The Chemistry Maths Book, Second Edition

384 Chapter 13Second-order differential equations. Some special functions

By dividing throughout by−A

2

α 22 mand noting thatα 1 = 1 mω2A, where

is the angular frequency of the oscillator, we obtain

This differential equation is identical to equation (13.34). The Hermite functions are

therefore the eigenfunctions of the quantum-mechanical harmonic oscillator problem.

13.6 The Laguerre equation

The Laguerre equation

4

xy′′ 1 + 1 (1 1 − 1 x)y′ 1 + 1 ny 1 = 10 (13.38)

where nis a real number, has a power-series solution that, when nis a positive integer

or zero, is a polynomial of degree ncalled a Laguerre polynomialL

n

(x):

(13.39)

forn 1 = 1 0, 1, 2, 3, =The first few of these are

L

0

(x) 1 = 11 L

1

(x) 1 = 111 − 1 x

L

2

(x) 1 = 121 − 14 x 1 + 1 x

2

L

3

(x) 1 = 161 − 118 x 1 + 19 x

2

1 − 1 x

3

(13.40)

The Laguerre polynomials satisfy the recurrence relation

L

n+ 1

(x) 1 − 1 (1 1 + 12 n 1 − 1 x)L

n

(x) 1 + 1 n

2

L

n− 1

(x) 1 = 10 (13.41)

from which, given L

0

and L

1

, all higher polynomials can be found.

0 Exercises 23, 24

Associated Laguerre functions

The associated Laguerre equation is

xy′′ 1 + 1 (m 1 + 111 − 1 x)y′ 1 + 1 (n 1 − 1 m)y 1 = 10 (13.42)

and has polynomial solution when both nand mare positive integers or zero, with

m 1 ≤ 1 n. These solutions are the associated Laguerre polynomialsL

m

n

(x). They are

related to the Laguerre polynomials by the differential formula

(13.43)

Lx

d

dx

Lx

n

m

m

m

n

()= ()

Lx x

n

x

nn

x

n

nn n n

() ( )

()

=− − (

!

• 
  

−

!

−+−

−−

1

1

1

2

1

2

1

22

2

 ))

n

n!

















′′+− + = = +

()

ψψ()120 ω,=,,,, 0123

2

1

2

zn if En...n

ω= km

4

Edmond Laguerre (1834–1886).