球谐函数

球谐函数是拉普拉斯方程的球坐标系形式解的角度部分。在古典場論、量子力学等领域广泛应用。

目录

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  1 函数的推导
  
  
  
  • 
    1.1 本微分方程的推导
    
  
  
  • 
    1.2 本征方程的求解
    
  
  
  
  


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  2 前几阶球谐函数
  


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  3 参见
  

函数的推导

本微分方程的推导

球坐标下的拉普拉斯方程式:

∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin2⁡θ∂2f∂φ2=0{displaystyle nabla ^{2}f={1 over r^{2}}{partial over partial r}left(r^{2}{partial f over partial r}right)+{1 over r^{2}sin theta }{partial over partial theta }left(sin theta {partial f over partial theta }right)+{1 over r^{2}sin ^{2}theta }{partial ^{2}f over partial varphi ^{2}}=0,!} 。

實值的球諧函數 Y_lm，l = 0 到 4 （由上至下），m=0 到 4（由左至右）。負數階球諧函數 Y_l,-m 可由正數階函數對 z-軸轉 90/m 度得到。

利用分离变量法，设定 f(r, θ, φ)=R(r)Y(θ, φ)=R(r)Θ(θ)Φ(φ){displaystyle f(r, theta , varphi )=R(r)Y(theta , varphi )=R(r)Theta (theta )Phi (varphi )} 。其中Y(θ, φ){displaystyle Y(theta , varphi )}代表角度部分的解，也就是球谐函数。

代入拉普拉斯方程，得到：

ΘΦr2ddr(r2dRdr)+RΦr2sin⁡θddθ(sin⁡θdΘdθ)+RΘr2sin2⁡θd2Φdφ2=0{displaystyle {Theta Phi over r^{2}}{d over dr}left(r^{2}{dR over dr}right)+{RPhi over r^{2}sin theta }{d over dtheta }left(sin theta {dTheta over dtheta }right)+{RTheta over r^{2}sin ^{2}theta }{d^{2}Phi over dvarphi ^{2}}=0,!}

分离变量后得：

{1Rddr(r2dRdr)=λ1Φd2Φdφ2=−m2λ+1Θsin⁡θddθ(sin⁡θdΘdθ)−m2sin2⁡θ=0{displaystyle {begin{cases}{dfrac {1}{R}}{dfrac {d}{dr}}left(r^{2}{dfrac {dR}{dr}}right)=lambda \{dfrac {1}{Phi }}{dfrac {d^{2}Phi }{dvarphi ^{2}}}=-m^{2}\lambda +{dfrac {1}{Theta sin theta }}{dfrac {d}{dtheta }}left(sin theta {dfrac {dTheta }{dtheta }}right)-{dfrac {m^{2}}{sin ^{2}theta }}=0end{cases}}} ，整理得{r2R″+2rR′−λR=0Φ″+m2Φ=0sin⁡θddθ(sin⁡θΘ′)+(λsin2⁡θ−m2)Θ=0{displaystyle {begin{cases}r^{2}R''+2rR'-lambda R=0\Phi ''+m^{2}Phi =0\sin theta {dfrac {d}{dtheta }}(sin theta Theta ')+(lambda sin ^{2}theta -m^{2})Theta =0end{cases}}}

本征方程的求解

这里，Φ{displaystyle Phi }是一个以2π{displaystyle 2pi }为周期的函数，即满足周期性边界条件Φ(φ)=Φ(φ+2π){displaystyle Phi (varphi )=Phi (varphi +2pi )}，因此m{displaystyle m}必须为整数。而且可以解出：

Φ=eimϕ{displaystyle Phi =e^{imphi }}，m∈Z{displaystyle min mathbb {Z} }

而对于Θ{displaystyle Theta }的方程，进行变量替换 t=cos⁡θ{displaystyle t=cos theta }，dt=−sin⁡θdθ{displaystyle dt=-sin theta dtheta }，|t|⩽1{displaystyle |t|leqslant 1}，得到关于t{displaystyle t}的伴随勒让德方程。方程的解应满足在[−1,1]{displaystyle [-1,1]}区间上取有限值，此时必须有λ=l(l+1){displaystyle lambda =l(l+1)}，其中l{displaystyle l}为自然数，且l⩾|m|{displaystyle lgeqslant |m|}。对应方程的解为Pℓm(t){displaystyle P_{ell }^{m}(t)}。即可以解出：

Θ=Pℓm(cos⁡θ){displaystyle Theta =P_{ell }^{m}(cos theta )}，l∈N,l⩾|m|{displaystyle lin mathbb {N} ,lgeqslant |m|}

