Classical phase space distribution

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]. 

Recently, Stock and Muller have proposed an alternative strategy to tackle the ZPE problem [102, 103, 224]. The theory is based on the observation that the unphysical flow of ZPE is a consequence of the fact that the classical phase-space distribution may enter regions of phase space that correspond to a... 

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. 

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. 

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... 

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... 

The distribution generated by surprisal analysis is meant to reproduce the results of actual interest, a typical example being the distribution of vibrational energy of products, which is of interest say for chemical laser action. [3] The distribution is not meant to reproduce the fully detailed distribution in classical phase space, which, as already noted, has a to be a highly correlated and complicated distribution. 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

The numbers

investigated statistically. For instance, one may calculate the probabihty distribution Pn function value (p if one randomly picks a mesh point j,k). Or one may look at correlations between

phase-space information can be projected directly from quantum wave functions. Thus, classical and quantum dynamics can be compared on an equal footing. All these questions are still under active investigation. 

The discovery, the linear surprisal, due to Kinsey, Bernstein, and Levine is about a rule on microcanonical rate constants ( /( /)) or the associated product distribution (p/(s/)) experimentally observed in a chemical reaction, in which a final state, for instance, in a vibrational level of a given energy Ej is specified. A statistically estimated product distribution pj (s ) corresponding to Pj(Ej) is called the prior distribution, which is usually evaluated in terms of the volume of a relevant classical phase space and is frequently represented in terms of energy parameters. Their remarkable finding [2-5] is an exponential form... 

This is the classical Lionville equation. An alternative derivation of this equation that sheds additional light on the nature of the phase space distribution function /(r, p / Z) is given in Appendix 1 A. 

Figure 9. Snapshots of the phase space distribution (PSD) obtained from classical trajectory simulations based on the fewest-switches surface-hopping algorithm of a 50 K initial canonical ensemble [46], Na atoms are indicated by black circles, and F atoms are indicated by gray crosses. Dynamics on the hrst excited state starting at the Cj structure (t = 0 fs) over the structure with broken Na-Na bond t = 90 fs) and subsequently over broken ionic Na-F bond (t = 220 fs) toward the conical intersection region (t = 400 fs), Dynamics on the ground state after branching of the PSD from the hrst excited state leads to strong spatial delocalization (t = 600 fs). The C2v isomer can be identihed at 800 fs in the center-of-mass distribution. See color insert.

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. 

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