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WuChong-shi ( )§24.1 Bessel Bessel Bessel J1/2(x) J1/2(x)=∞Xk=0(−)kk!Γ(k+3/2)x22k+1/2=r2πx∞Xk=0(−)k(2k+1)!x2k+1=r2πxsinx.!J1/2(x)#$% &’()*+J−1/2(x)=r2πxcosx.,-./Jν(x) 012*345671xddxxνJν(x)=xν−1Jν−1(x),−1xddxx−νJν(x)=x−(ν+1)Jν+1(x),89!:;x−n+1/2J−n+1/2(x)=1xddxnx1/2J1/2(x)=1xddxnr2πsinx,x−n−1/2Jn+1/2(x)=−1xddxnx−1/2J1/2(x)=−1xddxnr2πsinxx.=? 1 Bessel @#$% @#A BCD EF GHJn+1/2(x)IJ−(n+1/2)(x)#JKL3 W[Jn+1/2(x),J−(n+1/2)(x)]=(−)n+12πx.MNn+1/2(x)IJ−(n+1/2)(x)JKN3Nn+1/2(x)=cos(n+1/2)π·Jn+1/2(x)−J−(n+1/2)(x)sin(n+1/2)π=(−)n+1J−(n+1/2)(x).WuChong-shi§24.2OBesselPQR2S§24.2TBessel HelmholtzUV∇2u+k2u=0WXYZ4[\]^_‘abcd:;ef\UV1r2ddrr2dRdr+k2−λr2R=0.W ghi[λl=l(l+1),l=0,1,2,··· 8 j1UV klmn Fk=0 01JKL3l#rlBr−l−1 (op20q)Fk6=0 9r^sx=krBy(x)=R(r)tUV^u1x2ddxx2dydx+h1−l(l+1)x2iy(x)=0.j1UVvuXBesselUVw xyBBesselUVzeN{ FXBesselUV(|01 } 1#x=0~ } 1#x=∞z~ }(BBesselUVN& F=9! tw uBesselUV ;j1UVWx=0} ZUVρ(ρ−1)+2ρ−l(l+1)=0,M Zuρ1=lBρ2=−(l+1)BBesselUV Zρ=±ν & r^sy(x)=v(x)√x,j’9! v(x) f\UVWx=0} Z8 ^uρ=±l+12,BBesselUV } ’ j’v(x) f\UV8#1xddxxdvdx+1−(l+1/2)2x2v=0.~#l+1/2 BesselUV w 01JKL3l8#Jl+1/2(x)BNl+1/2(x) W= .89!tXBesselUV(17.84) JKL3l ujl(x)=rπ2xJl+1/2(x)=√π2∞Xn=0(−)nn!Γ(n+l+3/2)x22n+lnl(x)=(−)l+1j−l−1(x)=rπ2xNl+1/2(x)=(−)l+1√π2∞Xn=0(−)nn!Γ(n−l+1/2)x22n−l−1,\ vul XBessel BXNeumann WuChong-shi ( ) PQ( )R3S 1XBessel BXNeumann ( xo 24.1) y# j0(x)=sinxx,n0(x)=−cosxx,j1(x)=1x2 sinx−xcosx,n1(x)=−1x2 cosx+xsinx,j2(x)=1x3h 3−x2sinx−3xcosxi;n2(x)=−1x3h 3−x2cosx+3xsinxi.¡24.1¢Bessel£⁄jl(x)¥¢Neumann£⁄nl(x)ƒ§¤'“«‹›fifl'y=±1/x { (–9!†‡XHankel h(1)l(x)=jl(x)+inl(x),h(2)l(x)=jl(x)−inl(x).·24.1t eikrcosθ Legendre¶•y‚„ ”»eikrcosθ=∞Xl=0cl(kr)Pl(cosθ), ‚„4 cl(kr)=2l+12Z1−1eikrxPl(x)dx=2l+12∞Xn=0(ikr)nn!Z1−1xnPl(x)dx.…‰p19qp4 ¿8|cl(kr)=2l+12∞Xn=0(ikr)l+2n(l+2n)!Z1−1xl+2nPl(x)dx=2l+12il∞Xn=0(−)n(l+2n)!(kr)l+2n·(l+2n)!2l+2nn!√πΓ(n+l+3/2)=2l+12il√π∞Xn=0(−)nn!Γ(n+l+3/2)kr2l+2n=(2l+1)iljl(kr).WuChong-shi§24.2OBesselPQR4S! `8|‚„yeikrcosθ=∞Xl=0(2l+1)iljl(kr)Pl(cosθ).´ˆueikrcosθ=eikz#HelmholtzUV l ∇2+k2eikrcosθ=0, |eikrcosθ=∞Xl=0Aljl(kr)Pl(cosθ).˜W mn#¯˘†+4 Al˙Aljl(kr)=2l+12Z1−1eikrxPl(x)dx=2l+121ikreikrxPl(x) 1−1−1ikrZ1−1eikrxP0l(x)dx#=2l+121ikreikr−(−)le−ikr+O1r2. U¨jl(kr)=1krcoskr−12l+12π−π4+O1r2=1krcoskr−l+12π+O1r2=12kr(−i)l+1eikr+il+1e−ikr+O1r2=12kr(−i)l+1eikr−(−)le−ikr+O1r2,=Al(−i)l+12=2l+12i Al=(2l+1)il, `8:;‚„yeikrcosθ=∞Xl=0(2l+1)iljl(kr)Pl(cosθ).