18.9 HERMITE FUNCTIONS

Show that

I≡

∫∞

−∞

Hn(x)Hn(x)e−x

2

dx=2nn!

√

π. (18.132)

Using the Rodrigues’ formula (18.130), we may write

I=(−1)n

∫∞

0

Hn(x)

dn

dxn

(e−x

2

)dx=

∫∞

−∞

dnHn

dxn

e−x

2

dx,

where, in the second equality, we have integrated by partsntimes and used the fact that
the boundary terms all vanish. From (18.128) we see thatdnHn/dxn=2nn!. Thus we have

I=2nn!

∫∞

−∞

e−x

2

dx=2nn!

√

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<∞to be expanded in a series of the form

f(x)=

∑∞

n=0

anHn(x),

in which the coefficientsanare given by

an=

1

2 nn!

√

π

∫∞

−∞

f(x)Hn(x)e−x

2

dx.

We note that it is sometimes convenient to define theorthogonal Hermite functions

φn(x)=e−x

(^2) / 2
Hn(x); they also may be used to produce a series expansion of a
function in the interval−∞ ≤x<∞. Indeed,φn(x) is proportional to the
wavefunction of a particle in thenth energy level of a quantum harmonic
oscillator.
Generating function
The generating function equation for the Hermite polynomials reads
G(x, h)=e^2 hx−h
2

∑∞
n=0
Hn(x)
n!
hn, (18.133)
a result that may be proved using the Rodrigues’ formula (18.130).

Show that the functionsHn(x)in (18.133) are the Hermite polynomials.
It is often more convenient to write the generating function (18.133) as
G(x, h)=ex
2
e−(x−h)
2

∑∞

n=0

Hn(x)

n!

hn.