Quantum evolution operator

The Wigner form of the quantum evolution operator iLw X ) in (62) for the equation of motion for W X, X2,t) can be rewritten in a form that is convenient for the passage to the quantum-classical limit. Recalling that the system may be partitioned into S and S subspaces, the Poisson bracket operator A can be written as the sum of Poisson bracket operators acting in each of these subspaces as A Xi) = A xi) + A Xi). Thus, we may write... 

When analytical solutions are not known and the approximate analytical methods give results of limited applicability, the numerical methods may be a solution. Let us first discuss a method based on the diagonalization of the second-harmonic Hamiltonian [48,49]. As we have already said, the two parts of the Hamiltonian Ho and Hi given by (55), commute with each other, so they are both constants of motion. The //0 determines the total energy stored in both modes, which is conserved by the interaction ///. This means that we can factor the quantum evolution operator... 

The path-integral quantum mechanics relies on the basic relation for the evolution operator of the particle with the time-independent Hamiltonian H x, p) = -i- V(x) [Feynman and... 

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

The time evolution operator propagates an arbitrary quantum state thanks to the non-zero matrix elements (Ve0ph)j m ,jm- The set ( )j(q)Qm(Q) has all electronic base states corresponding to all possible chemical species in the sense discussed above only because the generalized electronic diabatic set is complete. 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

In this section we analyze the dynamics of a spin subject to a classical random field and derive the equation of motion for the spin dynamics (the spin-evolution operator), averaged over the fluctuations. Following the discussion of the case with quantum fluctuations, we first analyze the dynamics in a stationary field B and a random field exactly as in the quantum case one can reduce the analysis of the dissipative corrections to the Berry phase accumulated over a conic loop to the problem with a stationary field by going over to a rotating frame. 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

The DMC method achieves the lowest-energy eigenfunction by employing the quantum mechanical evolution operator in imaginary time [25], For an initial function expanded in eigenstates, one finds that contributions of the excited states decay exponentially fast with respect to the ground state. 

Boltzmann Density Operators in Both Representations The Evolution Operator of a Driven Quantum Harmonic Oscillator [59]... 

InEq. (113), I)] (tfv is the IP time-evolution operator of the driven quantum harmonic oscillator interacting with the thermal bath,... 

Another possibility is to extract the reduced time evolution operator from the analytical solution obtained by Louisell and Walker for the reduced time-dependent density operator of a driven damped quantum harmonic oscillator. 

Appendix D shows that the IP time evolution operator of a driven quantum harmonic oscillator is given by Eq. (D.23), that is,... 

From this viewpoint, which is the most fundamental, the line shape as a whole is the sum of the diagonal matrix elements of the time evolution operator of the driven damped quantum harmonic oscillator in the IP representation with respect to the diagonal part of the Hamiltonian of this oscillator. According to Eq. (120), each diagonal element is a sum of time-dependent terms... 

APPENDIX D THE EVOLUTION OPERATOR OF A DRIVEN QUANTUM HARMONIC OSCILLATOR [54]... 

As a consequence, because of Eqs. (D.7) and (D.23), the full-time evolution operator (D.2) of the driven quantum harmonic oscillator takes the form ... 

Here, pj is the Boltzmann density operator of the H-bond bridge viewed as a quantum harmonic oscillator, pe is the Boltzmann density operator of the thermal bath, and (t) are effective time-evolution operators governing the dynamics of the H-bond bridge depending on the excited-state degree k of the fast mode. They are given by Eq. (110), that is,... 

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