Tensors for Physics

9.4 Cartesian and Spherical Tensors 159

with thespherical components

a(m)=a·(e(m))∗=(− 1 )ma·e(−m). (9.19)

Explicitly, the relation between the spherical and Cartesian components is

a(^0 )=az, a(±^1 )=∓

1

√

2

(ax±iay). (9.20)

The unit vector̂r=r−^1 ris represented in terms of the spherical polar anglesθ
andφaccording to{cosφsinθ,sinφsinθ,cosθ}. Then the spherical components
of the position vectorrare

r(^0 )=rcosθ, r(±^1 )=∓

1

√

2

rsinθexp(±iφ). (9.21)

Apart from the numerical factor

√

4 π/3, the spherical components of̂r=r−^1 rare
equal to the first order spherical harmonicsY 1 m(̂r). Similarly, for any vectora=âa,
one has

a(m)=a

√

4 π

3

Y 1 m(̂a). (9.22)

The generalization of the interrelation between Cartesian and spherical components
to tensors of rank>1 is discussed next.

9.4.2 Spherical Components of Tensors

LetSμ 1 μ 2 ···μbe a symmetric traceless tensor of rank, given by its Cartesian com-

ponents. By analogy to (9.19) the pertaining spherical componentsS(m)are obtained
by the scalar multiplication with-fold product of the Cartesian components of the
vectorse(m)∗,viz.

S(m)=

∑^1

m 1 =− 1

...

∑^1

m=− 1

(− 1 )m

×Sμ 1 μ 2 ···μeμ(− 1 m^1 )e(μ− 2 m^2 )···eμ(−m)δ(m,m 1 +m 2 +...+m). (9.23)

Here, the notationδ(m,m 1 +m 2 +...+m)is used for the Kronecker delta symbol,
i.e.δ(···)=1,form 1 +m 2 +...+m=m,andδ(···)=0,otherwise.Bydefinition,
the possible values formare the integer numbersm=−,−+ 1 ,..., 0 ,...,−
1 ,. Clearly, the irreducible tensor of rankhas 2+1 spherical components.