kRvrFY7$$\"3%HL$e9S8&\\#F1 $\"3a$*fx7IJbgFY7$$\"3s++D1#=bq#F1$\"3cBF\"H*3'*yYFY7$$\"3\"HLL$3s?6HF 1$\"3QX>$HbXkQ$FY7$$\"3a***\\7`Wl7$F1$\"3]pdR)3zBD#FY7$$\"3enmmm*RRL$F 1$\"3$Qy]'*>)4G9FY7$$\"3%zmmTvJga$F1$\"3`j\\D\\%[UV)FD7$$\"3]MLe9tOcPF 1$\"3%=rI$fu4?ZFD7$$\"31,++]Qk\\RF1$\"3Bp3)*R`;NEFD7$$\"3![LL3dg6<%F1$ \"3!pmJFbHlF\"FD7$$\"3%ymmmw(GpVF1$\"3_\"zYysc&[jF47$$\"3C++D\"oK0e%F1 $\"3S'\\u^p=S'GF47$$\"35,+v=5s#y%F1$\"3i'fMO!HFt7F47$$\"\"&F*$\"35$ey$ )*GAc]F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F][m-%+AXESLABELSG6$Q\"x6\"Q! Fb[m-%%VIEWG6$;F(Fbjl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "The Fourier transforms of the Hermite functions " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 68 "We use Maple to find the Fourier transforms of the Hermite func tions" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "h[n](x) = 1 /sqrt(2^n*n!);" "6#/-&%\"hG6#%\"nG6#%\"xG*&\"\"\"F,-%%sqrtG6#*&)\"\"#F (F,-%*factorialG6#F(F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "H[n](x)*e xp(-x^2/2)" "6#*&-&%\"HG6#%\"nG6#%\"xG\"\"\"-%$expG6#,$*&F*\"\"#F1!\" \"F2F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 8 " is the " }{TEXT 285 1 "n" }{TEXT -1 23 " th Hermite polynomial." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "alias (Hm=orthopoly[H]):\nh := (n,x) -> 1/sqrt(2^n*n!)*Hm(n,x)*exp(-x^2/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6$%\"nG%\"xG6\"6$%)operato rG%&arrowGF)*(-%%sqrtG6#*&)\"\"#9$\"\"\"-%*factorialG6#F4F5!\"\"-%#HmG 6$F49%F5-%$expG6#,$*&#F5F3F5*$)F=F3F5F5F9F5F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Let's start with the case " }{XPPEDIT 18 0 "n = 3" "6#/%\"nG\"\"$" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "h(3,x );\ninttrans[fourier](%,x,omega):\nFh3 := unapply(simplify(%),omega); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#7F&*(\"\"$#F&\"\"# ,&*&\"\")F&)%\"xGF)F&F&*&F'F&F0F&!\"\"F&-%$expG6#,$*&F+F2F0F+F2F&F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Fh3Gf*6#%&omegaG6\"6$%)operator G%&arrowGF(*0^##\"\"\"\"\"$F/F0#F/\"\"#F2F1%#PiGF19$F/-%$expG6#,$*&#F/ F2F/*$)F4F2F/F/!\"\"F/,&F0F=*&F2F/F " 0 "" {MPLTEXT 1 0 24 "simplify(Fh3(x)/h( 3,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(\"\"##\"\"\"F$%#PiGF%^#F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F[3] (omega) = Fr*[h[3](x)]" "6#/-&%\"FG6#\"\"$6#%&omegaG*&%#FrG\"\"\"7#-&% \"hG6#F(6#%\"xGF-" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2*Pi)*i*h[3] (omega)" "6#*(-%%sqrtG6#*&\"\"#\"\"\"%#PiGF)F)%\"iGF)-&%\"hG6#\"\"$6#% &omegaGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "The corresponding " }{TEXT 261 10 "eigenvalue" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "sqrt(2*Pi)*i" "6#*&-%%sqrtG6#*&\"\"# \"\"\"%#PiGF)F)%\"iGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "This result seems to hold for all val ues of " }{TEXT 279 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "n = 11" "6#/%\"nG\"#6" }{TEXT -1 15 " we have . