Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS18.9.1 Properties of Hermite polynomialsThe Hermite polynomials and functions derived from them are important in theanalysis of the quantum mechanical behaviour of some physical systems. Wetherefore briefly outline their useful properties in this section.Rodrigues’ formulaThe Rodrigues’ formula for the Hermite polynomials is given byH n (x) =(−1) n e x2dx n (e−x2 ). (18.130)This can be proved using Leibnitz’ theorem.◮Prove the Rodrigues’ formula (18.130) for the Hermite polynomials.Letting u = e −x2 and differentiating with respect to x, we quickly find thatu ′ +2xu =0.Differentiating this equation n + 1 times using Leibnitz’ theorem then givesu (n+2) +2xu (n+1) +2(n +1)u (n) =0,which, on introducing the new variable v =(−1) n u (n) , reduces tov ′′ +2xv ′ +2(n +1)v =0. (18.131)Now letting y = e x2 v, we may write the derivatives of v asv ′ = e −x2 (y ′ − 2xy),v ′′ = e −x2 (y ′′ − 4xy ′ +4x 2 y − 2y).Substituting these expressions into (18.131), and dividing through by e −x2 , finally yieldsHermite’s equation,y ′′ − 2xy +2ny =0,thus demonstrating that y =(−1) n e x2 d n (e −x2 )/dx n is indeed a solution. Moreover, sincethis solution is clearly a polynomial of order n, it must be some multiple of H n (x). Thenormalisation is easily checked by noting that, from (18.130), the highest-order term is(2x) n , which agrees with the expression (18.128). ◭Mutual orthogonalityWe saw in section 17.4 that Hermite’s equation could be cast in Sturm–Liouvilleform with p = e −x2 , q =0,λ =2n and ρ = e −x2 , and its natural interval is thus[−∞, ∞]. Since the Hermite polynomials H n (x) are solutions of the equation andare regular at the end-points, they must be mutually orthogonal over this intervalwith respect to the weight function ρ = e −x2 ,i.e.∫ ∞−∞dnH n (x)H k (x)e −x2 dx =0 ifn ≠ k.This result may also be proved directly using the Rodrigues’ formula (18.130).Indeed, the normalisation, when k = n, is most easily found in this way.626