Numerical 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. 

How many of these structures can be resolved depends on the microwave power which controls the widths of these transitions. The lowest resonance frequencies are given by U2 = 0.375, W3 = 0.444 and W4 = 0.469. We check this simple picture with the help of numerical solutions of the time dependent Schrodinger equation (6.2.1). We choose a dimensionless microwave field strength of e = 0.01 and compute the ionization probabilities after 100 cycles of the microwave field for 70 microwave frequencies chosen as Uj = 0.2-f j — l)Au>, j = 1,..., 70, and Au = 0.006. The initial state is chosen to be the SSE ground state n = 1). 

The AO results may also be used for benchmark tests of simpler models. In this context we have also checked a simple non-perturbative model, the UCA. This model includes the main features of fast heavy-ion stopping, as is shown by comparison with large-scale AO results for the impact-parameter dependent electronic energy transfer. The computation of the energy loss within the UCA is much simpler and by many orders of magnitude faster than the full numerical solution of the time-dependent Schrodinger equation. 

The main ingredients in the numerical solution of the time-dependent Schrodinger equation are given as follows ... 

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]. 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

It should be stressed that we are discussing here numerically exact results obtained by the solution of the time-dependent Schrodinger equation for an isolated system. No assumptions or approximations leading to decay or dissipation have been introduced. The time evolution of the wave function (t) is thus fully reversible. The obviously irreversible time evolution of the electronic population probabilities in Figs. 2 and 3 arises from the reduction process, that is, the integration over part of the system [in this case, the nuclear degrees of freedom, cf. Eqs. (12) and (13)]. 

At this point we may abandon the Heisenberg picture and proceed to the Schrodinger representation. Of course, both representations are equivalent, as soon as the field problem has been reduced to studying a finite-dimensional quantum system. However, the most of numerous investigations of the time-dependent quantum oscillator, since Husimi s paper [285], were performed in the Schrodinger picture. So it is natural to use the known results. According to several studies [279,285,286], all the characteristics of the quantum oscillator are determined completely by the complex solution of the classical oscillator equation of motion... 

We now discuss the necessary details of the computational aspects to obtain the reaction probability by the scheme developed above. In the following we proceed with the coupled-surface calculations the uncoupled-surface calculations follow from them in an elementary way. The time-dependent Schrodinger equation (TDSE) is solved numerically in the diabatic electronic representation on a grid in the (i , r, 7) space using the matrix Hamiltonian in Eq. (11). For an explicitly time-independent Hamiltonian the solution reads... 

An accurate numerical description of molecular vibrations in the field of physical chemistry often requires explicit solutions of the time-dependent or time-independent Schrodinger equation. A full quantum mechanical treatment of all involved particles, i.e., all electrons and nuclei, however, is only possible for very small and rather simple systems such as H. For larger systems one must rely an approximations, because the demands on CPU time and memory of a numerically exact treatment quickly exceed today s numerical capacities. 

As described in Sec.3.1, standard wavepacket propagation schemes can be employed for the evaluation of fiux correlation functions. These methods employ multi-dimensional grids or basis sets to represent the wavefunction. Thus, the numerical effort of these schemes increases exponentially with the number of degrees of freedom. Given the computational resources presently available, only systems with up to four atoms can be treated accurately. The extension of numerically exact calculations towards larger systems therefore requires other schemes for the solution of the time-dependent or time-independent Schrodinger equation. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Equations (10.47) or (10.50) do not have a mathematical or numerical advantage over Eqs (10.43), however, they show an interesting analogy with another physical system, a spin particle in a magnetic field. This is shown in Appendix 1 OA. A more important observation is that as they stand, Eqs (10.43) and their equivalents (10.47) and (10.50) do not contain information that was not available in the regular time-dependent Schrodinger equation whose solution for this problem was discussed in Section 2.2. The real advantage of the Liouville equation appears in the description... 

Attosecond dynamics is now one of the most active fields in science [222, 476] (see also the introductory section of Ref. [499], which shows a concise list of the studies covering many phenomena and relevant studies). In particular, tracking electronic motions in chemical dynamics is a fundamentally important process. To monitor those electron wavepacket dynamics in an attosecond intense laser field [476], Bandrauk and his coworkers have developed the theory of attosecond-scale time-resolved photoelectron spectroscopy [499, 500], Photoionization dynamics of small molecules like has been studied so far, for which direct numerical integrations of the related (low-dimensional) time-dependent Schrodinger equations are possible. The photoelectron signals are extracted from those numerical solutions... 

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