Tensors for Physics

234 12 Integral Formulae and Distribution Functions

The shear flow induced distortion of the pair correlation function implies a corre-
sponding anisotropy of the static structure factor. This anisotropy has been observed
in computer simulations [59], in light scattering [60] and in neutron scattering exper-
iments [61].

12.5 Selection Rules for Electromagnetic Radiation

12.5.1 Expansion of the Wave Function

LetΨ =Ψ(t,r)be the wave function, in spatial representation, which obeys the
Schrödinger equationfor a quantum mechanical single particle problem, e.g. the
electronboundbytheproton,inthehydrogenatom.ThepertainingHamiltonoperator
is the sum of the operator for the kinetic energy, viz.

H

op

kin=−

^2

2 m

Δ=−

^2

2 m

Δr−

^2

2 m

r−^2 LμLμ,

cf. Sect.7.6.4, and the potential energyV = V(r). For the radial partΔrof
the Laplace operator see (7.91). The angular part of the Laplacian involves the
differential operatorLμ=εμλνrλ∂/∂rν,cf.(7.80).
To describe the angle dependence of the wave functionΨ(t,r)an expansion can
be made with respect to spherical harmonics, as found in text books on Quantum
Mechanics, or with respect to Cartesian basis tensors, as presented here. With the
help of the normalized basis functions, see also (12.14),

φμ 1 ···μ≡

√

( 2 + 1 )!!

!

̂rμ 1 ···̂rμ, (12.136)

the expansion is written as

Ψ(t,r)=

∑∞

= 0

cμ 1 ···μ(t,r)φμ 1 ···μ(̂r). (12.137)

In general, the moment tensorscμ 1 ···μ, depend on the timetand onr=|r|. There
is no explicit time dependence whenΨis a solution of the stationary Schrödinger
equation. Being symmetric traceless, theth rank tensorcμ 1 ···μhas 2+1 inde-
pendent components, in accord with the same number ofm-values of the spherical
components.