Mathematical Methods for Physics and Engineering - Matematica.NET

18.9 HERMITE FUNCTIONS◮Show thatI ≡∫ ∞−∞H n (x)H n (x)e −x2 dx =2 n n! √ π. (18.132)Using the Rodrigues’ formula (18.130), we may write∫ ∞∫ ∞I =(−1) n H n (x) dnd n H n0 dx n (e−x2 ) dx = e −x2 dx,−∞ dx nwhere, in the second equality, we have integrated by parts n times and used the fact thatthe boundary terms all vanish. From (18.128) we see that d n H n /dx n =2 n n!. Thus we haveI =2 n n!∫ ∞−∞e −x2 dx =2 n n! √ π,where, in the second equality, we use the standard result for the area under a Gaussian(see section 6.4.2). ◭The above orthogonality and normalisation conditions allow any (reasonable)function in the interval −∞ ≤ x