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "h(11,x);\ninttrans[fourier](%,x,om ega):\nFh11 := unapply(simplify(%),omega);\nsimplify(Fh11(x)/h(11,x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"(g\"[NF&*(\"$a\"#F&\" \"#,.*&\"%[?F&)%\"xG\"#6F&F&*&\"&?j&F&)F0\"\"*F&!\"\"*&\"'!)o]F&)F0\" \"(F&F&*&\"(!3u%% Fh11Gf*6#%&omegaG6\"6$%)operatorG%&arrowGF(*.^##\"\"\"\"&?x#F/\"#x#F/ \"\"#%#PiGF29$F/-%$expG6#,$*&#F/F3F/*$)F5F3F/F/!\"\"F/,.\"&&R5F>*&\"&] Y$F/F=F/F/*&F0F/)F5\"\"%F/F>*&\"%?zF/)F5\"\"'F/F/*&\"$!))F/)F5\"\")F/F >*&\"#KF/)F5\"#5F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(\"\" ##\"\"\"F$%#PiGF%^#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "To e laborate on the discussion of this phenomenon, consider the differenti al equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/ (d*x^2) = (x^2-2*n-1)*y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F( !\"\"*&,(*$F+F&F(*&F&F(%\"nGF(F,F(F,F(F'F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 5 " f \"(" } {TEXT 265 1 "x" }{TEXT -1 4 ") - " }{XPPEDIT 18 0 "x^2*f(x)+(2*n+1)*f( x) = 0;" "6#/,&*&%\"xG\"\"#-%\"fG6#F&\"\"\"F+*&,&*&F'F+%\"nGF+F+F+F+F+ -F)6#F&F+F+\"\"!" }{TEXT -1 14 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Using the differentiation formulas:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 263 2 "Fr" }{TEXT -1 7 " [ f '(" }{TEXT 264 1 "x" } {TEXT -1 6 ") ] = " }{XPPEDIT 18 0 "i*omega*Fr*[f(x)]" "6#**%\"iG\"\" \"%&omegaGF%%#FrGF%7#-%\"fG6#%\"xGF%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Fr*[x*f(x)] = i;" "6#/*&%#Fr G\"\"\"7#*&%\"xGF&-%\"fG6#F)F&F&%\"iG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/(d*omega)" "6#*&%\"dG\"\"\"*&F$F%%&omegaGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Fr*[f(x)];" "6#*&%#FrG\"\"\"7#-%\"fG6#%\"xGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr" "6#%#FrG" }{TEXT -1 7 " [ f \"(" } {TEXT 280 1 "x" }{TEXT -1 2 ")]" }{XPPEDIT 18 0 "`` = -omega^2*F(omega );" "6#/%!G,$*&%&omegaG\"\"#-%\"FG6#F'\"\"\"!\"\"" }{TEXT -1 9 ", whe re " }{XPPEDIT 18 0 "F(omega) = Fr*[f(x)];" "6#/-%\"FG6#%&omegaG*&%#Fr G\"\"\"7#-%\"fG6#%\"xGF*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[x^2*f(x )];" "6#*&%#FrG\"\"\"7#*&%\"xG\"\"#-%\"fG6#F(F%F%" }{TEXT -1 8 " = -F \+ \"(" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 2 ")." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Hence, from (i), w e obtain the transformed equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-omega^2*F(omega)+``;" "6#,&*&%&omegaG\"\"#-%\"FG6#F %\"\"\"!\"\"%!GF*" }{TEXT -1 4 "F \"(" }{XPPEDIT 18 0 "omega" "6#%&ome gaG" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``+(2*n+1)*F(omega) = 0;" "6#/,&% !G\"\"\"*&,&*&\"\"#F&%\"nGF&F&F&F&F&-%\"FG6#%&omegaGF&F&\"\"!" }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 5 " F \"(" }{XPPEDIT 18 0 "omega" "6#%&omegaG" } {TEXT -1 1 ")" }{XPPEDIT 18 0 "``-omega^2*F(omega)+(2*n+1)*F(omega) = \+ 0;" "6#/,(%!G\"\"\"*&%&omegaG\"\"#-%\"FG6#F(F&!