In the previous problem, the solution of the eigenvalue problem is
A) $\lambda = n ( n + 1 ) , R = r ^ { n } , n = 1,2,3 , \ldots$
B) $\lambda = n ( n - 1 ) , R = r ^ { n } , n = 1,2,3 , \ldots$
C) $\lambda = n ( n - 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
D) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
E) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$

Question

Quizplus · Accepted Answer

The correct solution for the eigenvalue problem in spherical coordinates, which typically arises in the context of solving Laplace's equation or the Schrödinger equation for a central potential, involves eigenvalues $\lambda = n(n+1)$ and eigenfunctions that are the Legendre polynomials $P_n(\cos\theta)$ for the angular part. This corresponds to choice D.

In the Previous Problem, the Solution of the Eigenvalue Problem λ=n(n+1),R=rn,n=1,2,3,…\lambda = n ( n + 1 ) , R = r ^ { n } , n = 1,2,3 , \ldotsλ=n(n+1),R=rn,n=1,2,3,…

In the Previous Problem, the Solution of the Eigenvalue Problem $\lambda = n ( n + 1 ) , R = r ^ { n } , n = 1,2,3 , \ldots$