Liouville equation quantum form

The classical free streaming Liouville operators are iCf1 1 = Jy and = for the light ( ) subsystem particles and (h) heavy environmental particles, respectively. The quantum-classical Wigner-Liouville equation (9) can be written in a more compact form,... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

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. 

If the thermal equilibration of states 2 and 3 is very slow, the osciUatimis between states 1 and 2 continue indefinitely (Fig. 10.4A). As the time cmistant for conversion of state 2 to state 3 is decreased, the oscillations are damped and state 3 is formed more rapidly (Fig. 10.4B-D). But when Ti becomes much less than hlH2i, the rate of formation of state 3 decreases again (Fig. 10.4E,F) This quantum mechanical effect is completely contrary to what one would expect from a classical kinetic model of a two-step process, where increasing the rate constant for conversion of the intermediate state to the final product can only speed up the overall reaction (Box 10.2). In the stochastic Liouville equation, the slowing of the overall process results from very rapid quenching of the off-diagonal terms of p by the stochastic decay of state 2. This is essentially the same as the slowing of equilibration of two quantum states when is much less than h/Hi2, which we saw in Fig. 10.3. 

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

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... 

The classical Liouville equation does have an equivalent in quantum mechanics, which is needed for a consistent description of quantum statistical mechanics the quantum Liouville equation. Equilibrium quantum statistical mechanics requires the introduction of the density operator on an appropriate Hilbert space, and the quantum liouvUle equation for the density operator is a logical and necessary extension of the Schrodinger equation. The quantum Liouville equation can even be written, formally at least, in a form that resembles its classical counterpart. It allows for some weak and almost internally consistent form of dissipative dynamics, known as the Redfield theory, which finds its main use in relating NMR relaxation times to spectral densities arising from solvent fluctuations, although in recent... 

In this section, which should also be viewed as an introduction to the much harder quantum mechanical problems, we wiU briefly outline some of the essential properties of the dynamical phase space density, going back to what we already treated in this chapter, and show how those ideas can be cast in the form called the classical Liouville equation and the Fokker-Planck equation. 

Interaction between quantum systems and classical flelds is not problematic. It is the basis of almost all forms of optical spectroscopy where the transition dipole operator of the system interacts with the electric and magnetic flelds of light. It is a necessary ingredient of linear response theory, and also of the Redfleld relaxation mechanism. The starting point for all these examples is the quantum Liouville equation... 

Equation (37) is the quantum statistical analogue of Liouville s equation. To find the quantum analogue of the classical principle of conservation of phase density the solution to (37) is written in the form... 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

We outline briefly in this section how to link the theory of quantum resonances to statistical physics and thermodynamics by extending the concept of effective Hamiltonian as recently discussed in Ref. [60]. The quantum Liouville-von Neumann equation is written in the form... 

Introducing the formal deflnition of the quantum Liouville operator t, we can cast Eq. (236) in a form that resembles the equation of motion for classical dynamical variables. More specifically, we can write... 

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