(9!˚¸j1‚„y 1 ˝l˛ ˇ— — j#u ††N ‘ ue−iωt rBθuXYZ .y # θ=0( ~z )U Æ ¨ªª ukM • jl(kr) |X¨ª N jl(kr)∼1krsinkr−lπ2.WuChong-shi ( ) PQ( )R5S ( )ŁØŒº ( );˜Wu abæd ˝ ı f\UV†lmnłø kljœ†lmn |ßU \]^_ j U H| † ‰ ¯ kUVB†l @#JK =†lmn l | K Wp15q abcd F kl V\ j l †lmn k #cd +(o15.1 )j U # ) ‰ kl f\UV†lmnW˝ . [ 1mn 1. !#$ %2. & ’()*+ , -./0 !#$ -. 12&%3. -. 3 456 7j q„8ab8 7˝ .9:j 1mn7Mu\]^_ ;† 1, ˝ H=?@jAłø (B#C\ W g ˝mn jœ #) §24.1DEFGH FG1.IJKIJLM»W NK.†‡ nOPQLMVw RS(PQ)‰x,y,··· T 9!/COU_V U_ WX YZ*[;nOU_V u= †‡nOU_ IJ ,nOU_V ( Ku, N)W\† ] {ei,i=1,2,···,n}^`V ? 1U_x@9!‰wWj ] . _‘(YZ)x1,x2,···,xn Tx=x1e1+x2e2+···+xnen=nXi=1xiei. V U_xBy eo ab†‡u(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi.j# 1, GH|(x,y)=(y,x)B(x,x)≥0,c d x=0‘e|(x,x)=0 W= .89!†‡U_x fgkxkkxk=(x,x)1/2. EnOU_V ¯¿hij.kab†‡lmn+j‘ U_fg89) #,WuChong-shi§24.1opqrsPQqrR6S u itU_fgh#, uWitfg†‡ v[/ab†‡w5u(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyi,x x∗i#xi Eyz GHWEU_V (x,y)=(y,x)∗.j’ abYZGH#COU_ Zb {|*[ }– B~ #U_ ab G \ ab H 7ab 9)†‡ ~ +w 7M + 1 ˝ ab†‡(!`8vuab ˝) 24.1(†‡W, E NK. )U_V U_xBy ab(x,y)#wb Z_ 1.(x,y)=(y,x)∗%2.(αx+βy,z)=α∗(x,z)+β∗(y,z) α β . K & %3. x(x,x)≥0% x=0 (x,x)=0 ·24.1 x=x1x2...xnBy=y1y2...yn#, N. U_Pu( † ) D D Pii u~, 9†‡U_xBy abu(x,y)=x1,x2,···,xnP110···00P22···0......00···Pnny1y2...yn.·24.2,^_t |E4 ¶•y FW¶•y !¡¶•yBE ¢ [£7 1EU_V u⁄»0≤t≤1 x(t)By(t)#=U_V 01U_( ¶•y) wb ab9!†‡u(x,y)=Z10x∗(t)y(t)ρ(t)dt,x æ¥ ρ(x)≥0 6≡0 WuChong-shiƒ§¤'“(«)‹›fi(fl) 7–w hx#ρ(x)≡1(x,y)=Z10x∗(t)y(t)dt.†‡· ¶• &‚1„”’(»…‰ ¿& `&´ ˆ˜ ¯´ ˘˙¨&· ¿.˚¸‚3„” &‰ ˝˛ ˇ— W= .8/(x,x)1/2=kxkvuU_x ( U_x fg ) 7.¨ab ˝ p1Bp2 k9:(x,αy)=α(x,y).=kαxk=(αx,αx)1/2=hαα∗(x,x)i1/2=|α|kxk.˘ 1z U_ !w 87u | fg U_ vu U_xkxk,xkxk=1.F†‡ ab U_V vuabV F |ab ,U_V vu A V (Euclideanspace)%F |ab EU_V vu V (unitaryspace) 2. W Æ ab†‡`89! ªU_ YZ F d (x,y)=0‘0U_x,y~ F U_B˘U_@~ 24.2 | iBj(xi,xj)=δij vU_]{x1,x2,···}# 45 &´ 6 Ł&˚ ØŒº ˘ 6, æ´ α1x1+α2x2+α3x3+···=0, αj=0,j=1,2,3,···. (n ´ ˆ˜ & ,n¯45 ´ ı’( ˆ˜&łøŒ œ( ø ß œ) 45 ł • ¿ ı3 & „6 WuChong-shi§24.1opqrsPQqrR8S3. 24.3W| OU_V ¯¿ ]~ U_(vu 1 PQ )c W 1 ~ U_ ^ v ~ U_ # FW| O U_V 1 U U V F !!#$%&’( ) *+$,-./0 12 34 ( ( 156 789:;( 1)=?) +@AB ( ) 1C FD EF) V G 1 34 ( {xi,i=1,2,···,k},,