故球谐函数可以表达为：

Yℓm(θ,φ)=NΦ(φ)Θ(θ)=NeimφPℓm(cos⁡θ){displaystyle Y_{ell }^{m}(theta ,varphi )=NPhi (varphi )Theta (theta )=N,e^{imvarphi },P_{ell }^{m}(cos {theta }),!} ，l∈N,m=0,±1,±2,…±l{displaystyle lin mathbb {N} ,m=0,pm 1,pm 2,ldots pm l}；

其中N 是归一化因子。

經過歸一化後，球谐函数表達為：

Yℓm(θ, φ)=(−1)m(2ℓ+1)4π(ℓ−|m|)!(ℓ+|m|)!Pℓm(cos⁡θ)eimφ{displaystyle Y_{ell }^{m}(theta , varphi )=(-1)^{m}{sqrt {{(2ell +1) over 4pi }{(ell -|m|)! over (ell +|m|)!}}},P_{ell }^{m}(cos {theta }),e^{imvarphi },!} ；

这里的 Yℓm{displaystyle Y_{ell }^{m},!} 称为 ℓ{displaystyle ell ,!} 和 m{displaystyle m,!} 的球谐函数。以上推导过程中，i{displaystyle i,!} 是虛數單位， Pℓm{displaystyle P_{ell }^{m},!} 是伴随勒让德多项式 。

其中Pℓm(x){displaystyle P_{ell }^{m}(x),!} 用方程式定義為：

Pℓm(x)=(1−x2)|m|/2 d|m|dx|m|Pℓ(x){displaystyle P_{ell }^{m}(x)=(1-x^{2})^{|m|/2} {frac {d^{|m|}}{dx^{|m|}}}P_{ell }(x),} ；

而 Pℓ(x){displaystyle P_{ell }(x),!} 是 l{displaystyle l} 階勒讓德多項式，可用羅德里格公式表示為：

Pℓ(x)=12ℓℓ!dℓdxℓ(x2−1)l{displaystyle P_{ell }(x)={1 over 2^{ell }ell !}{d^{ell } over dx^{ell }}(x^{2}-1)^{l}} 。

