The Chemistry Maths Book, Second Edition

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

(b)

In this case, bothP

1

(x)andP

3

(x)are odd functions so that the orthogonality of

the functions is a new property, not a consequence of even 2 odd parity.

0 Exercise 19

The corresponding property of the associated Legendre functions is

(13.26)

In addition whenl 1 = 1 l′,

(13.27)

and this result is used to construct the set of normalizedassociated Legendre

functions (and normalized Legendre polynomials whenm 1 = 10 ),

(13.28)

with property

(13.29)

Whenx 1 = 1 cos 1 θthese functions form part of the solutions of the Schrödinger

equation for the hydrogen atom (Section 14.6).

EXAMPLE 13.9Show that (i)P

1

1

is orthogonal toP

3

1

, (ii)P

2

2

is orthogonal toP

2

3

.

(i)

IPxPxdx==−−xxd

−

+

−

+

ZZ

1

1

1

1

3

1

1

1

22

3

2

() () ( )( 151 )xx

Px x Px x x

1

1212

3

12122

1

3

2

() ( ) , () ( ) (=− = − 151 −)

Z

−

+

, ′,,′

==

= ′

≠ ′

1

1

1

0

ΘΘ

lm l m ll

xxdx

ll

ll

() () δ

if

if











Θ

lm l

m

x

llm

lm

Px

,

||

=

+−!

+!

()

()(||)

(||)

()

21

2

Z

−

+

||

( )

=

• 
  

+!

−!

1

1

2

2

21

Pxdx

l

lm

lm

l

m

()

()

(||)

(||)

Z

−

+

||

′

||

=≠′

1

1

PxPxdx 0 ll

l

m

l

m

() () when

ZZ

−

+

−

+

−

+

=−=

1

1

13

1

1

42

1

1

1

2

53

1

2

P x P x dx() () (x x dx) xxx

53

1

2

−=−00 0













=















