Time-dependent equation representation

Time evolution in quantum mechanics is described, in the Schrodinger representation, by the Schrodinger time-dependent equation... 

Although they are able to use the progress curves for analysis of the data obtained from experimental assays, many biochemists prefer to use linear representations of enz5mie kinetics. Instead of using the time-dependent solution (1.26), they rearrange the time-dependent equation (1.21) to a sort of time-dependent regression expression, for example, the reciprocal double plot equation ... 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

This representation is slightly inconvenient since Ey and 2 in equation (Al.6.56) are explicitly time-dependent. For a monocln-omatic light field of frequency oi, we can transfonn to a frame of reference rotating at the frequency of the light field so that the vector j s a constant. To completely remove the time dependence... 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

The following treatment starts with the complete quantal equations and introduces an eikonal representation which allows for a formally exact treatment. It shows how a time-dependent eikonal treatment can be combined with TDHF... 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

As transpires from equation (2.2), a steady state is established by mutual compensation of diffusion and chemical reaction. The concentration profile is indeed the product of a time-dependent function, i, by a space-dependent function in the exponential. The conditions required for the system to be in zone KP, K small and A large, will often be termed pure kinetic conditions in following analyses. Besides its irreversibility, the main characteristics of the cyclic voltammetric wave in this zone can be derived from its dimensionless representation in Figure 2.2b and its equation (see Section 6.2.1),4 where... 

In complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

Let us emphasize that we have made no approximations yet. Equation (3.13) is a set of simultaneous differential equations for the coefficients cm that determine the state function (3.13) is fully equivalent to the time-dependent Schrodinger equation. [The column vector c(/) whose elements are the coefficients ck in (3.8) is the state vector in the representation that uses the tyj s as basis functions. Thus (3.13) is a matrix formulation of the time-dependent Schrodinger equation and can be written as the matrix equation ihdc/dt = Gc, where dc/dt has elements dcmf dt and G is the square matrix with elements exp(.iu>mkt)H mk. ... 

Similarly, in equation (6.9) for the covariances the matrices A and B are now time-dependent. We define the interaction representation by setting... 

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