6-1 球坐标系下亥姆霍斯方程分离变量(4)

数学物理方法中偏微分方程求解

1 d 2 dR (r ) R(r ) S (θ , ) (r ) S (θ , ) + 2 (sin θ ) 2 r dr dr r sin θ θ θR(r ) 2 S (θ , ) + 2 2 + λ R(r ) S (θ , ) = 0 2 r sin θ 1 d 2 dR (r ) 1 1 S (θ , ) (r )+ [ (sin θ ) R (r ) dr dr S (θ , ) sin θ θ θ

r2 R(r ) S (θ , )

1 2 S (θ , ) + 2 ] + λr 2 = 0 sin θ 21 1 S (θ , ) 1 2 S (θ , ) [ (sin θ )+ 2 ] = 2 θ S (θ , ) sin θ θ sin θ