Quantum density

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

The system (reduced) quantum density current can be derived from this expression, being... 

This derived expression satisfies conditions a-d mentioned above and based on numerical computatiotf 6-2 seems to bound the exact result from above. It is similar but not identical to Wigner s original guess. The quantum phase space function which appears in Eq. 52 is that of the symmetrized thermal flux operator, instead of the quantum density. 

One possible definition of a classical-like quantum density is given by... 

This approximate inversion formula is quite accurate for bell shaped or mono-tonically increasing functions f (E). It can be substantially improved by iteration. One Laplace transforms the function f 1 [E) and then applies STILT to the difference function f(P) - fi( p). The iterated inversion formula is exact for the class of functions Eme aE. As shown in Ref. 227 it is stable with respect to noise. It has been applied successfully for obtaining quantum densities of states in Ref. 226. 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

W. Kohn, L.J. Sham, Quantum density oscillations in an inhomogeneous electron gas. Phys. Rev. 137, A1697-A1705 (1965)... 

The multipole (or polarization) moments introduced according to (2.14) present a classical analogue of quantum mechanical polarization moments [6, 73, 96,133, 304]. They are obtained by expanding the quantum density matrix [73, 139] over irreducible tensor operators [136, 140, 379] and will be discussed in Chapters 3 and 5. 

All the aforesaid also applies to the classical limit of the quantum density operator p, namely to the probability density p(0, tensor operators to which the spherical functions Ykq 0, different ways of expansion are used by different authors, both in the quantum approach and in the classical limit. This complicates considerably comparison of the results obtained by these authors, including experimental data. 

At this stage, we are confident that a clear connection between Levy statistics and critical random events is established. We have also seen that non-Poisson renewal yields a class of GME with infinite memory, from within a perspective resting on trajectories with jumps that act as memory erasers. The non-Poisson and renewal character of these processes has two major effects. The former will be discussed in detail in Section XV, and the latter will be discussed in Section XVI. The first problem has to do with decoherence theory. As we shall see, decoherence theory denotes an approach avoiding the use of wave function collapses, with the supposedly equivalent adoption of quantum densities becoming diagonal in the pointer basis set. In Section XV we shall see that the decoherence theory is inadequate to derive non-Poisson renewal processes from quantum mechanics. In Section XVI we shall show that the non-Poisson renewal properties, revealed by the BQD experiments, rule out modulation as a possible approach to complexity. 

At the same time very often the real optical field interacting with atoms ha.s rather broad spectral profile, width of which is broader or comparable with the inhomogeneous width of the atomic transition. In this case, a broad spectral line approximation for quantum density matrix approach has proved to be verj- rewai d-ing. This approximation was introduced in the 1960s by C. Cohcn-Taimoudji for excitation of atoms with ordinai-y light sources [10]. This was an era before lasers. Later on it was adjusted for application for exedtation of atoms wdth multimode lasers [11] and for excitation of molecules in the case of large angular momentum states [3, 12]. 

This quantity represents the ratio of a population at the critical region to the density of states before reaction. The variable is meant to be the total internal energy that is the sum of A (the energy of the mode A) and sB (that of B). ilA (eA), for instance, is the quantum density of states in the A mode at an energy eA, which is defined as... 

Linearization methods start from a path integral representation of the forward and backward propagators in expressions for time correlation function, and combine them to describe the overall time evolution of the system in terms of a set of classical trajectories whose initial conditions are sampled from a quantity related to the Wigner transform of the quantum density operator. The linearized expression for a correlation function provides a powerful tool for describing systems in the condensed phase. The rapid decay of... 

A balance equation for the quantum density q in the laser may be written with regard to the processes shown in (13). 

So with the values of a 10-16 — 10 18 quoted above and a total loss coefficient Ri 1 2 T2=0.3, the total threshold inversion is ANo / 1018— 1018 [cm-2]. For the somewhat idealized conditions of a photochemical iodine laser (Chapter 6.1) the balance equations (16) have been solved numerically in 25>. If a fast pumping rate Pi (<) is assumed, the change of the inversion and quantum density might appear as shown in Fig. 8. After threshold is reached, the inversion AN starts dumped oscillations around the threshold inversion ANo and later reaches stationary conditions. Thus, the part of the inversion AN which is available for laser output is that in excess of ANo. 

Fig. 8. Calculated time-dependent quantum density (upper plot) and corresponding inversion (lower plot) of a photochemical iodine laser with given linewidth and resonator parameters. After threshold is reached the inversion exhibits oscillations and shows a steady-state behavior later on...

It is seen that the bifurcations are nearly removed and that now the motion is much more in favor of a counterclockwise rotation. In fact, one finds that 83% of the rotational wavepackets move in this direction. The classical trajectories exhibit a dynamics similar to that of the quantum densities. This is true for the ones moving freely but also for the trapped trajectories which do not have enough energy to escape the potential wells for a discussion of similar features and their relation to femtosecond pump-probe signals, see Ref. 214. 

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