Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

and hence that theHn(x) satisfy the Hermite equation

y′′− 2 xy′+2ny=0,

wherenis an integer≥0.

Use Φ to prove that

(a) Hn′(x)=2nHn− 1 (x),

(b)Hn+1(x)− 2 xHn(x)+2nHn− 1 (x)=0.

18.6 A charge +2qis situated at the origin and charges of−qare situated at distances
±afrom it along the polar axis. By relating it to the generating function for the
Legendre polynomials, show that the electrostatic potential Φ at a point (r, θ, φ)
withr>ais given by

Φ(r, θ, φ)=

2 q

4 π 0 r

∑∞

s=1

(a

r

) 2 s

P 2 s(cosθ).

18.7 For the associated Laguerre polynomials, carry through the following exercises.

(a) Prove the Rodrigues’ formula

Lmn(x)=

exx−m

n!

dn

dxn

(xn+me−x),

taking the polynomials to be defined by

Lmn(x)=

∑n

k=0

(−1)k

(n+m)!

k!(n−k)!(k+m)!

xk.

(b) Prove the recurrence relations

(n+1)Lmn+1(x)=(2n+m+1−x)Lmn(x)−(n+m)Lmn− 1 (x),

x(Lmn)′(x)=nLmn(x)−(n+m)Lmn− 1 (x),

but this time taking the polynomial as defined by

Lmn(x)=(−1)m

dm

dxm

Ln+m(x)

or the generating function.

18.8 The quantum mechanical wavefunction for a one-dimensional simple harmonic
oscillator in itsnth energy level is of the form
ψ(x)=exp(−x^2 /2)Hn(x),

whereHn(x)isthenth Hermite polynomial. The generating function for the

polynomials is

G(x, h)=e^2 hx−h

2

=

∑∞

n=0

Hn(x)

n!

hn.

(a) FindHi(x)fori=1, 2 , 3 ,4.

(b) Evaluate by direct calculation

∫∞

−∞

e−x

2

Hp(x)Hq(x)dx,

(i) forp=2,q= 3; (ii) forp=2,q= 4; (iii) forp=q= 3. Check your

answers against the expected values 2pp!

√

πδpq.