Rotational wave equation

The function r, which is a function of the Eulerian angles 6, < >, and x (Appendix I) describing the orientation in space of the rotating coordinate system, is obtained by solution of the rotational wave equation. Some results from the rotational problem are summarized in Appendix XVI. 

This condition, together with the expressions for the rotating waves. Equation (13), defines the Hopf locus in parameter space. (The determinant of (15)... 

Let us consider a diatomic molecule in such a crystal. As a first approximation we may neglect the translational oscillations of the molecule under consideration and both the translational and rotational motion of the other molecules in the crystal. The wave equation then may be written... 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

Solution of the wave equation for rotation of a rigid diatomic molecule leads to the following expression for rotational energy ... 

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

As stressed in Section II, the coefficients y, M, are to be determined from suitable solvability conditions. One finds that to order e2 the solvability conditions yield y, = 0. Thus, the amplitudes p, p2 cannot be determined to this order. To order e3 one obtains a nontrivial result, in the form of two coupled cubic equations for p, and p2. Among the possible solutions of these equations one obtains rotating wave solutions. Setting p2 = 0 one finds a clockwise wave ... 

Equation (8.16), in the rotating wave approximation, is then given by... 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

Note that, because Euler angles 0 and (f>, there is no contribution from the term in J2. Substituting (2.138) into (2.136), we obtain the wave equation for the rotation vibration wave functions in the Born adiabatic approximation ... 

The corresponding rotation-vibration wave equation in the Born-Oppenheimer approximation, in which all coupling of electronic and nuclear motions is neglected, is... 

It will be recalled that our use of the Bom adiabatic approximation in section 2.6 enabled us to separate the nuclear and electronic parts of the total wave function. This separation led to wave equations for the rotational and vibrational motions of the nuclei. We now briefly reconsider this approximation, with the promise that we shall study it at greater length in chapters 6 and 7. 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

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