EBK quantization

Systems in the collinear eZe configuration which have tori would be the antiproton-proton-antiproton (p-p-p) system, the positronium negative ion (Pr- e-e-e)), which corresponds to the case of Z= 1, = 1, and If these systems have bound states, we can see the effect of our finding in the Fourier transform of the density of states for the spectrum. For a positronium negative ion, the EBK quantization was done [34]. Stable antisymmetric orbits were obtained and were quantized to explain some part of the energy spectrum. As hyperbolic systems, H and He have been already analyzed in Refs. 11 and 17, respectively. Thus, Li+ is the next candidate. We might see the effect of the intermittency for this system in quantum defect as shown for helium [14]. 

Figure 5. Island chain in the Base (M oiodel problem. The presence of such classical resonances prevented Hase from carrying out EBK quantization for this model of H-C-C dynamics above dissociation. See Table III. Reproduced from ref. ( ). Copyright 1983, American Chemical Society.

The first task is to construct semiclassical wave function in the classically allowed region. As is well known, the KAM torus exists according to the Kolmogorov-Arnold-Moser (KAM) [57,58] and this integral system can be quantized by the Einstein-Brillouin-Keller (EBK) quantization rule [58] as... 

Before 1925/26 quantum mechanics was based on ad hoc quantization procedures such as the Bohr-Sommerfeld quantization method. The disadvantage of this method is that it is coordinate dependent, and usually works only in Cartesian coordinates. In 1917 Einstein suggested a coordinate independent quantization procedure, an important improvement over the Bohr-Sommerfeld quantization scheme. Einstein s quantization method was subsequently extended and improved by Brillouin (1926) and Keller (1958). Therefore, this quantization scheme is referred to as EBK... 

One can Immediately ask, what If there Is not such an EBK torus for the coupled system The empirical result Is that the method may well work anyway Table 111 shows results (26) obtained by adiabatic quantization of the Hase ( ) HCC two-degrees-of-free-dom problem. Away from the 5 2 resonances (see Fig. 5) the adiabatic and Hase results are In accord, but the adiabatic method also successfully quantizes the resonance zones. Another Illustration Is given In Table IV where a two-degrees-of-freedom model of HOD Is adlabatlcally quantized above the classical dissociation threshold... 

Table III. Adlabatlcally obtained semlclasslcal resonance energies for the H-C-C system of Ref. (O. Results of BR (Borondo, Reinhardt (26)) are compared with EBK results of Hase (O. As discussed In the text, the adiabatic method quantizes the classical resonances, where, as Indicated by Hase was unable to obtain results due to nonlinear resonance.

The preceding normal-mode/rigid-rotor sampling assumes the vibrational-rotational levels for the polyatomic reactant are well described by separable normal modes and separability between rotation and vibration. However, if anharmonicities and mode-mode and vibration-rotation couplings are important, it may become necessary to go beyond this approximation and use the Einstein-Brillouin-Keller (EBK) semiclassical quantization conditions [32]... 

The adiabatic switching semiclassical quantization method [60-62] may also be used to choose initial conditions for polyatomic reactants. This approach does not require an explicit determination of the topologically independent paths Ct and actions Jt for Eq. (3.36) and, in principle, may be more easily applied to larger polyatomics than the EBK semiclassical quantization approach described above. However, what is required is a separable zero-order Hamiltonian H0 that gives rise to the same kind of intramolecular motion as does the complete Hamiltonian [63,64]. 

EBK semiclassical quantization is applicable if the classical motion is quasiperiodic. For N coupled modes, the quantum conditions are... 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

When classical trajectories are calculated for the H2O model with two O—H stretches discussed in section 4.3.2 (p. 76), local-mode type motion is found for which energy is trapped in individual O—H bonds (Lawton and Child, 1979, 1981). The trajectories are quasiperiodic and application of the EBK semiclassical quantization condition [Eq. (2.72)] results in pairs of local-mode states in which there are n quanta in one bond and m in the other or vice versa. The pair of local-mode states (n,m) and (m,n) have symmetry-related trajectories which have the same energy. The local-mode trajectory for the (5,0) state is depicted in Figure 4.6d. 

The Bohr theory was extended in various ways, particularly by Arnold Sonunofeld. This old quantum theory is still used as a tool to understand quantum mechanics and to obtain LQ)proximate solutions. The old quantum theory deals with, for example, the Bohr-Sonunerfeld quantum condition (quantization of action along a path) EBK theory of A. Einstein, L. Brillouin, and J. B. KeUer and JWKB theory of H. Jeffreys, G. Wenzel, H. A. Kramers, and L. Brillouin. Unfortunately, there is no possibiUty of giving justice to the various aspects of old quantum theory in this book. 

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