the angular momentum of the rigid rotor. With respect to the body-fixed frame it has the components , and can be shown to be related to the angular velocity, This angular momentum is a conserved (time-independent) quantity if viewed from a stationary space-fixed frame. Since the body-fixed frame moves (depends on time) the components are not time independent. If we were to represent with respect to the stationary space-fixed frame, we would find time independent expressions for its components. The kinetic energy is expressed in terms of the angular momentum by Hamilton form The Hamilton form of the kinetic energy is written in terms of generalized momenta where it is used that the is symmetric. In Hamilton form the kinetic energy is, with the inverse metric tensor given by This inverse tensor is needed to obtain the Laplace-Beltrami operator, which (multiplied by ) gives the quantum mechanical energy operator of the rigid rotor. The classical Hamiltonian given above can be rewritten to the following expression, which is needed in the phase integral arising in the classical statistical mechanics of rigid rotors, Quantum mechanical rigid rotor As usual quantization is performed by the replacement of the generalized momenta by operators that give first derivatives with respect to its canonically conjugate variables (positions). Thus, and similarly for and . It is remarkable that this rule replaces the fairly complicated function of all three Euler angles, time derivatives of Euler angles, and inertia moments (characterizing the rigid rotor) by a simple differential operator that does not depend on time or inertia moments and differentiates to one Euler angle only. The quantization rule is sufficient to obtain the operators that correspond with the classical angular momenta. There are two kinds: space-fixed and body-fixed angular momentum operators. Both are vector operators, i.e., both have three components that transform as vector components among themselves upon rotation of the space-fixed and the body-fixed frame, respectively. The explicit form of the rigid rotor angular momentum operators is given here (but beware, they must be multiplied with ). The body-fixed angular momentum operators are written as . They satisfy anomalous commutation relations. The quantization rule is not sufficient to obtain the kinetic energy operator from the classical Hamiltonian. Since classically commutes with and and the inverses of these functions, the position of these trigonometric functions in the classical Hamiltonian is arbitrary. After quantization the commutation does no longer hold