前几阶球谐函数

主条目：球谐函数表

l{displaystyle l}	m{displaystyle m}	Φ(φ){displaystyle Phi (varphi )}	Θ(θ){displaystyle Theta (theta )}		極坐標中的表達式	直角坐標中的表達式	量子力學中的記号
0	0	12π{displaystyle {frac {1}{sqrt {2pi }}}}	12{displaystyle {frac {1}{sqrt {2}}}}		12π{displaystyle {frac {1}{2{sqrt {pi }}}}}	12π{displaystyle {frac {1}{2{sqrt {pi }}}}}	s{displaystyle {mbox{s}},}
1	0	12π{displaystyle {frac {1}{sqrt {2pi }}}}	32cos⁡θ{displaystyle {sqrt {frac {3}{2}}}cos theta }		123πcos⁡θ{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}cos theta }	123πzr{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}{frac {z}{r}}}	pz{displaystyle {mbox{p}}_{z},}
1	+1	12πexp⁡(iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(ivarphi )}	32sin⁡θ{displaystyle {frac {sqrt {3}}{2}}sin theta }	{{displaystyle {Bigg {}}	123πsin⁡θcos⁡φ{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}sin theta cos varphi }	123πxr{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}{frac {x}{r}}}	px{displaystyle {mbox{p}}_{x},}
1	-1	12πexp⁡(−iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-ivarphi )}	32sin⁡θ{displaystyle {frac {sqrt {3}}{2}}sin theta }		123πsin⁡θsin⁡φ{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}sin theta sin varphi }	123πyr{displaystyle {frac {1}{2}}{sqrt {frac {3}{pi }}}{frac {y}{r}}}	py{displaystyle {mbox{p}}_{y},}
2	0	12π{displaystyle {frac {1}{sqrt {2pi }}}}	1252(3cos2⁡θ−1){displaystyle {frac {1}{2}}{sqrt {frac {5}{2}}}(3cos ^{2}theta -1)}		145π(3cos2⁡θ−1){displaystyle {frac {1}{4}}{sqrt {frac {5}{pi }}}(3cos ^{2}theta -1)}	145π2z2−x2−y2r2{displaystyle {frac {1}{4}}{sqrt {frac {5}{pi }}}{frac {2z^{2}-x^{2}-y^{2}}{r^{2}}}}	d3z2−r2{displaystyle {mbox{d}}_{3z^{2}-r^{2}}}
2	+1	12πexp⁡(iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(ivarphi )}	152sin⁡θcos⁡θ{displaystyle {frac {sqrt {15}}{2}}sin theta cos theta }	{{displaystyle {Bigg {}}	1215πsin⁡θcos⁡θcos⁡φ{displaystyle {frac {1}{2}}{sqrt {frac {15}{pi }}}sin theta cos theta cos varphi }	1215πzxr2{displaystyle {frac {1}{2}}{sqrt {frac {15}{pi }}}{frac {zx}{r^{2}}}}	dzx{displaystyle {mbox{d}}_{zx},}
2	-1	12πexp⁡(−iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-ivarphi )}	152sin⁡θcos⁡θ{displaystyle {frac {sqrt {15}}{2}}sin theta cos theta }		1215πsin⁡θcos⁡θsin⁡φ{displaystyle {frac {1}{2}}{sqrt {frac {15}{pi }}}sin theta cos theta sin varphi }	1215πyzr2{displaystyle {frac {1}{2}}{sqrt {frac {15}{pi }}}{frac {yz}{r^{2}}}}	dyz{displaystyle {mbox{d}}_{yz},}
2	+2	12πexp⁡(2iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(2ivarphi )}	154sin2⁡θ{displaystyle {frac {sqrt {15}}{4}}sin ^{2}theta }	{{displaystyle {Bigg {}}	1415πsin2⁡θcos⁡2φ{displaystyle {frac {1}{4}}{sqrt {frac {15}{pi }}}sin ^{2}theta cos 2varphi }	1415πx2−y2r2{displaystyle {frac {1}{4}}{sqrt {frac {15}{pi }}}{frac {x^{2}-y^{2}}{r^{2}}}}	dx2−y2{displaystyle {mbox{d}}_{x^{2}-y^{2}}}
2	-2	12πexp⁡(−2iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-2ivarphi )}	154sin2⁡θ{displaystyle {frac {sqrt {15}}{4}}sin ^{2}theta }		1415πsin2⁡θsin⁡2φ{displaystyle {frac {1}{4}}{sqrt {frac {15}{pi }}}sin ^{2}theta sin 2varphi }	1215πxyr2{displaystyle {frac {1}{2}}{sqrt {frac {15}{pi }}}{frac {xy}{r^{2}}}}	dxy{displaystyle {mbox{d}}_{xy},}
3	0	12π{displaystyle {frac {1}{sqrt {2pi }}}}	1272(5cos3⁡θ−3cos⁡θ){displaystyle {frac {1}{2}}{sqrt {frac {7}{2}}}(5cos ^{3}theta -3cos theta )}		147π(5cos3⁡θ−3cos⁡θ){displaystyle {frac {1}{4}}{sqrt {frac {7}{pi }}}(5cos ^{3}theta -3cos theta )}	147πz(2z2−3x2−3y2)r3{displaystyle {frac {1}{4}}{sqrt {frac {7}{pi }}}{frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}}}	fz(5z2−3r2){displaystyle {mbox{f}}_{z(5z^{2}-3r^{2})}}
3	+1	12πexp⁡(iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(ivarphi )}	14212(5cos2⁡θ−1)sin⁡θ{displaystyle {frac {1}{4}}{sqrt {frac {21}{2}}}(5cos ^{2}theta -1)sin theta }	{{displaystyle {Bigg {}}	14212π(5cos2⁡θ−1)sin⁡θcos⁡φ{displaystyle {frac {1}{4}}{sqrt {frac {21}{2pi }}}(5cos ^{2}theta -1)sin theta cos varphi }	14212πx(5z2−r2)r3{displaystyle {frac {1}{4}}{sqrt {frac {21}{2pi }}}{frac {x(5z^{2}-r^{2})}{r^{3}}}}	fx(5z2−r2){displaystyle {mbox{f}}_{x(5z^{2}-r^{2})}}
3	-1	12πexp⁡(−iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-ivarphi )}	14212(5cos2⁡θ−1)sin⁡θ{displaystyle {frac {1}{4}}{sqrt {frac {21}{2}}}(5cos ^{2}theta -1)sin theta }		14212π(5cos2⁡θ−1)sin⁡θsin⁡φ{displaystyle {frac {1}{4}}{sqrt {frac {21}{2pi }}}(5cos ^{2}theta -1)sin theta sin varphi }	14212πy(5z2−r2)r3{displaystyle {frac {1}{4}}{sqrt {frac {21}{2pi }}}{frac {y(5z^{2}-r^{2})}{r^{3}}}}	fy(5z2−r2){displaystyle {mbox{f}}_{y(5z^{2}-r^{2})}}
3	+2	12πexp⁡(2iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(2ivarphi )}	1054cos⁡θsin2⁡θ{displaystyle {frac {sqrt {105}}{4}}cos theta sin ^{2}theta }	{{displaystyle {Bigg {}}	14105πcos⁡θsin2⁡θcos⁡2φ{displaystyle {frac {1}{4}}{sqrt {frac {105}{pi }}}cos theta sin ^{2}theta cos 2varphi }	14105πz(x2−y2)r3{displaystyle {frac {1}{4}}{sqrt {frac {105}{pi }}}{frac {z(x^{2}-y^{2})}{r^{3}}}}	fz(x2−y2){displaystyle {mbox{f}}_{z(x^{2}-y^{2})}}
3	-2	12πexp⁡(−2iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-2ivarphi )}	1054cos⁡θsin2⁡θ{displaystyle {frac {sqrt {105}}{4}}cos theta sin ^{2}theta }		14105πcos⁡θsin2⁡θsin⁡2φ{displaystyle {frac {1}{4}}{sqrt {frac {105}{pi }}}cos theta sin ^{2}theta sin 2varphi }	12105πxyzr3{displaystyle {frac {1}{2}}{sqrt {frac {105}{pi }}}{frac {xyz}{r^{3}}}}	fxyz{displaystyle {mbox{f}}_{xyz},}
3	+3	12πexp⁡(3iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(3ivarphi )}	14352sin3⁡θ{displaystyle {frac {1}{4}}{sqrt {frac {35}{2}}}sin ^{3}theta }	{{displaystyle {Bigg {}}	14352πsin3⁡θcos⁡3φ{displaystyle {frac {1}{4}}{sqrt {frac {35}{2pi }}}sin ^{3}theta cos 3varphi }	14352πx(x2−3y2)r3{displaystyle {frac {1}{4}}{sqrt {frac {35}{2pi }}}{frac {x(x^{2}-3y^{2})}{r^{3}}}}	fx(x2−3y2){displaystyle {mbox{f}}_{x(x^{2}-3y^{2})}}
3	-3	12πexp⁡(−3iφ){displaystyle {frac {1}{sqrt {2pi }}}exp(-3ivarphi )}	14352sin3⁡θ{displaystyle {frac {1}{4}}{sqrt {frac {35}{2}}}sin ^{3}theta }		14352πsin3⁡θsin⁡3φ{displaystyle {frac {1}{4}}{sqrt {frac {35}{2pi }}}sin ^{3}theta sin 3varphi }	14352πy(3x2−y2)r3{displaystyle {frac {1}{4}}{sqrt {frac {35}{2pi }}}{frac {y(3x^{2}-y^{2})}{r^{3}}}}	fy(3x2−y2){displaystyle {mbox{f}}_{y(3x^{2}-y^{2})}}

