13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 251
is perpendicular tou. This also applies to the rotational momentum operatorJ,
thus one hasJ·u=0. The eigenvalues ofJ·Jarej(j+ 1 )wherejcan have the
integer values 0, 1 , 2 , 3 ,...for hetero-nuclear molecules like HD or CO. For homo-
nuclear molecules,jmust be even or odd, depending on the nuclear spins, e.g. one
hasj= 0 , 2 , 4 ,...for para-hydrogen, where the total spin of the two protons is
zero, andj= 1 , 3 ,...for ortho-hydrogen the total nuclear spin is 1, in units of.
Magnetic quantum numbersm, again in units of, assume the values−j,−j+1,
...j− 1 ,j. When thez-axis is identified with the quantization direction, one has,
in ‘bra-ket’ notation,
Jz|jm〉=m|jm〉, J·J|jm〉=j(j+ 1 )|jm〉.Here|jm〉indicates the quantum mechanical state vector. Despite of this name, the
quantity|jm〉is not a vector in the sense of being a tensor of rank 1, as defined in
Sect.2.5. The components ofJobey the angular momentum commutation relations
JμJν−JνJμ≡[Jμ,Jν]−=iεμνλJλ. (13.51)In contradistinction to the spin operatorss,cf.13.1, the magnitude of the rotational
angular momentum is not fixed and it cannot have half-integer eigenvalues.
13.6.2 Projection into Rotational Eigenstates, Traces.
A state with a fixed eigenvaluejis obtained with the help of the projection operator
Pj=∑jm=−j|jm〉〈jm|. (13.52)Applications may require the projection of an observableO=O(u)depending on
u, into rotational eigenstates. This means, expressions of the form
Ojj′
≡PjO(u)Pj′
, (13.53)are needed. The casesj= j′and j = j′are often called ‘diagonal’ and ‘non-
diagonal’, without mentioning that these terms just refer to the rotational quantum
numbers and not to the magnetic quantum numbers. Examples forOare: the electric
dipole moment parallel tou, the electric quadrupole moment or the anisotropic part
of a molecular polarizability tensor, proportional touu. The pertaining applications
are dipole and quadrupole radiation, birefringence and light scattering.