Wigner phase-space

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

Here, we give here a brief outline of the methods as introduced in Refs. 43, 44, and 47. Suppose that the initial state of the system is tpoili, ,<]n) From ipo, the Wigner phase-space distribution D(qi,..., qn pi,. ..,Pn] is computed. This distribution is used to sample initial positions and momenta, . .., for a classical trajectory simulation of the process of... 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

D. J. Tannor I also have a question to Prof. Zewail on the doublepeak structure in Nal. Would the double-peak structure be characteristic of (li olti i (/)>. where j/t is the first excited state of the oscillator, as can be seen from Wigner phase-space pictures (cf. Fig. 1) ... 

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

G. J. Martyna, J. Chem. Phys. (in press, 1996). In this paper, an effective set of molecular dynamics equations are specified that provide an alternative path-integral approach to the calculation of position and velocity time correlation functions. This approach is essentially based on the Wigner phase-space function. For general nonlinear systems, the appropriate MD mass in this approach is not the physical mass, but it must instead be a position-dependent effective mass. 

To convert the Ehrenfast equations of motion, Eqs. (D.14) and (D.16), into the dynamics of p(t), let us consider the involved Green s function or propagator G(t t ) in the Wigner phase space by which... 

As it does no longer depend on the variables qt,Pt, t, the above equation for the Green s function is also valid for the reduced density operator in the Wigner phase space. Therefore,... 

In this limit, the momentum is no longer a dynamics variable and does not correlate with system. By the time of interest the momentum of solvation mode has already equilibrated. This would correspond to the Wigner phase space distribution of ... 

Heller EJ () Wigner phase space method Analysis for semiclassical applications. J Chem Phys 65 1289... 

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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