Quantum phase space distribution

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

Extension to the multidimensional case is trivial. Wigner developed a complete mechanical system, equivalent to quantum mechanics, based on this distribution. He also showed that it satisfies many properties desired by a phase-space distribution, and in the high-temperature limit becomes the classical distribution. 

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Suppose we are given an arbitrary system Hamiltonian H(x, p) in terms of the dynamical variables x and p we will be more specific regarding the precise meaning of x and p later. The Hamiltonian is the generator of time evolution for the physical system state, provided there is no coupling to an environment or measurement device. In the classical case, we specify the initial state by a positive phase space distribution function fci(x,p) in the quantum case, by the (position-representation) positive... 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

As an example. Fig. 18 shows the diabatic electronic population probability for Model I. The quantum-mechanical results (thick line) are reproduced well by the QCL calculations, which have assumed a localization time of to = 20 fs. The results obtained for the standard QCL (thin full line) and the energy-conserving QCL (dotted line) are of similar quality, thus indicating that the phase-space distribution p]](x, p) at to = 20 fs is similar for the two schemes. Also shown in Fig. 18 are the results obtained for a standard surface-hopping calculation (dashed line), which largely fail to match the beating of the quantum reference. 

In 1965, Joseph E. Mayer (Sidebar 13.5) and co-workers published a paper [M. Baur, J. R. Jordan, P. C. Jordan, and J. E. Mayer. Towards a Theory of Linear Nonequilibrium Statistical Mechanics. Ann. Phys. (NY) 65, 96-163 (1965)] in which the vectorial character of the thermodynamic formalism was suggested from a statistical mechanical origin. Although this paper attracted little attention at the time, its results suggest how thermodynamic geometry might be traced to the statistics of quantum mechanical phase-space distributions. 

With the LAND-map approach, due to our focus in this example on computing bath subsystem operators which are diagonal in the quantum subsystem states, and because we chose the initial density to be localized in state 1, the correlation function in (43) contains only two terms (a = a = 1 and / = / = ), and a = a = 1 and / = / = 2). For this situation the system starts out with a phase space distribution determined by the value a = a = 1 and evolves along trajectories whose as3unptotic properties are governed by forces obtained from (44) that depend on the value of / = / . ... 

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

In this section we advocate a far more advantageous route to studying conceptual features of the classical-quantum correspondence, and indeed for each mechanics independently, in which phase space distributions are used in both classical and quantum mechanics, that is, classical Liouville dynamics50 in the former and the Wigner-Weyl representation in the latter. This approach provides, as will be demonstrated, powerful conceptual insights into the relationship between classical and quantum mechanics. The essential point of this section is easily stated using similar mathematics in both quantum and classical mechanics results in a similar qualitative picture of the dynamics. 

QM1 and QM2 indicates that the coupled channel expansion may not have converged for these vibrationally inelastic processes 78 within this level of uncertainty, however, there is good agreement between the semiclassical and quantum calculations. Also shown in Fig. 12 is a phase space distribution... 

Park, J. L., Band, W., Yourgrau, W. (1980). Simultaneous measurement, phase-space distributions, and quantum state determination. . Pl s. 492,189-199. 

Morante S, Rossi GC, Testa M (2006) The stress tensor of a molecular system an exercise in statistical mechanics. J Chem Phys 125 034101 66. Nelson DF, Lax M (1976) Asymmetric total stress tensor. Phys Rev B 13 1770-1776 Das A (1978) Stress tensor in a class of gauge theraies. Phys Rev D 18 2065-2067 Cohen L (1979) Local kinetic energy in quantum mechanics. J Chem Phys 70 788-789 Cohen L (1984) Representable local kinetic tmergy. J Chem Phys 80 4277-4279 Cohen L (1996) Local values in quantum mechanics. Phys Lett A 212 315-319 Ayers PW, Parr RG, Nagy A (2002) Local kinetic tmergy and local temperature in the density-functional theory of electronic structure. Int J Quantum Chem 90 309-326 Cohen L (1966) Generalized phase-space distribution functions. J Math Phys 7 781-786 Cohen L (1966) Can quantum mechanics be formulated as classical probability theory. Philos Sci 33 317-322... 

We see that the density operator p t), the quantum analog of the classical phase space distribution f r, p -,t), is different from other operators that represent dynamical variables. The same difference in time evolution properties was already encountered in classical mechanics between dynamical variables and the distribution function, as can be seen by comparing Eq. (10.8) with (1.104) and Eq. (10.9) with (1.99). This comparison also emphasizes the correspondence between the classical and quantum Liouville operators. 

The third method is to turn the wavepacket < ) and cj)(t) into Wigner phase space distributions and approximate the quantum dynamics by classical mechanics. The wavefunctions (j) and (j)(t)... 

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