球谐函数表(完整版).pdf

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1、TableofsphericalharmonicsThisisatableoforthonormalizedsphericalharmonicsthatemploytheCondon-Shortleyphaseuptodegreel=10.Someoftheseformulasgivethe“Cartesian”version.Thisassumesx,y,z,andrarerelatedtoandφthroughtheusualspherical-to-Cartesiancoordinatetransformation:x=rsincosφy=rsinsinφz=rcos

2、1Sphericalharmonics1.1l=0[1]√011Y0(;φ)=21.2l=1[1]√√113iφ13(xiy)Y1(;φ)=
e
sin=
2222r√√01313zY1(;φ)=
cos=
22r√√113iφ13(x+iy)Y1(;φ)=
e
sin=
2222r1.3l=2[1]√√115115(xiy)2Y2(;φ)=
e2iφ
sin2=
24242r2√√1115iφ115(xiy)zY2(;φ)=
e
sin
cos=
22222r√√1515(2z2x2y2)Y0(;φ)=
(

3、3cos21)=
244r2√√1115iφ115(x+iy)zY2(;φ)=
e
sin
cos=
22222r√√115115(x+iy)2Y2(;φ)=
e2iφ
sin2=
24242r2121SPHERICALHARMONICS1.4l=3[1]√√135135(xiy)3Y3(;φ)=
e3iφ
sin3=
388r3√√11051105(xiy)2zY2(;φ)=
e2iφ
sin2
cos=
34242r3√√121121(xiy)(4z2x2y2)Y1(;φ)=
eiφ
sin
(5cos2

4、1)=
388r3√√1717z(2z23x23y2)Y0(;φ)=
(5cos33cos)=
344r3√√121121(x+iy)(4z2x2y2)Y1(;φ)=
eiφ
sin
(5cos21)=
388r3√√11051105(x+iy)2zY2(;φ)=
e2iφ
sin2
cos=
34242r3√√135135(x+iy)3Y3(;φ)=
e3iφ
sin3=
388r31.5l=4[1]√√335335(xiy)4Y4(;φ)=
e4iφ
sin4=
4162162r4√√335335(

5、xiy)3zY3(;φ)=
e3iφ
sin3
cos=
488r4√√3535(xiy)2
(7z2r2)Y2(;φ)=
e2iφ
sin2
(7cos21)=
48282r4√√3535(xiy)
z
(7z23r2)Y1(;φ)=
eiφ
sin
(7cos33cos)=
488r4√√3131(35z430z2r2+3r4)Y0(;φ)=
(35cos430cos2+3)=
41616r4√√3535(x+iy)
z
(7z23r2)Y1(;φ)=
eiφ
sin
(7cos33cos)

6、=
488r4√√3535(x+iy)2
(7z2r2)Y2(;φ)=
e2iφ
sin2
(7cos21)=
48282r4√√335335(x+iy)3zY3(;φ)=
e3iφ
sin3
cos=
488r4√√335335(x+iy)4Y4(;φ)=
e4iφ
sin4=
4162162r41.6l=5[1]√53775iφ5Y5(;φ)=
e
sin32√433854iφ4Y5(;φ)=
e
sin
cos162√313853iφ32Y5(;φ)=
e
sin
(9cos1)32√211155

7、2iφ23Y5(;φ)=
e
sin
(3coscos)821.7l=63√11165iφ42Y5(;φ)=
e
sin
(21cos14cos+1)162√011153Y5(;φ)=
(63cos70cos+15cos)16√11165iφ42Y5(;φ)=
e
sin
(21cos14cos+1)162√2111552iφ23Y5(;φ)=
e
sin
(3coscos)82√313853iφ32