Schroedinger equation time dependent

They add up to one. At later times, the density operator can be constructed from solutions of the time-dependent Schroedinger equation,... 

Scheme for Propagating the Time Dependent Schroedinger Equation. 

The dipole-dipole interaction V has a diagonal term, which slightly shifts the energies, and an off diagonal term, which induces the collisional transitions. Since Vdepends on r, Vis time dependent, and the wavefunction is given by the solution to the time dependent Schroedinger equation... 

For the past several years we have been developing and applying quantum dynamics (QD) and quantum-classical molecular dynamics (QCMD) methods, which are based on the explicitly time-dependent Schroedinger equation. For an overview of the models and simulation results see e.g.4 and the references cited... 

As pointed out in the review by C )erly and Alder ( 31) the diffusion interpretation of the Schroedinger equation has an extensive history. The diffusion analogy becomes apparent if one writes the time-dependent Schroedinger Equation (one electron for simplicity) in terms of imaginary time, t. 

A reaction-path based method is described to obtain information from ab initio quantum chemistry calculations about the dynamics of energy disposal in exothermic unimolecular reactions important in the initiation of detonation in energetic materials. Such detailed information at the microscopic level may be used directly or as input for molecular dynamics simulations to gain insight relevant for the macroscopic processes. The semiclassical method, whieh uses potential energy surface information in the broad vicinity of the steepest descent reaction path, treats a reaction coordinate classically and the vibrational motions perpendicular to the reaction path quantum mechanically. Solution of the time-dependent Schroedinger equation leads to detailed predictions about the energy disposal in exothermic chemical reactions. The method is described and applied to the unimolecular decomposition of methylene nitramine. 

We solve the time-dependent Schroedinger equation using the following Hamiltonian for the quantum degrees of freedom. 

The time-dependent Schroedinger equation leads to the following equations for the expansion coefficients... 

A natural way to introduce equations for excited states into a quantum chemical approach is to consider stimulating the molecule by a time-varying electric field to which the molecule can respond by excitation, and derive solutions from the time-dependent Schroedinger equation. Analysis then leads to equations for the excitation energies and properties of the excited state eigensolutions like transition moments. In particular, such an approach, after a Fourier transformation from time to frequency, will yield the dynamic polarizability whose spectral expansion is... 

Consider a Brillouin Zone, such as that defined for CsCl, and an energy band (k), defined within that zone. Further, imagine a single electron within that band. If its wave function is an energy eigenstate, the time-dependent Schroedinger equation, Eq. (1-17), tells us that... 

Substituting Eq. (3) into the time dependent Schroedinger equation and adopting the standard use of first order perturbation theory and the rotating wave approxima-tion(3,4) gives ... 

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