and the order of operators and functions in the Hamiltonian (energy operator) becomes a point of concern. Podolsky proposed in 1928 that the Laplace-Beltrami operator (times ) has the appropriate form for the quantum mechanical kinetic energy operator. This operator has the general form (summation convention: sum over repeated indices—in this case over the three Euler angles ): where is the determinant of the g-tensor: Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution. (Note: The corresponding eigenvalue equation gives the Schrödinger equation for the rigid rotor in the form that it was solved for the first time by Kronig and Rabi (for the special case of the symmetric rotor). This is one of the few cases where the Schrödinger equation can be solved analytically. All these cases were solved within a year of the formulation of the Schrödinger equation.) Nowadays it is common to proceed as follows. It can be shown that can be expressed in body-fixed angular momentum operators (in this proof one must carefully commute differential operators with trigonometric functions). The result has the same appearance as the classical formula expressed in body-fixed coordinates, The action of the on the Wigner D-matrix is simple. In particular so that the Schrödinger equation for the spherical rotor () is solved with the degenerate energy equal to . The symmetric top (= symmetric rotor) is characterized by . It is a prolate (cigar shaped) top if . In the latter case we write the Hamiltonian as and use that Hence The eigenvalue is -fold degenerate, for all eigenfunctions with have the same eigenvalue. The energies with |k| > 0 are -fold degenerate. This exact solution of the Schrödinger equation of the symmetric top was first found in 1927. The asymmetric top problem () is not soluble analytically, but it can be solved numerically. Direct experimental observation of molecular rotations For a long time, molecular rotations could not be directly observed experimentally. Only measurement techniques with atomic resolution made it possible to detect the rotation of a single molecule. At low temperatures, the rotations of molecules (or part thereof) can be frozen. This could be directly visualized by Scanning tunneling microscopy i.e., the stabilization could be explained at higher temperatures by the rotational entropy. The direct observation of rotational excitation at single molecule level was