1000 Solved Problems in Modern Physics

3.1 Basic Concepts and Formulae 135

Hetero-nuclear diatomic molecules such as CO and linear polyatomic molecule
such as HCN, do possess dipolemoment and so also rotational spectrum.
The selection rule is

ΔJ= 0 ,± 1. (3.20)

The occurrence of a rotational Raman spectrum depends on the change in polar-
izibility. Such a change can occur in symmetric top molecules. The selection rule is

ΔJ= 0 ,± 2. (3.21)

Required properties of the wave function

Ψshould be finite, single-valued and continuous. These requirements follow from
the interpretation of|ψ|^2 as the probability density.

Ladder operators

Raising operator:

J+|jm>=

√

(j+m+1)(j−m)|j,m+ 1 >

Lowering operator:

J−|jm>=

√

(j−m+1)(j+m)|j,m− 1 > (3.23)

The Klein–Gordon equationfor the relativistic free motion of a spinless particle
of rest massm.

∇^2 φ−

1

c^2

∂^2 φ

∂t^2

=

m^2 c^2 φ

^2

(3.24)

If the quantum of nuclear field is assumed to have massm, then its range

R∼



mc

(3.25)

The variation method

It consists of evaluating the integrals on the RHS of the inequality

E 0 

∫

ψ∗Hψdτ

∫

|ψ|^2 dτ

(3.26)