Introduction to Quantum Mechanics · Problem 2.64

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · legendre_differential_equation

Problem

Legendre's differential equation reads

\left(1 - x^2\right)\frac{d^2 f}{dx^2} - 2x\frac{df}{dx} + \ell(\ell+1)\,f = 0, \tag{2.208}

where

\ell

is some (non-negative) real number. (a) Assume a power series solution,

f(x) = \sum_{n=0}^{\infty} a_n\,x^n,

and obtain a recursion relation for the constants

a_n

. (b) Argue that unless the series truncates (which can only happen if

\ell

is an integer), the solution will diverge at

x = 1

. (c) When

\ell

is an integer, the series for one of the two linearly independent solutions (either

f_{\text{even}}

f_{\text{odd}}

depending on whether

\ell

is even or odd) will truncate, and those solutions are called Legendre polynomials

P_\ell(x)

. Find

P_0(x)

P_1(x)

P_2(x)

, and

P_3(x)

from the recursion relation. Leave your answer in terms of either

a_0

a_1