Feynman model

Feynman model. The Feynman approach, or LCAO (hnear combination of atomic orbitals) method, assumes that a wavefunction of valence electrons i// in a metal is a linear combination of atomic functions ... 

FIG. 13 Herringbone order parameter and total energy for N2 (X model with Steele s corrugation). Quantum simulation, full line classical simulation, dotted line quasiharmonic theory, dashed line Feynman-Hibbs simulation, triangles. The lines are linear connections of the data. (Reprinted with permission from Ref. 95, Fig. 4. 1993, American Physical Society.)... 

A more elegant model of quantum mechanical computers was introduced by Feynman ([feyii85], [feynSb]). Although the formalism is similar to Benioff s - both approaches seek to define an appropriate quantum mechanical Hamiltonian H whose time evolution effectively represents the execution of a desired computation... 

Feynman s approach is more powerful because it gives an explicit construction for this Hamiltonian. Peynman s motivation for constructing his model was his belief that certain ciuantum phenomena fundamentally cannot be simulated by purely classical computers ([feyii82],[feyii87]). 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

Margolus (margfiOb] generalizes Feynman s formalism - which applies to strictly serial computation - to describe deterministic parallel quantum computation in one dimension. Each row in Margolus model is a tape of a Turing Machine, and adjacent Turing Machines can communicate when their tapes arc located at the same coordinate. Extension of the formalism to more than one dimension remains an open problem. 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

A dynamic description of the effect of relaxation on the probability of the adiabatic transition may be performed using various methods, e.g., a Feynman path integral approach similar to that presented in Section III (see also Refs. 81-84). Here we shall present the results for a simple model obtained by another method.85... 

Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature...

The Feynman-Hibbs and QFH models perform quite well in free energy calculations as long as the quantum corrections are modest. The conditions for validity of the approximations are given above. 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

To describe the fully compactified model, with Euclidean coordinates, say Xi, restricted to segments of length Li (i = 1,2,. D) and the field tp(x) satisfying anti-periodic (bag model) boundary conditions, the Feynman rules should be modified following the Matsubara prescription... 

Hirshfeld (1984) found the electrostatic charge balance at the F nuclei, based on the experimental deformation density, to be several times more repulsive (i.e., anti-bonding) than that of the promolecule. Very sharp dipolar functions at the exocyclic C, N, and F atoms, oriented along the local bonds, were introduced in a new refinement in which the coefficients of the sharp functions were constrained to satisfy the electrostatic Hellmann-Feynman theorem (chapter 4). The electrostatic imbalance was corrected with negligible changes in the other parameters of the structure. The model deformation maps were virtually unaffected, except for the innermost contour around the nuclear sites. 

Or should it be the other way around - advanced social science from an elementary standpoint In that case, my model would be a short and wonderful book by Richard Feynman. QED, an introduction to quantum electrodynamics for the general public. The comparison is not as presumptuous as one might think. On the one hand, Feynman s ability to go to the core of a subject, without technicalities but also without loss of rigor, may be unsurpassed in the history of science and is in any case beyond mine. On the other, quantum electrodynamics is more arcane than any of the topics discussed here. On balance, therefore, the reader may find my exposition just as intelligible. 

The existence of spin-1/2 particles is evidence that the projective-space model is correct. For a description of the relevant experiments with Stern-Gerlach machines, see the Feynman Lectures [FLS, Vol. Ill, Chapter 6]. 

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