Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSand hence that the H n (x) satisfy the Hermite equationy ′′ − 2xy ′ +2ny =0,where n is an integer ≥ 0.Use Φ to prove that(a) H n(x) ′ =2nH n−1 (x),(b) H n+1 (x) − 2xH n (x)+2nH n−1 (x) =0.18.6 A charge +2q is situated at the origin and charges of −q are situated at distances±a from it along the polar axis. By relating it to the generating function for theLegendre polynomials, show that the electrostatic potential Φ at a point (r, θ, φ)with r>ais given byΦ(r, θ, φ) =2q ∞∑ ( a) 2sP2s (cos θ).4πɛ 0 r rs=118.7 For the associated Laguerre polynomials, carry through the following exercises.(a) Prove the Rodrigues’ formulaL m n (x) = ex x −m d nn! dx n (xn+m e −x ),taking the polynomials to be defined byn∑L m n (x) = (−1) k (n + m)!k!(n − k)!(k + m)! xk .k=0(b) Prove the recurrence relations(n +1)L m n+1(x) =(2n + m +1− x)L m n (x) − (n + m)L m n−1(x),x(L m n ) ′ (x) =nL m n (x) − (n + m)L m n−1(x),but this time taking the polynomial as defined byL m n (x) =(−1) m dmdx L n+m(x)mor the generating function.18.8 The quantum mechanical wavefunction for a one-dimensional simple harmonicoscillator in its nth energy level is of the formψ(x) =exp(−x 2 /2)H n (x),where H n (x) isthenth Hermite polynomial. The generating function for thepolynomials is∞∑ HG(x, h) =e 2hx−h2 n (x)= h n .n!n=0(a) Find H i (x) fori =1, 2, 3, 4.(b) Evaluate by direct calculation∫ ∞−∞e −x2 H p (x)H q (x) dx,(i) for p =2,q = 3; (ii) for p =2,q = 4; (iii) for p = q = 3. Check youranswers against the expected values 2 p p! √ πδ pq .642