Quantum paths

Beck, T. L., Quantum path integral extension of Widom s test particle method for chemical potentials with application to isotope effects on hydrogen solubilities in model solids, J. Chem. Phys. 1992, 96, 7175-7177... 

Diffusion constants are enhanced with the approximate inclusion of quantum effects. Changes in the ratio of diffusion constants for water and D2O with decreasing temperature are accurately reproduced with the QFF1 model. This ratio computed with the QFF1 model agrees well with the centroid molecular dynamics result at room temperature. Fully quantum path integral dynamical simulations of diffusion in liquid water are not presently possible. 

Joachim C (1987) Control of the quantum path-target state distance bistable-like characteristic in a small tight-binding system. J Phys A 20 L1149... 

Fig. 4.1. An illustration of two quantum paths of electrons that contribute the HHG in a mixed gas. The variation on quantum path <5r generates the phase difference of harmonics. Notice that A/p expresses the difference in energy between the two potentials...

The first order term in A/p comes from the difference of the potential energy and the higher order terms should be included when AIP/UP is not small enough. The phases, which the freed electrons accumulate during their different quantum paths, are transferred to the harmonics through the coherent process of HHG and lead to the interferences (Fig. 4.1). 

Chandler D, Wolynes PG (1981) J Chem Phys 74 4078. Quantum Path Integral Monte Carlo can also provide a way to evaluhte the rate in non-adiabatic reactive flux correlation theory Wolynes PG (1987) J Chem Phys 87 6559... 

Figure 7. Comparison of SH (thin solid line), MFT (dashed line), and quantum path-integral (solid line with dots) calculations (Ref. 198) obtained for Model Va describing electron transfer in solution. Shown is the time-dependent population probability Pf t) of the initially prepared diabatic electronic state.

Figure 28. Time-dependent (a) adiabatic and (b) diabatic electronic excited-state populations as obtained for Model Vb describing electron transfer in solution. Quantum path-integral results [199] (big dots) are compared to mapping results for the limiting cases y = 0 (dashed lines) and Y = 1 (dotted lines) as well as ZPE-adjusted mapping results for Yi p, = 0.3 (full lines).

Figure 29. Comparison of quantum path-integral results (thick tines) and ZPE-corrected mapping results (thin lines) for the diabatic electronic populations of a three-state electron transfer model describing (a) sequential and (b) superexchange electron transfer.

Sofar the imaging results of Fig. 3.1 were discussed in very classical terms, using the notion of a set of trajectories that take the electron from the atom to the detector. However, this description does not do justice to the fact that atomic photoionization is a quantum mechanical proces. Similar to the interference between light beams that is observed in Young s double slit experiment, we may expect to see the effects of interference if many different quantum paths exist that connect the atom to a particular point on the detector. Indeed this interference was previously observed in photodetachment experiments by Blondel and co-workers, which revealed the interference between two trajectories by means of which a photo-detached electron can be transported between the atom and the detector [33]. The current case of atomic photoionization is more complicated, since classical theory predicts that there are an infinite number of trajectories along which the electron can move from the atom to a particular point on the detector [32,34], Nevertheless, as Fig. 3.2 shows, the interference between trajectories is observable [35] when the resolution of the experiment is improved [36], The number of interference fringes smoothly increases with the photoelectron energy. 

In order to treat this system theoretically, the Bloch states of the strong optical lattice must be considered. In this picture we develop a model to explain the collisional decay of the oscillations as a two-particle collision of Bloch-states (and no longer free atoms). There are various quantum paths for this collision (since every lattice momentum has several relevant branches), leading to a destructive interference of the central s-wave collisional sphere, and a splitting in the resulting collisional shell, related to the observed spectral splitting. The matrix elements for this process lead to a suppression of the outward driven shell and enhance the centrally driven collisional shell, which is no longer... 

Figure 4. The bridging between Bose-Einstein condensation in the low-density, weak interaction region and in the high-density, strong interaction region [124, 125]. Data for Tc/r , where is the critical temperature and T is the critical temperature in an ideal Bose-Einstein gas, were calculated from quantum path integral Monte-Carlo simulations for a hard-sphere many-boson model [124, 125], The effective dimensionless interaction parameter is pa, where p is the density and CT is the hard-core sphere diameter. The two open circles (o) represent experimental data for bulk liquid He.

The Onset of the Superfluid Transition in the Finite System [50, 65, 66, 155]. This transition is referred to as the X point in the bulk system. What is the analogy in a finite system Pioneering quantum path integral Monte-Carlo simulations [65, 66] established the appearance of a rounded-off (smeared) X transition in finite (He) (N = 64 and 128) clusters. This was manifested by a maximum in the temperature dependence of the specific heat (Fig. 6), which... 

Figure 8. The cluster size and temperature dependence of the superfluid relative density pj p for ( He)jy clusters (A = 8,16,32,64) [155]. Data obtained from quantum path integral Monte-Carlo simulations [155]. Finite size scaling of pjp, according to Eq. (46), is presented in the insert.

It is of considerable interest to use the electron bubble as a probe for elementary excitations in finite boson quantum systems—that is, ( He)jy clusters [99, 128, 208, 209, 243-245]. These clusters are definitely liquid down to 0 K [46 9] and, on the basis of quantum path integral simulations [65, 155], were theoretically predicted (see Chapter II) to undergo a rounded-off superfluid phase transition already at surprising small cluster sizes [i.e., Amin = 8-70 (Table VI)], where the threshold size for superfluidity and/or Bose-Einstein condensation can be property-dependent (Section II.D). The size of the ( He)jy clusters employed in the experiments of Toennies and co-workers [242-246] and of Northby and coworkers [208, 209] (i.e., N lO -lO ) are considerably larger than Amin- In this large cluster size domain the X point temperature depression is small [199], that is, (Tx — 2 X 10 — 2 X 10 for V = lO -lO. Thus for the current... 

Equilibrium properties can be determined from the partition function Zq and this quantity can, in turn, be computed using Feynman s path integral approach to quantum mechanics in imaginary time [86]. In this representation of quantum mechanics, quantum particles are mapped onto closed paths r(f) in imaginary time f, 0 f )8ft. The path integral expression for the canonical partition function of a quantum particle is given by the P 00 limit of the quantum path discretized into P segments. 

If there are quantum paths which are contained completely in the interior of the shell, and if n(i) is the principal quantum number of the largest of them, then... 

Before proceeding, it is necessary to define a centroid-constrained imaginary-time propagator [3] (i.e., the correlation function of quantum path fluctuations with respect to the position of the centroid variable... 

Since the centroids of the paths q T) in this correlation function are constrained to be at q, the paths can be rewritten as q T) = q + q(r), where g(r) is the quantum path fluctuation variable with respect to the centroid. A Fourier decomposition of the paths (t) can now be introduced such that... 

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