Hamiltonian operator time-dependent

One may attempt to approximate to such an experimental situation by considering a subsystem with small dimensions in the direction of the flow, so that a single temperature may be sufficiently precise in describing it. In this model one would have to provide a time-dependent hamiltonian operating in such a way as to feed energy into the system at one boundary and to remove energy from the other boundary. We would therefore be obliged to discuss systems with hamiltonians that are explicitly functions of time, and also located on the boundaries of the macrosystem. 

This approximation is numerically stable since it is unitary. The method is suitable for short time dynamics and time-dependent Hamiltonian operators. To reduce... 

The time-dependent Hamiltonian operator H describing the sum of kinetic and potential energies in the system it is a function of the coordinates of the... 

We next consider the case where a collection of harmonic oscillators are driven by nonlinear potentials, keeping however only terms quadratic in the s, within the expansion of the driving potentials. The time-dependent Hamiltonian operator is then written... 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

At this point we have expressed the Hamiltonian, the density operator and the evolution operator in Fourier space. We have introduced an effective Hamiltonian, defined in the Hilbert space of the same dimension 2 as the total time-dependent Hamiltonian itself, and we have shown how to transform operators between the two representations. The definition of the effective Hamiltonian enables us to predict the overall evolution of the spin system, despite the fact that we can not find time-points for synchronous detection, f, where Uint f) = exp —iWe//t In actual experiments the time dependent signals are monitored and after Fourier transformation they result in frequency sideband... 

T+ being the time-ordering operator. In the derivation of Eq. (168) we assumed that B(t)) = 0. It needs to be stressed that Eq. (168) generalizes previously known master equations to arbitrary time-dependent hamiltonians, Hs t) for the system and Hi(t) for system-bath coupling, [Cohen-Tannoudji 1992], Henceforth, we explicitly consider a driven TLS undergoing decay, whose resonant frequency and dipolar coupling to the reservoir are dynamically modulated, so that... 

The M(x, t ) operator is defined for a general time-dependent Hamiltonian by... 

In this chapter, we turn to problems of quantum chemistry and of many-electron atomic and molecular physics for which fhe desideratum is the quantitative knowledge and easy conceptual understanding of dynamical processes and phenomena thaf depend explicifly on time. We focus on a theoretical and computational approach which computes q>(q,t) by solving nonperturbatively the many-electron TDSE for unstable states of atoms and small molecules. The time evolution of fhese states is caused either by the time-independent Hamiltonian, Ham ( -g-/ case of time-resolved autoionization—see below) or by the time-dependent Hamiltonian, H t) = Ham + Vext(f), where Vext(f) is the sum of the identical one-electron operators that couple the field of a strong pulse of radiation to the electronic and nuclear moments of N-electron atomic or molecular states of inferest, thereby producing, during and at the end of the interaction, final stafes in the ionization or the dissociation continua. 

The case of the parabolic barrier can be solved in a similar fashion however, the algebraic procedure becomes cumbersome due to the fact that the corresponding ladder operators are not adjoint to each other. A better approach is to use the concept of the Bargmann-Segal space, whereby avoiding long algebraic derivations [32]. We exemplify this method in the case of the harmonic oscillator. Let us consider the time-dependent Hamiltonian ... 

This last equation opens the discussion about the evolution operator expression (or solution) if instead of a constant (stationary) Hamiltonian Ho one has to treat a time-dependent Hamiltonian, H(t). At this point two possibilities may be approached. 

The problem with this way of treating the evolution operator solution for the time-dependent Hamiltonian, although elegant, is IMe practicable since the theory work only if the series is entirely considered if only few terms are considered then, practically an infinity of terms from the global Hamiltonian are omitted and the description blows up ... 

Now, within U-picture, the (time dependent) statistical operator, the (time-dependent) Hamiltonian and a general state become ... 

Consistent with time-independent Hartree-Fock theory the main approximation in time-dependent Hartree-Fock theory is, that the system is represented by a single Slater determinant, which now is composed of time-dependent single-particle wavefunctions. The time-dependent Schrodinger equation that has to be solved is given in eqn (1). The time-dependent Hamiltonian consists of a static Hamiltonian and an additional time-dependent operator describing the time-dependent perturbation, e.g. an electric field, which is a sum of time-dependent single-particle potentials ... 

The zero-order Hamiltonian is not the same as the zero-order Hamilton in Section 19.3. It is the complete time-independent Hamiltonian operator of the molecule in the absence of radiation. The perturbation term H describes the interaction between the molecule and the electric field of the radiation, and is time-dependent because of the oscillation of the radiation. 

At the beginning of this section it is important to point out that the theoretical treatment we will briefly discuss now is sometimes called semiclassical just because the correlation functions are classical. A quantum mechanical treatment has been proposed (2, p. 284) which presents many formal similarities to the semiclassical treatment but which fundamentally differs from it by the way the correlation functions are defined (in terms of time-dependent operators and not of time functions). This distinction is rarely done in review articles about relaxation where the semiclassical approach is generally presented as the only way to handle the problem. To introduce the correlation functions and spectral densities, we will consider a spin system characterized by eigenstates a, b, c. and corresponding energies E, E. A perturbation, time-dependent Hamiltonian H(t) acts on this system. H(t) corresponds to the coupling of the spin system with the lattice. We shall make the assumption that H(t) can be written as a product A f(t). 

The same considerations can be expressed in the framework of the SCRF formalism this means reconsidering equations (1) and (2) when an explicit time dependence is introduced. Actually, the introduction of this time dependence can be realized in two different ways. First, we can take into account an external oscillating field which acts as a time-dependent perturbation operator, V (t), to be added to the standard one due to the solvent (Vim). Secondly, the introduction of time may derive from the varying field induced by a solute system which undergoes a chemical reaction, in this case the solute Hamiltonian, and consequently also the solvent perturbator, which depends on time. The effective Hamiltonians corresponding to the two different systems are shown below ... 

The Chebyshev procedure consists of expanding the quantum operators in terms of orthogonal Chebyshev polynomials. It is considered to be an efficient and reliable method, since the convergence of the expansion is guaranteed. According to the method, the time-dependent hamiltonian is treated as a constant operator within the time slice dt. Thus, the time propagator is expanded in a series of Chebyshev polynomials for a time t within dt as... 

The product of three exponential operators in this expression can be interpreted as propagation under the time-dependent Hamiltonian (37)... 

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