Phase-space representation

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... 

Another way to study the quantum dynamics of a system is to consider quantum phase space representations, that can be compared directly with classical results. Although there is no unique way to define a phase space representation of quantum mechanics, the most popular are the Wigner (Wigner, 1932) and Husimi (Husimi, 1940) functions. The Wigner transform... 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

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 ... 

For such phase space representations with one variable, or indeed with any number of independent concentrations, there are a number of rules which the trajectories must obey. In particular, trajectories cannot cross themselves, except at singular points (the stationary states) or if they form closed orbits (such as limit cycles or some other forms we will introduce later). Also, the trajectories cannot pass over singular points. The first of these rules is perhaps most easily shown for two-dimensional systems, where we have a two-dimensional phase plane. Let us assume that the rate equations for the two independent concentrations (or concentration and temperature), x and y, can be written in the form... 

Fig. 13.4. (a) Phase space representation of a period-2 limit cycle for a three-variable system (b) the corresponding evolution of oine variable in time. 

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... 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

The second point is that the new phase-space representation permits the definition of a true dividing surface in phase space which truly separates the reactant and product sides of a reaction. Traditional transition state theory of chemical reactions, based simply on coordinate-space definitions of the degrees of freedom, required an empirical correction factor, the transmission... 

We employ the frequency space to visualize resonances, and besides we use phase space representation to get features of the frequency space. 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

A third requirement is less absolute but still provide a useful consistency check for models that reduce to simple Brownian motion in the absence of external potentials The dissipation should be invariant to translation (e.g. the resulting friction coefficient should not depend on position). Although it can be validated only in representations that depend explicitly on the position coordinate, it can be shown that Redfield-type time evolution described in such (position or phase space) representations indeed satisfies this requirement under the required conditions. 

The phase space representation of the Fourier method is of a rectangular shape. The volume in phase space covered by the Fourier representation is calculated as follows The length of the spatial dimension in phase is L, and the maximum momentum is pmax. Therefore, the represented volume becomes Y = 2L-pmax, where the factor of two appears because the momentum range is from -pmax to + pmax. Using the fact that p = ftk, the phase space volume can be expressed as... 

Figure 10 shows a schematic representation of the phase space representation imposed by the optimal and suboptimal grids. 

Figure 13 Phase-space representation of the n = 8 vibrational eigenstate of H2. Notice that at the outer classical turning point, the momentum distribution becomes narrow due to the slowing down of the particle at the outer region of the potential. Notice the unused phase-space area for large p and q.

The limelight now shifts to the time-energy phase space. A detailed description is beyond the scope of the present chapter has been recently reviewed (15). The present focus will therefore be on the interrelations between the position-momentum phase space representation and the propagator representing operators in the time-energy phase space. 

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

The main difference between the Hamiltonian and dissipative systems arises from the conservation condition that applies to the former. In Hamiltonian systems, the total energy is fixed. A trajectory with a given initial condition and energy will continue with that same energy for the remainder of the trajectory. In the phase space representation, this will result in a stable trajectory that does not pull in toward an attractor. A periodic trajectory in a Hamiltonian system will have an amplitude and position in the phase space that is determined by the initial conditions. In fact, the phase space representation of a Hamiltonian system often includes many choices of initial conditions in the same phase space portrait. The Poincare section, to be described below, likewise contains many choices of initial conditions in one diagram. 

Figure 3.15 Phase space representation of the states of the computational basis of two qubits (a) 00), (b) 01), (c) 110) and (d) 111). The states are well defined in position (horizontal axis) and undefined in momentum (vertical axis). Amplitudes vary from —0.125 (white) to +0.125 (black). Due to the periodic boundary conditions, an interference pattern appears as the black and white stripes.

The solution of time evolution problems for classical systems is facilitated by introducing a classical phase space representation that plays a role in the description of classical systems in a matmer that is formally analogous to the role played by the coordinate and momentum representations in quantum mechanics. The state vectors T ) of this representation enumerate all of the accessible phase points. The phase function /(E ) is given by / (f ) = (f I/), which can be thought to represent a component of the vector f) in the classical phase space representation. The application of the classical Liouville operator (f ) to the phase function /(f ) is defined by (f )/(f ) = (f I/), where is an abstract op-... 

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