Tensors for Physics

190 11 Isotropic Tensors

Δrrμ 1 rμ 2 ···rμ =(+ 1 )r−^2 rμ 1 rμ 2 ···rμ,

andconsequentlytheresult(11.31)isrecovered.This,incidentally,provesthevalidity
of the-dependent factors in (11.26) and (11.29).

11.3.4 Consequences for the Orbital Angular Momentum

Operator

In spatial representation, the orbital angular momentum operatorL, in units of,is
related to the differential operatorLby

Lμ=

1

i

Lμ,

cf. (7.86). The action of the operatorLon irreducible tensors rμ 1 rμ 2 ···rμ is
immediately inferred from (11.29). Thanks to the imaginary unitioccurring here,
application ofL·L=LλLλonrμ 1 rμ 2 ···rμ yields

LλLλrμ 1 rμ 2 ···rμ =(+ 1 )rμ 1 rμ 2 ···rμ. (11.32)

Thus the irreducible tensors of rank, constructed from the components ofror of its
unit vector̂r, as well as the multipole potential tensorsXμ 1 μ 2 ···μare eigenfunctions
of the square of the angular momentum with the eigenvalues(+ 1 ). Sinceis the
rank of a tensor, the possible values forare non-negative, integer numbers. This
underlies the quantization of the magnitude of the orbital angular momentum.
Spherical components of irreducible tensors, cf. Sect.9.4.2, which involve the
components of the position vectorr, likerμ 1 rμ 2 ···rμ, the multipole potentials

X···, or the spherical harmonicsY(m), are eigenfunctions of thez-componentLzof
the orbital angular momentum operatorL.For=1, with the components ofr
denoted by{x,y,z}, this is inferred fromLz(x±iy)=±ix−y, which implies
Lz(x±iy)=±(x±iy), andLzz= 0 = 0 ·z. Obviously, the eigenvalues occurring
here arem=±1 andm=0. For general, the eigenvalue equation is

LzY(m)=mY(m). (11.33)

Due toLz=( 1 /i)∂/∂φ, this relation follows fromY(m)∼exp(miφ),cf.(9.29).
Since the operatorLdoes not affect the magnituderofr, but only the angles, the

spherical components ofrμ 1 rμ 2 ···rμ and of the multipole potentialsX···intro-
duced in Sect.10.1.1, also obey the eigenvalue equation (11.33).