Free Quantum Evolution

Moving to the treatment of the free quantum evolution, the average of the first order coordinate is assumed as vanishing 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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

Note that Eqs. (4.557) and (4.558) parallel the statistical behavior of error in measurements that being vanishing in the first case as mean deviation, is manifested in the second as squared deviation (dispersion), 

through recalling the referential Eq. (4.541)— the step (ii) in above algorithm—the average of the second order coordinate provides now the expression 

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Through characterizing the numerical results of Eq. (4.562), one firstly observes that they practically start from where the PAV function of Eq. (4.556) approaches its highest output. In other words, this tell us remarkable information according to which the observed and free quantum evolutions are continuous realities, being smoothly accorded in the point of precise measurement ( = 0). Another very interesting observation is that the PAV ratio symmetrically spans in Eq. (4.562) the existence domain either for wave PAVe [0.952, 1) or particle PAV ( 1, 1.048] manifestations around their exact equivalence PAV = 1. However, the precise wave-particle equivalence is two-fold, namely in the socalled omega ( 2) and alpha (a) points of Eq. (4.562) characterized by the extended HUR versions of Eq. (4.561) written, respectively, as... 

It is very instructive to present in a unitary manner the observed and free quantum evolution cases in the chart of Figure 4.6 by linking the HUR shapes of Eqs. (2.99) and (4.561) with the particle/wave ratios values of Eqs. (4.556) and (4.562), respectively. The PAV contribution spreads from the exclusively undulatory quantum manifestation (PAV = 0) in the observed domain of quantum evolution until the particle dominance (PAV > 1) in the free domain of quantum evolution. 

FIGURE 4.6 The chart of Heisenberg Uncertainty Relationship (HUR) appearance for observed and free quantum evolutions covering the complete scale of the particle to wave ratios as computed from the Eqs. (4.556) and (4.562), respectively the points Q and a correspond to wave-particle precise equivalence and to the special extended-HURs of Eqs. (4.563) and (4.564), respectively (Putz, 2010c). 

However, the wave-particle duality matches perfectly and always with HUR in its standard (Schrodinger) formulation of Eq. (2.99) on the other side, the wave-particle exact equivalence (PAV = 1) may be acquired only in the free evolution regime that, in turn, it is driven by modified HUR as given by Eq. (4.563). In other words, it seems that any experiment or observation upon a quantum object or system would destroy the PAV balance specific for free quantum evolution towards the undulatory manifestation through measurement. 

Yet, having the analytical expressions for both observed and free quantum evolutions may considerably refine our imderstanding of macro- and micro-universe. For instance, with various (PAV)... 

The symmetries of wavepackets viewed on a progressively finer scale offer a temperature-robust way of encoding several qubits of information [62-64], The encoding and often the full control over the quantum evolution of the wavepacket [65] can be implemented by alternating periods of free motion with phase kicks imposed by coordinate-dependent Stark shifts. Distinguishing odd from even wave forms is the essence of the decoding of the qubits of information encoded in the wavefunction by the dynamics of atoms in a trap. The calculations below demonstrate the possibility to distinguish between the even wave form/(°+) and the odd onef K... 

Thus, the CMD method is isomorphic to classical time evolution of the phase space centroids on the quantum centroid potential of mean force, Vomd. It should be noted that in the harmonic, classical, and free particle limits, the CMD representation for the QDO [Eq. (50)] is also exact. Furthermore, it should also be noted that the approximation in Eq. (50) does not rely on any kind of mean field approximation. 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

Note that entanglement occurs independently of any classical interaction of the particles. In other words, entanglement occurs for free particles as well as for particles exerting forces on one another. To put it yet another way, the possibility of entanglement arises from the quantum mechanical state space itself, not from any differential equation or differential operator used to describe the evolution of the system. 

The quantum dynamics under the control-free Hamiltonian Hq is assumed to be incapable of producing the desired evolution - Thus, a suitable control interaction Vc(0 is introduced... 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

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