Solution of the Time-Dependent Schrodinger Equation

But the unadorned Eqs. (9.1.5) and (9.1.6) do not make sufficiently clear the physical significance of the nonstationary,initially-localized, f (O) states that are most easily created by a short, Fourier transform limited pulse of electromagnetic radiation. Short is not an absolute quality. For a sufficiently short pulse, the nature of the initial localization prepared and the specific dynamical processes sampled depend primarily on the duration of the preparation pulse and secondarily on its spectral content (Heller, et ai, 1982 Johnson, et ai, 1996). 

A bright state is prepared at t = 0. Lee and Heller (1979) derive a rigorous description of the wavepacket that is actually prepared when the excitation pulse is of finite duration. It is not a single eigenstate. It evolves in time subject to the spectroscopic time-independent Heff, as specified by the time-dependent Schrodinger equation Eq. (9.1.2). 

In the frequency domain, it is natural to think of the spectrum that originates from eigenstate V fe. 

Although the eigenstate-to-eigenstate and autocorrelation function formulations of the spectrum, Ik(w), are mathematically equivalent, they focus attention on complementary features. The most readily interpretable features in the autocorrelation function picture are early-time features (initial decay rate, the times at which partial recurrences occur, the magnitudes of the earliest and largest partial recurrences) which primarily sample the potential surface at the highly localized and a priori known initial position of the wavepacket, I (O), before it has had time to explore the entire dynamically accessible region of the potential surface. This early time information is encoded in the broad envelope (low resolution) of the Ik(u ) spectrum (see Fig. 9.2). 

The autocorrelation function picture of a frequency domain spectrum (Heller, 1981) is derived as follows. Consider the e,v e — g, t electronic absorption spectrum. In the Franck-Condon limit (constant transition moment) and neglecting rotation, the eigenstate spectrum observed from the v vibrational level 

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A1.6.2.1 WAVEPACKETS SOLUTIONS OF THE TIME-DEPENDENT SCHRODINGER EQUATION... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

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. 

All of the methods for designing laser pulses to achieve a desired control of a molecular dynamical process require the solution of the time-dependent Schrodinger equation for the system interacting with the radiation field. Normally, this equation must be solved many times within an iterative loop. Different possible approaches to the solution of these equations are discussed in Section V. 

We next address the question as to whether equation (3.59) is actually the most general solution of the time-dependent Schrodinger equation. Are there other solutions that are not expressible in the form of equation (3.59) To... 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

The linear combinations of the solutions. Equations (7.15) and (7.16), are also good solutions of the time-dependent Schrodinger equation, Eq. (7.6). For example, there is a state symmetric with respect to the median plane ... 

Allison and Truhlar have compared TST and VTST to accurate solution of the time-dependent Schrodinger equation for a number of three-atom chemical reactions (it is only for such small systems that the accurate solution of the time-dependent Schrodinger equation is practical) and those results are listed in Table 15.1. On the high-quality surfaces available for this comparison, VTST is typically accurate to within 50% at temperatures above 600 K. 

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schrodinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

The quantitative theory of resonance charge exchange has been developed by Firsov [18]. The basis for calculations was the fact that, when the energies of the colliding particles relative motion are small, the solution of the time-dependent Schrodinger equation appears to be the wave function ... 

If we have a single particle of mass m moving under the influence of a potential U, then we concern ourselves in quantum chemistry with solutions of the time-dependent Schrodinger equation... 

If we fix a realization of the path Q(t), then, when performing the path integration over q, the particle may be treated as if it were subject to a time-dependent potential Vint[<2(r), q]. From traditional quantum mechanics it is clear that this integration is equivalent to the solution of the time-dependent Schrodinger equation in imaginary time ... 

J Bengtsson, E. Lindroth, S. Selsta, Solution of the time-dependent Schrodinger equation using uniform complex scaling, Phys. Rev. A 78 (3) (2008) 032502. 

The formal solution of the time-dependent Schrodinger equation is given by... 

The TDSCF approximation is a good starting point for a mixed quantum mechanical/classical treatment. Let us assume that R is the classical and r the quantum mechanical mode. Then, the wavefunction r(r t) describing the vibration of the fragment molecule is a solution of the time-dependent Schrodinger equation... 

Dateo, C.E., Engel, V., Almeida, R., and Metiu, H. (1991). Numerical solutions of the time-dependent Schrodinger equation in spherical coordinates by Fourier transform methods, Computer Physics Communications 63, 435-445. 

On the theoretical side, advances have also been made both in methodology and in concepts. For example, new and powerful techniques for the solution of the time-dependent Schrodinger equation (see Section 1.1) have been developed. New concepts for laser control of chemical reactions have been introduced where, for example, one laser pulse can create a non-stationary nuclear state that can be intercepted or redirected with a second laser pulse at a precisely timed delay. 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

A Volkov state is obtained from the solution of the time-dependent Schrodinger equation for a free particle in an external plane-wave laser field. Such states were first derived by Volkov, in a relativistic context [21]. 

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