The generating function, g(x, t), for the Legendre polynomials, Pn(x) is given by:

(a) Use this generating function to derive explicit expressions for Pn(x) with 0 ≦ n ≦ 3.

(b) Show from the generating function, without using explicit expressions for Pn(x), that Pn(−x) = (−1)nPn(x).

The electrostatic potential at a distance r from a point charge Q is equal, in some units, to Q/r. A charge +2Q is placed at the origin and charges −Q are placed on the z-axis at z = ±a as shown below.

(c) Find the leading order term in the electrostatic potential at a remote point P with polar coordinates r, θ such that r ＞＞ a. ［Hint: Use the expansion of g(x, t) above with t = a/r. ］

(d) Now, assuming that the point P is close to the origin, r ＜＜ a, find the potential at this point neglecting contributions proportional to powers of r/a.

Consider a space V of real functions on an interval [−1, 1], where the inner product,

(a) Write the eigenvalue problem, M(y) = λy, in Sturm–Liouville form and hence determine the weight function w(x).

(b) Formulate the conditions for the Sturm–Liouville operator, L = w(x)M, to be self-adjoint in the space V , and hence state boundary conditions that must be satisfied by any function f(x) ∈ V .

(c) Use the generalised Rodrigues formula (see Lecture 17) for orthogonal polynomials generated by the operator M to find its first four eigenfunctions and all its eigenvalues.

(d) Find a matrix representation of M in the monomial basis {|un＞} ≡ {xn} for 0 ≦ n ≦ 3.

(e) Use the matrix representation found in part (d) to determine the eigenvalues and eigenfunctions of M for n ≦ 3, and compare with those found in part (c).

You do not need to normalise the eigenfunctions that should coincide with those of part (c) up to an overall factor .

The Schrödinger equation for the ground state of a particle of mass m in a potential V (x) is

(a) Sketch the potential V (x).