Important Partial Differential Equations in curvilinear cdts Legendre

Important Partial Differential Equations- in curvilinear cdts

Legendre polynomials (scale to Pn (1) = 1) are the solution of the Legendre equation Rodrigues's formula Orthogonality (1 -x 2)y’’ – 2 xy’ = [(1 -x 2)y’]’

Pj, Pk are solutions of

Generating function and properties generating function Symmetry F(x) = F(-x) even function F(x) = -F(-x) odd function

Taking partial derivative w/r to r Taking partial derivative w/r to x Finally the case, i = j = n

Rigid Rotor in Quantum Mechanics Transition from the above classical expression to quantum mechanics can be carried out by replacing the total angular momentum by the corresponding operator: Wave functions must contain both θ and Φ dependence: are called spherical harmonics

Schrondinger equation

Two equations

Solution of second equation

Solution of First equation Associated Legendre Polynomial

Associated Legendre Polynomial

For l=0, m=0

First spherical harmonics Spherical Harmonic, Y 0, 0

l= 1, m=0

l= 1, m=0 θ cos 2θ 0 1 30 3/4 45 1/2 60 1/4 90 0

l=2, m=0 θ 0 cos 2θ 1 3 cos 2θ-1 2 30 3/4 (9/4 -1)=5/4 45 1/2 (3/2 -1)=1/2 60 1/4 (3/4 -1)=-1/4 90 0 -1

l = 1, m=± 1 Complex Value? ? If Ф 1 and Ф 2 are degenerateeigenfunctions, their linear combinations are also an eigenfunction with the same eigenvalue.

l=1, m=± 1 Along x-axis

Application: Find the potential v (a) interior to and (b) exterior to a hollow sphere of unit radius if half of its surface is charged to potential v 0 and the other half to potential zero. V=V 0 V=0