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数学物理方法公式（11）：Spherical Harmonics

2023-03-22 20:19 作者:打电动的阿伟嘻嘻嘻  | 我要投稿

Laplace方程在球坐标系形式解中的角向部分Y_%7Blm%7D(%5Ctheta%2C%5Cphi)%2C%5C%20l%5Cin%5C%7B0%2C1%2C2%2C%5Ccdots%5C%7D%2C%5C%20m%3D0%2C%5Cpm1%2C%5Cpm2%2C%5Ccdots%2C%5Cpm%20l.%0A

%5Chat%7BL%5E2%7D%3D-%5B%5Cfrac%7B1%7D%7B%5Csin%5Ctheta%7D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20%5Ctheta%7D(%5Csin%5Ctheta%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20%5Ctheta%7D)%2B%5Cfrac%7B1%7D%7B%5Csin%5E2%5Ctheta%7D%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%5Cphi%5E2%7D%5D%2C%5C%20%5Chat%7BL_z%7D%3D-i%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%5Cphi%7D.

%5Chat%7BL%5E2%7DY_%7Blm%7D(%5Ctheta%2C%5Cphi)%3Dl(l%2B1)Y_%7Blm%7D(%5Ctheta%2C%5Cphi)%2C%5C%20%5Chat%7BL_z%7DY_%7Blm%7D(%5Ctheta%2C%5Cphi)%3DmY_%7Blm%7D(%5Ctheta%2C%5Cphi).

Y_%7Blm%7D(%5Ctheta%2C%5Cphi)%3D%5Csqrt%7B%5Cfrac%7B2l%2B1%7D%7B4%5Cpi%7D%5Cfrac%7B(l-m)!%7D%7B(l%2Bm)!%7D%7DP%5Em_l(%5Ccos%5Ctheta)e%5E%7Bim%5Cphi%7D%2C%5C%20P%5Em_l(x)%3D(1-x%5E2)%5E%7B%5Cfrac%7Bm%7D%7B2%7D%7D%5Cfrac%7Bd%5Em%7D%7Bdx%5Em%7DP_l(x)%2C

后者称为 Associated Legendre函数 P%5Em_l(x)%3D%5Cfrac%7B(-1)%5Em%7D%7B2%5Ell!%7D(1-x%5E2)%5E%7B%5Cfrac%7Bm%7D%7B2%7D%7D%5Cfrac%7Bd%5E%7Bl%2Bm%7D%7D%7Bdx%5E%7Bl%2Bm%7D%7D(x%5E2-1)%5El.

性质：

（1）：P%5Em_l(x)%3D(-1)%5Em%5Cfrac%7B(l%2Bm)!%7D%7B(l-m)!%7DP%5E%7B-m%7D_%7Bl%7D(x).

（2）：%5Cint%5E%7B%5Cpi%7D_%7B%5Ctheta%3D0%7D%5Cint%5E%7B2%5Cpi%7D_%7B%5Cphi%3D0%7D%7CY_%7Blm%7D(%5Ctheta%2C%5Cphi)%7C%5E2%5Csin%5Ctheta%20d%5Ctheta%20d%5Cphi%3D1.

（3）：Y_%7Blm%7D(%5Ctheta%2C%5Cphi)%3D(-1)%5Em(Y_%7Blm%7D(%5Ctheta%2C%5Cphi))%5E%7B%5Cast%7D.

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