Tensors for Physics

158 9 Irreducible Tensors

with

N=

!

( 2 − 1 )!!

=

1 · 2 · 3 ···(− 1 )·

1 · 3 · 5 ···( 2 − 3 )·( 2 − 1 )

. (9.11)

For a proof of this relation see Sect.10.3.5. The special casea = bleads to
P(a,a)=a^2 N, sinceP( 1 )=1.
Examples for= 1 , 2 , 3 ,4 are listed explicitly:

P 1 (a,b)=abP 1 (̂a·̂b), P 1 (x)=x, (9.12)

P 2 (a,b)=

2

3

a^2 b^2 P 2 (̂a·̂b), P 2 (x)=

3

2

(

x^2 −

1

3

)

, (9.13)

P 3 (a,b)=

2

5

a^3 b^3 P 3 (̂a·̂b), P 3 (x)=

5

2

(

x^3 −

3

5

x

)

, (9.14)

P 4 (a,b)=

8

35

a^4 b^4 P 4 (̂a·̂b), P 4 (x)=

35

8

(

x^4 −

6

7

x^2 +

3

35

)

. (9.15)

Some general properties of Legendre polynomials, in particular the prescription for
their evaluation via a generating function, are presented in Sect.10.3.5.

9.4 Cartesian and Spherical Tensors

9.4.1 Spherical Components of a Vector

Lete(x),e(y),e(z)be unit vectors parallel to the coordinate axes. A vectorais given
by the linear combinationa=axe(x)+aye(x)+aze(x).Theax,ay,azare the standard
Cartesian components. Thespherical unit vectors

e(^0 )=e(z), e(±^1 )=∓

1

√

2

(

e(x)∓ie(y)

)

, (9.16)

which have the properties

(

e(m)

)∗

=(− 1 )me(−m),

(

e(m)

)∗

·e(m

′)

=δmm′, m,m′=− 1 , 0 , 1 , (9.17)

can as well be used as basis vectors. Then the vectorais represented by

a=a(^1 )e(^1 )+a(^0 )e(^0 )+a(−^1 )e(−^1 )=

∑^1

m=− 1

a(m)e(m), (9.18)