Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

18.9.1 Properties of Hermite polynomials

The Hermite polynomials and functions derived from them are important in the

analysis of the quantum mechanical behaviour of some physical systems. We

therefore briefly outline their useful properties in this section.

Rodrigues’ formula

The Rodrigues’ formula for the Hermite polynomials is given by

Hn(x)=(−1)nex

2 dn

dxn

(e−x

2

). (18.130)

This can be proved using Leibnitz’ theorem.

Prove the Rodrigues’ formula (18.130)for the Hermite polynomials.

Lettingu=e−x

2

and differentiating with respect tox, we quickly find that

u′+2xu=0.

Differentiating this equationn+ 1 times using Leibnitz’ theorem then gives

u(n+2)+2xu(n+1)+2(n+1)u(n)=0,

which, on introducing the new variablev=(−1)nu(n), reduces to

v′′+2xv′+2(n+1)v=0. (18.131)

Now lettingy=ex

2

v, we may write the derivatives ofvas

v′=e−x

2

(y′− 2 xy),

v′′=e−x

2

(y′′− 4 xy′+4x^2 y− 2 y).

Substituting these expressions into (18.131), and dividing through bye−x^2 , finally yields
Hermite’s equation,

y′′− 2 xy+2ny=0,

thus demonstrating thaty=(−1)nex

2

dn(e−x

2
)/dxnis indeed a solution. Moreover, since
this solution is clearly a polynomial of ordern, it must be some multiple ofHn(x). The
normalisation is easily checked by noting that, from (18.130), the highest-order term is
(2x)n, which agrees with the expression (18.128).

Mutual orthogonality

We saw in section 17.4 that Hermite’s equation could be cast in Sturm–Liouville

form withp=e−x

2

,q=0,λ=2nandρ=e−x

2

, and its natural interval is thus

[−∞,∞]. Since the Hermite polynomialsHn(x) are solutions of the equation and

are regular at the end-points, they must be mutually orthogonal over this interval

with respect to the weight functionρ=e−x

2

,i.e.

∫∞

−∞

Hn(x)Hk(x)e−x

2

dx=0 ifn=k.

This result may also be proved directly using the Rodrigues’ formula (18.130).

Indeed, the normalisation, whenk=n, is most easily found in this way.