Phys. 6124 Assignment 10: Helmholtz Equation and Hermite Polynomials, Assignments of Physics

Download Phys. 6124 Assignment 10: Helmholtz Equation and Hermite Polynomials and more Assignments Physics in PDF only on Docsity! Phys. 6124 Assignment 10 Problem 1 Separate variables in the Helmholtz equation in spherical polar coordinates splitting off the radial dependence first. Show that your separated equations have the same form as the equations we obtained in class by splitting off the azimuthal angle dependence as the first step. Problem 2 Several types of orthogonal polynomials frequently occur in various physics problem. For instance, Hermite polynomi- als Hn(x) arise in the quantum harmonic oscillator problem. We will try to find them here via an eigenvalue problem for the following differential operator: L̂ = d2 dx2 − 2x d dx . This operator is defined on a space of functions (in our case, polynomials) such that L̂Ψ(x) = Ψ′′(x) − 2xΨ′(x) for any function Ψ(x). The Hermite polynomials are known to be the eigenvectors (or, more accurately, eigenfunctions) of this operator. We want to find a few of these eigenfunctions and the corresponding eigenvalues by calculating the matrix elements of this operator and then diagonalizing it. a) For the formalism to work we have to ensure that the space of polynomials is a vector space and the operator is linear. Please prove that this is indeed the case. This operator is infinite-dimensional, so the corresponding matrix will also be inifinite-dimensional. We don’t know how to diagonalize inifinite-dimensional matrices. To circumvent this problem let us restrict our attention to a subspace of low order polynomials (up to order three). As our first step, let’s introduce a basis: e1 = 1, e2 = x, e3 = x2, and e4 = x3. b) Calculate the matrix elements Lmn of L̂ in this basis. c) Find the eigenvalues and eigenvectors of Lmn and thus determine the coefficients of the first four Hermite polyno- mials. If you are feeling adventurous, you can find quite a few more higher order polynomials this way, but the procedure quickly becomes too cumbersome due to the necessity of evaluating the determinant and solving algebraic equations of high order. Next problem explores an alternative route of generating sets of orthogonal polynomials.