Averaging Quantum Fluctuations

3 EXTENDING HEISENBERG UNCERTAINTY 4.7.3.1 Averaging Quantum Fluctuations 

We like to identify the general quantum fluctuation conditions in which the HUR is valid and when it is eventually extended. We already note that, whereas the momentum dispersion computation is fixed by relations (4.544)-(4.546), the evaluation of the coordinate dispersion has more freedom in its internal working machinery, namely (Putz, 2010c)  

Quantum Nanochemistty-Volume I Quantum Theory and Observability 

The present algorithm may be naturally supplemented with the analysis of the wave-particle duality. This is accomplished by means of considering further averages over the quantum fluctuations for the mathematical obj ects exp(-/lx) and exp(-A x ) that are most suited to represent the waves and particles, due to their obvious shapes, respectively. Such computations of averages are best performed employing the Fourier -transformation as resulted from the de Broglie packet, Eq. (4.537) with Eq. (4.537), equivalently rewritten successively as (Putz, 2009 Putz, 2010c)  

With the rule (4.548) one may describe the average behavior of the wave and particle, respectively as 

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In this case the preparation of the barrier is performed mainly by the quantum fluctuations of the tunneling particle in the transverse direction. Note that the width of the distribution here is l/ /2 of that in the distribution function for the coordinates qp. This is due to the fact that in this case the fluctuations of the particle are of quantum character and a coherent averaging of the resonance... 

J. Troe Prof. Schinke, in your HCO results, when you compare the fluctuating quantum results with RRKM results, you observe that the RRKM curve is above the average quantum data. Can the reason be that you use an inadequate p( ) for the continuum energy range ... 

The process p + Pi —> P3 + Pa then involves the conservation of momentum, for there is no creation of any averaged momentum from the virtual quantum fluctuation. This process can be examined within the Coulomb gauge V A = 0. The field equation is then... 

In this section we analyze the dynamics of a spin subject to a classical random field and derive the equation of motion for the spin dynamics (the spin-evolution operator), averaged over the fluctuations. Following the discussion of the case with quantum fluctuations, we first analyze the dynamics in a stationary field B and a random field exactly as in the quantum case one can reduce the analysis of the dissipative corrections to the Berry phase accumulated over a conic loop to the problem with a stationary field by going over to a rotating frame. 

It must be emphasized that such phenomena are to be expected for a statistical system only in the regime of low level densities. Theories like RRKM and phase space theory (PST) (Pechukas and Light 1965) are applicable when such quantum fluctuations are absent for example, due to a large density of states and/or averaging over experimental parameter such as parent rotational levels in the case of incomplete expansion-cooling and/or the laser linewidth in ultrafast experiments. However, in the present case, it is unlikely that such phenomena can be invoked to explain why different rates are obtained when using ultrafast pump-probe methods that differ only in experimental detail. 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

If all sources of technical noise are eliminated, the signal-to-noise ratio (for repeated measurements) is fundamentally limited by the quantum fluctuations in the number of atoms which are observed to be in the 11) state. These fluctuations can be called quantum projection noise [6S], If spectroscopy is performed on N initially uncorrelated atoms (for example, T(t=0) = llj i )j), the imprecision in a determination of the frequency of the transition is limited by projection noise to (Aw) = l/fNTjtT) where t is the total averaging time. If the atoms can be initially prepared in entangled states, it is possible to achieve <... 

The identity between expressions (2.113) and (2.115) leaves with the important idea that the smearing operation produces in fact the average of quantum fluctuation for the ground state equilibrium. For the Coulomb interaction, say on the Hydrogen (H), either expression produces the working form... 

For calculating the quantum fluctuation paths averages one has to understand their inner nature in order reconciliation of free and harmonic features be achieved the so-called quantum current J t) is introduced (and presumed to appear in reality too as causing/driving the quantum... 

The actual philosophy is to introduce appropriately the quantum fluctuation information a = a(x ) respecting the average of the observed coordinate (Xg), by the Fe3mman integration rule founded in the ordinary quantum... 

It is obvious that the Eqs. (4.536) and (4.537) fulfill the necessary (natural) condition according which the average of the coordinate over the quantum fluctuations recovers the observed quantity of Eq. (4.535),... 

The relation with quantum fluctuation may be nevertheless gained by the average of the second order of the Feynman centroid-considered under the form... 

Being the quantum fluctuation factor crucial for assessing the free and observed quantum behavior, it should be noted it may discriminate between these two quantum sides of motion, however, based solely on experimental measures of classical and quantum paths, since one considers their squared averages and, respectively, as ... 

A particular general shortcoming of the alloy analogy approximation is that it cannot describe quantum fluctuations such as the zero point fluctuations of spin waves. Evidently, these can be important at and near T = 0. Moreover, an ensemble average of static fluctuations depicted by the alloy configurations will, within the CPA, inevitably lead to quasi-particles with finite lifetime even at T = 0. Namely, the ground state is genetically not that of a Fermi liquid as it mostly should be. In what follows we shall summarize briefly the current state of conceptual framework that needs to be invoked to deal with these issues. 

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