\"\"*&,&*&F)F&%\"nGF&F& F&F&F&-F+6#F(F&F&\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 87 "which is effectively the same equatio n as (i), except that the independent variable is " }{XPPEDIT 18 0 "om ega" "6#%&omegaG" }{TEXT -1 12 " instead of " }{TEXT 281 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Define the differential operator " }{TEXT 266 3 "Hrm " }{TEXT -1 4 " by " }{XPPEDIT 18 0 "Hrm*[f(x)] = ``;" "6#/*&%$HrmG\" \"\"7#-%\"fG6#%\"xGF&%!G" }{TEXT -1 4 "f \"(" }{TEXT 282 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``-x^2*f(x);" "6#,&%!G\"\"\"*&%\"xG\"\"#-%\"f G6#F'F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Then equa tion (i) has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Hrm*[f(x)] = -(2*n+1)*f(x)" "6#/*&%$HrmG\"\"\"7#-%\"fG6#%\"xGF&, $*&,&*&\"\"#F&%\"nGF&F&F&F&F&-F)6#F+F&!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 "while equation (ii) has the form " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Hrm*[F(omega)] = -(2*n+1)*F(ome ga)" "6#/*&%$HrmG\"\"\"7#-%\"FG6#%&omegaGF&,$*&,&*&\"\"#F&%\"nGF&F&F&F &F&-F)6#F+F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 39 " is a \+ solution of (i), it follows that " }{XPPEDIT 18 0 "Hrm*[H[n](x)] = -(2 *n+1)*H[n](x);" "6#/*&%$HrmG\"\"\"7#-&%\"HG6#%\"nG6#%\"xGF&,$*&,&*&\" \"#F&F,F&F&F&F&F&-&F*6#F,6#F.F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 70 "In addition to being eigenfunctions of the Fourier tra nsform operator " }{TEXT 271 2 "Fr" }{TEXT -1 24 ", the Hermite functi ons " }{XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 10 " are also " }{TEXT 261 14 "eigenfunctions" }{TEXT -1 30 " of the diff erential operator " }{XPPEDIT 18 0 "Hrm;" "6#%$HrmG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 79 "Indeed, we have the following commutativ e diagram, that is, when the operators " }{TEXT 283 2 "Fr" }{TEXT -1 5 " and " }{TEXT 284 3 "Hrm" }{TEXT -1 32 " are applied in either orde r to " }{XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 26 ", the result is the same. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 267 2 "Fr" }{TEXT -1 12 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "H[n](x)" "6#-&%\"HG6 #%\"nG6#%\"xG" }{TEXT -1 35 " ------------------------------> " } {XPPEDIT 18 0 "sqrt(2*Pi)*i*H[n](omega);" "6#*(-%%sqrtG6#*&\"\"#\"\"\" %#PiGF)F)%\"iGF)-&%\"HG6#%\"nG6#%&omegaGF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 " " }}{PARA 256 "" 0 "" {TEXT -1 85 " \+ | | \+ " }}{PARA 256 "" 0 "" {TEXT -1 71 " | \+ | " }}{PARA 256 "" 0 "" {TEXT 269 3 "Hrm" }{TEXT -1 63 " | \+ | " }{TEXT 270 3 "Hrm" }{TEXT -1 8 " " } }{PARA 256 "" 0 "" {TEXT -1 71 " | \+ | " }}{PARA 256 "" 0 "" {TEXT -1 71 " \+ | | \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-(2*n+1)*H[n](x)" "6#,$*&,&*&\"\"#\"\"\"%\"nGF(F(F(F(F (-&%\"HG6#F)6#%\"xGF(!\"\"" }{TEXT -1 25 " ---------------------> " } {XPPEDIT 18 0 "-(2*n+1)*sqrt(2*Pi)*i*H[n](omega);" "6#,$**,&*&\"\"#\" \"\"%\"nGF(F(F(F(F(-%%sqrtG6#*&F'F(%#PiGF(F(%\"iGF(-&%\"HG6#F)6#%&omeg aGF(!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " " } {TEXT 268 2 "Fr" }{TEXT -1 14 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }