Time-dependent Schrodinger equation TDSE

The central equation of (non-relativistic) quantum mechanics, governing an isolated atom or molecule, is the time-dependent Schrodinger equation (TDSE) ... 

The discretized adiabatic procedure, and its analog with STIRAP, is but one possibility for achieving broadband response of an optical device. An alternative, which we discuss, relies on the analogy between the Jones vector description of an optical beam and the two-state time-dependent Schrodinger equation (TDSE). This equation has two commonly used solutions. One is rapid adiabatic passage (RAP), produced by swept detuning (a chirp), and the other is Rabi oscillations, specifically a pi pulse. The RAP has theoretical connection with STIRAP, but the pi pulses have no such connections. We describe application of a procedure that has been used to extend the traditional pi pulses to broadband excitation. This can accomplish the present goal of PAP, under complementary conditions. 

Basic spectral features of high harmonics can be obtained from the HHG calculation of a single atom. Calculations of the time-dependent Schrodinger equation (TDSE) for a single atom may reveal the characteristic features of high harmonic generation, such as the plateau and cutoff in the high harmonic... 

Abstract Recent advances achieved in the numerical resolution of the Time-Dependent Schrodinger Equation (TDSE), have made possible to address difficult problems in the analysis of highly nonlinear processes taking place when an atom is submitted to an ultra-intense laser pulse. We discuss the main properties of the photoelectron spectra obtained when a high frequency harmonic field is also present in addition to the laser field. This class of processes is believed to serve as a basis to explore new secnarios to achieve a coherent control of atomic photoionization. 

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

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... 

The quantum mechanical description of a molecule requires the solution of the time-dependent Schrodinger equation (TDSE) associated with the Hamiltonian operator i/(r, R) = T u (R) + Tel (f) + y(f, R) described above... 

In the standard method for the solution of the nuclear time-dependent Schrodinger equation (TDSE), the wavefunction is expanded in a basis of time-independent functions called a primitive basis. Specifically, considering a molecule with / degrees of freedom (dofs), abasis of Nk, one-dimensional functions xjf (< ) with k= 1,..., / can be defined for each dof and the total nuclear wavefunction is expanded in the product basis composed of Hartree products of these one-dimensional functions... 

Armed with a formal theorem, we can then define time-dependent Kohn-Sham (TDKS) equations that describe noninteracting electrons that evolve in a time-dependent Kohn-Sham potential but produce the same density as that of the interacting system of interest. Thus, just as in the ground-state case, the demanding interacting time-dependent Schrodinger equation (TDSE) is replaced by a much simpler set of equations. The price of this enormous simplification is that the exchange-correlation piece of the Kohn-Sham potential has to be approximated. 

In the diabatic picture, the transition is caused by a coupling term 12(2, r), and hence the time-dependent Schrodinger equation (TDSE) which has to be solved involves two diabatic wavefunctions... 

At higher energies where many quantum states are energetically accessible and where also chemical reaction (dissociation) can take place, it is advantageous (see discussion below) to turn to a time-dependent description, where the time-dependent Schrodinger equation (TDSE) is solved. That is, instead of eq. (5.2) we would have... 

---