Quantum Liouville equation

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

The evolution of an isolated system is then given by the classical and quantum Liouville equations for the fine-grained distribution functions (i.e., the evolution is entropy-preserving) ... 

In this scheme, for a system with hamiltonian H, the starting point is the quantum Liouville equation for the density matrix, p(t),... 

We may take the Wigner transform of the quantum Liouville equation to obtain an alternate formulation of the equation of motion. The Wigner transforms of the density matrix and an operator A are defined, respectively,... 

This chapter focuses on the time-dependent Schrodinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrodinger equation alone a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. 

THE QUANTUM MECHANICAL DENSITY OPERATOR AND ITS TIME EVOLUTION QUANTUM DYNAMICS USING THE QUANTUM LIOUVILLE EQUATION... 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

In Section 2.2 we have used the two coupled states model as a simple playground for investigating time evolution in quantum mechanics. Here we refonnulate this problem in the density matrix language as an example for using the quantum Liouville equation... 

It is actually simple to find a formal time evolution equation in P space. This formal simplicity stems from the fact that the fundamental equations of quantum dynamics, the time-dependent Schrodinger equation or the Liouville equation, are linear. Starting from the quantum Liouville equation (10.8) forthe overall system— system and bath. 

Let us now follow a different route, starting from the quantum Liouville equation for the time evolution of the density operator p... 

Eigendistributions of Lw(p, q), which span the Hilbert space, are readily constructed from eigenstates j> of H (H > = E and from the operators py = l iX jl that satisfy the time-independent quantum Liouville equation... 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

This equation can be derived from the quantum Liouville equation by scaling the variables so that the momenta of the heavy particles have the same magnitude as those of the light particles, /iP, where n =, and measuring... 

The quantum Liouville equation can be brought into a form that more closely resembles the classical Liouville equation by introducing the quantum Liouville operator... 

If the initial density operator p(0) is known, the density operator p(t) at time t can be determined by using Eq. (223). This equation is the formal solution of the version of the quantum Liouville equation given by Eq. (225). In view of the equivalence of Eqs. (225) and (228), we can also write... 

The basic properties of the classical Liouville equation and the troublesome questions they raise are shared by the quantum Liouville equation. For the quantum case, we smmnarize these properties as follows (i) The canonical density operator is stationary with respect to the quantum Liouville equation, (ii) The quantum Liouville operator t is Hemtitiaa (iii) The quantum Liouville equation is time-reversal invariant, (iv) The Gibbs entropy S t) is time independent when S t) is determined using the formal solution p(t) of the quantum Liouville equatioa Given the density operator p(t), we can determine the average value d t)) of the quantum dynamical variable 0 at time t by using the relation... 

Matrix representations of the quantum Liouville equation and Heisenberg s equation of motion can be obtained by sandwiching both sides of Eqs. (225) and (236) or Eqs. (228) and (237) between the vectors 4>j and 4>k), where ( y) and k) are members of the orthonormal basis ( ). Tins procedure yields... 

Although the above matrix representations of the quantum Liouville equation and Heisenberg s equation of motion are formally correct, the solution of time evolution problems for quantum systems can be more readily accomplished by working in a representation called the superstate representation. The basis vectors of this representation are the superstates A y/t). These states are associated with the operators Njk= j) 4 k formed from the basis vectors < y) used in the formulation of Eqs. (239) and (240). The matrix element A(j,k) = (pj A (l)k) of the operator A is given by A j, k) = Njk A), which can be thought to represent a component of the vector A) in the superstate representation. We define the inner product A B) of A and 5) by A B) —Tr The matrix elements C(Jk, Im)... 

With the superstate representation at our disposal, we can rewrite the quantum Liouville equation and Heisenberg s equation of motion as vector equations of motion that are identieal in form to the vector equations of motion given by Eqs. (215) and (216) for elassieal systems. The only differenee between the quantum and elassieal veetor equations of motion is the manner in which the matrix elements of and the eomponents of 1) and ) are determined. Nonetheless, the expressions for the average of a qrrantrrm dynamieal variable differ from the eorrespond-ing expressions for elassieal systems. [See Eqs. (219a)-(219c).] For the qrrantttm ease, we have... 

Let us consider the problem of describing the response of a system to some probe, such as an electric or magnetic field, that can be treated as a classical external force. This problem can be dealt with by starting with the following version of the quantum Liouville equation ... 

The quantum analogs of the phase space distribution function and the Liouville equation discussed in Section 1.2.2 are the density operator and the quantum Liouville 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... 

---