Phase space quantum spectrum

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

The electric field envelope of the femtosecond pump pulse which is short compared to the period of the oscillations in Fig. 15.3 (b) covers a frequency range much broader than the energy spacing of individual levels of the low-frequency mode. In other words, the pump spectrum overlaps with several lines of the vibrational progression depicted in Fig. 15.1 (b). As a result, impulsive dipole excitation from the Vqd = 0 to 1 state creates a nonstationary superposition of the wavefunc-tions of low-frequency levels in the Vqd = 1 tate with a well-defined mutual phase. This quantum-coherent wavepacket oscillates in the Vqd = 1 state with the frequency Q of the low-frequency mode and leads to a modulation of O-H stretching absorption which is measured by the probe pulses. In addition to the wavepacket in the Vqd = 1 state, impulsive Raman excitation within the spectral envelope of... 

Trajectories in action/angle polyad phase space convey all of the most important qualitative relationships between a quantum spectrum and classical intramolecular dynamics. However, coordinate space trajectories are both more easily visualized and more directly comparable to quantum probability densities, ip(Qi,Q2,... ( 3jv-6) 2 Xiao and Kellman (1989) describe how the action/angle phase space trajectories for each eigenstate may be converted into a coordinate space trajectory. The key to this is the exact relationship between Morse oscillator displacement coordinates, rit and the action, angle variables, Ii,4>i (Rankin and Miller, 1971). Figure 9.17 shows, for the 6 eigenstates in the I = 3 (N = vs + va = 5) polyad of H20, the correspondences between the phase space trajectories, the coordinate space trajectories, and the probability densities. The resemblance between the classical coordinate space trajectories and the quantum probability densities is striking ... 

An alternative route is based on time-dependent approaches, where the standard statistical mechanics formalism relies on Fourier transform of the time correlation of vibrational operators [54—57]. These approaches can provide a complete description of the experimental spectrum, that is, the characterization of the real molecular motion consisting of many degrees of freedom activated at finite temperature, often strongly coupled and anharmonic in namre. However, computation of the exact quantum dynamics evolution of the nuclei on the ab initio potential surface is as prohibitive as the quantum/stationary-state approaches. In fact, even a semiclassical description of the time evolution of quanmm systems is usually computationally expensive. Therefore, time correlation methods for realistic systems are usually carried out by sampling of the nuclear motion in the classical phase space. In this context, summation over i in Eq. 11.1 is a classical ensemble average furthermore, the field unit vector e can be averaged over all directions of an isotropic fluid, leading to the well-known expression... 

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

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