Tensors for Physics

12.5 Selection Rules for Electromagnetic Radiation 235

The normalization condition

∫

|Ψ|^2 d^3 r=1 implies

∑∞

= 0

|c|^2 = 1 , |c|^2 = 4 π

∫

c∗ν 1 ···νcν 1 ···νr^2 dr. (12.138)

In the following, it is assumed that the part

Ψ=cμ 1 ···μφμ 1 ···μ

of the wave function associated with orbital angular momentumis a solution of the
stationary Schrödinger equationHΨ=EΨwith a radially symmetric potential
V. The functionc..then obeys the equation

[

−

^2

2 m

Δr+(+ 1 )

^2

2 m

r−^2 +V(r)

]

cμ 1 ···μ=Ecμ 1 ···μ,

and the appropriate boundary and integrability conditions. For a radially symmetric
interaction potential, the tensor functionsc..are the product of a scalar radial wave
functionR(r)and a tensorCμ 1 ···μ, which is complex, in general., viz.

cμ 1 ···μ=R(r)Cμ 1 ···μ.

The radial functionsR(r)are characterized by additional quantum numbers, like
the main quantum number of the H-atom. Explicit expressions are not needed for the
discussion of the selection rules. AssumingC∗ν 1 ···νCν 1 ···ν=1, the normalization
condition (12.138) implies

∑∞

= 0

4 π

∫

|R(r)|^2 r^2 dr= 1.

The orientational properties of a state described by a wave function is determined by
its angle dependent partφ(̂r). Assuming that both the radial and the angular parts
are appropriately normalized, the expectation value of an operatorO=O(L), which
is a function of the angular momentum operatorL, is given by

〈O(L)〉=( 4 π)−^1

∫

Φ∗O(L)Φd^2 ̂r,Φ=Cμ 1 ···μφμ 1 ···μ. (12.139)

The tensorsC..are complex, in general. Examples forOare the vector polarization

〈Lμ〉and the tensor polarization〈LμLν〉.Exercise12.5deals with an application.