1000 Solved Problems in Modern Physics

1.1 Basic Concepts and Formulae 5

First few Hermite’s polynomials are:

Ho(x)= 1 ,H 1 (x)= 2 x,H 2 (x)= 4 x^2 − 2

H 3 (x)= 8 x^3 − 12 x,H 4 (x)= 16 x^4 − 48 x^2 + 12 (1.23)

Generating function:

e^2 tx−t

2

=

∑∞

n= 0

Hn(x)tn

n!

(1.24)

Recurrence formulas:

Hn′(x)= 2 nHn− 1 (x)

Hn+ 1 (x)= 2 xHn(x)− 2 nHn− 1 (x) (1.25)

Orthonormal properties:

∫∞

−∞

e−x

2

Hm(x)Hn(x)dx= 0 m =n (1.26)

∫∞

−∞

e−x

2

{Hn(x)}^2 dx= 2 nn!

√

π (1.27)

Legendre functions:

Differential equation of ordern:

(1−x^2 )y′′− 2 xy′+n(n+1)y= 0 (1.28)

whenn= 0 , 1 , 2 ,...we get Legendre polynomialsPn(x).

Pn(x)=

1

2 nn!

dn

dxn

(x^2 −1)n (1.29)

First few polynomials are:

Po(x)= 1 ,P 1 (x)=x,P 2 (x)=

1

2

(3x^2 −1)

P 3 (x)=

1

2

(5x^3 − 3 x),P 4 (x)=

1

8

(35x^4 − 30 x^2 +3) (1.30)

Generating function:

1

√

1 − 2 tx+t^2

=

∑∞

n= 0

Pn(x)tn (1.31)