l=0{displaystyle l=0}

Y00(θ,φ)=121π{displaystyle Y_{0}^{0}(theta ,varphi )={1 over 2}{sqrt {1 over pi }}}

l=1{displaystyle l=1}

Y1−1(θ,φ)=1232πsin⁡θe−iφ{displaystyle Y_{1}^{-1}(theta ,varphi )={1 over 2}{sqrt {3 over 2pi }},sin theta ,e^{-ivarphi }}

Y10(θ,φ)=123πcos⁡θ{displaystyle Y_{1}^{0}(theta ,varphi )={1 over 2}{sqrt {3 over pi }},cos theta }

Y11(θ,φ)=−1232πsin⁡θeiφ{displaystyle Y_{1}^{1}(theta ,varphi )={-1 over 2}{sqrt {3 over 2pi }},sin theta ,e^{ivarphi }}

l=2{displaystyle l=2}

Y2−2(θ,φ)=14152πsin2⁡θe−2iφ{displaystyle Y_{2}^{-2}(theta ,varphi )={1 over 4}{sqrt {15 over 2pi }},sin ^{2}theta ,e^{-2ivarphi }}

Y2−1(θ,φ)=12152πsin⁡θcos⁡θe−iφ{displaystyle Y_{2}^{-1}(theta ,varphi )={1 over 2}{sqrt {15 over 2pi }},sin theta ,cos theta ,e^{-ivarphi }}

Y20(θ,φ)=145π(3cos2⁡θ−1){displaystyle Y_{2}^{0}(theta ,varphi )={1 over 4}{sqrt {5 over pi }},(3cos ^{2}theta -1)}

Y21(θ,φ)=−12152πsin⁡θcos⁡θeiφ{displaystyle Y_{2}^{1}(theta ,varphi )={-1 over 2}{sqrt {15 over 2pi }},sin theta ,cos theta ,e^{ivarphi }}

Y22(θ,φ)=14152πsin2⁡θe2iφ{displaystyle Y_{2}^{2}(theta ,varphi )={1 over 4}{sqrt {15 over 2pi }},sin ^{2}theta ,e^{2ivarphi }}

参见

• 勒让德多项式


• 伴随勒让德多项式


• 施图姆-刘维尔理论


• 柱谐函数


• 向量球諧函數

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