Quantum scaling

Our renormalization procedure is internally consistent in that the physical value of the tunneling amplitude depends on the scaling variable—the bare coupling Aq—only logarithmically. This bare coupling must scale with the only quantum scale in the problem—the Debye frequency, as pointed out in the first section. 

M. A. Continentino, Quantum Scaling in Many-Body Systems, World Scientific Lecture Notes in Physics, Vol. 67, World Scientific, Singapore,... 

Note again the characteristic linear quantum scaling with the number of particles r/pi,. We often refer to the variance of the coherent light state as the shot noise level. 

A novel system has been found, which is ideally suited to explore this frontier clusters containing small or large numbers of atoms can now be made. These are new objects whose properties evolve from the free atom limit to that of the solid as a function of size, or of the number of atoms they contain. Such objects are of quantum scale when they are small, but achieve macroscopic dimensions as their size increases. Thus, one can study the evolution of properties which persist from quantum to macroscopic sizes, or else search for the earliest appearance of solid state properties, for example plasmon oscillations in solids, as a function of the number of atoms in the cluster. 

Spectroscopy remains the routine experimental application of quantum mechanics. A typical spectroscopy experiment uses large numbers of photons to probe the properties of individual atoms and molecules. Each photon interacts with only one molecule at a time, bringing the experiment to the nanometer distances—the quantum scale—of single molecules. By measuring how a beam of photons is affected by its interaction with matter, we can now deduce the quantum properties of the substance. As we develop the theory of atomic and molecular quantum mechanics over the next several chapters, we will keep coming back to the practical issue of how our results can be supported by laboratory spectroscopy. 

The question mark in the scheme above prompts us to think about the nature of the fabric in which a single quantum particle exists. Which kind of world has no space and time associated with it One glimpse of this world reveals itself in the inherent spatial non-locality of the quantum world, as deduced from Bell s inequality and experimentally confirmed by the work of Aspect and others. Another gHmpse is perhaps the lack of temporal order at the very small quantum scale, where quantum particles are thought of as moving forward as well as backward in time. By the word "time in spacetime I always referred to the macroscopic irreversible time, in the sense of Pri-gogine s non-equilibrium thermodynamics or Eddington s "arrow of time . 

Another relevant area is the study of order/disorder phenomena, acknowledging that microscopically tiny fluctuations can be somewhat immediately amplified to a macroscopic scale. What seems to be a purely random event on one level can appear to be deterministically lawful behavior on some other level. Quantum mechanics may serve as another example where the question of measurement is actually the eminent question of interpreting macroscopic images of the quantum-scale events. Factually we construct things on the basis of information, which we may call information transducers. 

Figure A3.9.9. Dissociation probability versus incident energy for D2 molecules incident on a Cu(l 11) surface for the initial quantum states indicated (u indicates the mitial vibrational state and J the initial rotational state) [100], For clarity, the saturation values have been scaled to the same value irrespective of the initial state, although in reality die saturation value is higher for the u = 1 state.

Strain M C, Scuseria G E and Frisch M J 1996 Linear scaling for the electronic quantum coulomb problem Science 271 51-3... 

The scope of this section restricts the discussion. One omitted topic is the collision and interaction of molecules with surfaces (see [20, 21] and section A3.9). This topic coimects quantum molecular dynamics in gas and condensed phases. Depending on the time scales of the interaction of a molecule witli a surface, the... 

Moiseyev N 1998 Quantum theory of resonances calculating energies, widths and cross-sections by complex scaling Rhys. Rep. 302 212... 

Finally, tlie ability to optically address single molecules is enabling some beautiful experiments in quantum optics. The non-Poissonian photon arrival time distributions expected tlieoretically for single molecules have been observed directly, botli antibunching at short times [112] and bunching on longer time scales [6, 112 and 113]. The fluorescence excitation spectra of single molecules bound to spherical microcavities have been examined as a probe... 

Atomic-scale devices already projected pose design challenges at tlie quantum mechanical level. The framework of quantum computing is now being discussed in research laboratories [48, 49]. 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

Figure 3. Low-energy vibronic spectrum in a. 11 electronic state of a linear triatomic molecule, computed for various values of the Renner parameter e and spin-orbit constant Aso (in cm ). The spectrum shown in the center of figure (e = —0.17, A o = —37cm ) corresponds to the A TT state of NCN [28,29]. The zero on the energy scale represents the minimum of the potential energy surface. Solid lines A = 0 vibronic levels dashed lines K = levels dash-dotted lines K = 1 levels dotted lines = 3 levels. Spin-vibronic levels are denoted by the value of the corresponding quantum number P P = Af - - E note that E is in this case spin quantum number),...

The second aspect, predicting reaction dynamics, including the quantum behaviour of protons, still has some way to go There are really two separate problems the simulation of a slow activated event, and the quantum-dynamical aspects of a reactive transition. Only fast reactions, occurring on the pico- to nanosecond time scale, can be probed by direct simulation an interesting example is the simulation by ab initio MD of metallocene-catalysed ethylene polymerisation by Meier et al. [93]. 

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