Schrodinger equation, time-dependent solution

The difficulty of evaluating the quantity in Eq. (17) is that it requires the time-dependent solution of the Schrodinger equation in which the Hamiltonian is a function of all the solvent, ion, and metal nuclei, as well as of the metal electrons. In the two-state approximation, the Hamiltonian can be written as... 

This chapter focuses on the time-dependent Schrodinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrodinger equation alone a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. 

The second approach to the approximate description of the dynamic solvation effects is based on the semiempirical account for the time-dependent electrical polarization of the medium in the field of the solute molecule. In this case, the statistical averaging over the solute-solvent intennolecular distances and configurations is presumed before the solution of the SchrOdinger equation for the solute and correspondingly, the solvent is described as a polarizable dielectric continuum. The respective electrostatic solvation energy of a solute molecule is given by the following equation[13]... 

J26 True or false (a) Every linear combination of solutions of the time-dependent Schrodinger equation is a solution of this equation, (b) Every linear comlnnation of solutions of the time-independent Schrddinger equation is a solution of this equation, (c) Ihe nondegenerate perturbation-theory formula applies only to the ground state, (d) The exact... 

Although considerable research focuses on the development of high quality trial functions, the accuracy of the VMC method will always be limited by the flexibility and form of the approximate trial wave function. Highly accurate solutions to the Schrodinger equation can be computed by DMC which is rooted in the time-dependent Schrodinger equation and its solutions. 

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. 

The vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrodinger equation and depend only on the internal coordinates q = q q-,. .. of all electrons and the internal coordinates Q = Qi, Qi,. . . of all nuclei. 

To find a time-dependent solution of the Schrodinger equation we apply perturbation theory. We assume that a transition between two energy levels is caused by a time-dependent influence, which we represent as a small additive potential, v x, y, z, t). The Schrodinger equation is then... 

Listing 12.18. Code segment for the time dependent solution of Schrodinger s wave equation in a box. 

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. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

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. 

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. 

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