Schrodinger, time-dependent

Time evolution in quantum mechanics is described, in the Schrodinger representation, by the Schrodinger time-dependent equation... 

In the previous section, we showed that the RRGM can be used to calculate individual state-to-state transition probabilities, Ptft). Another way of studying the IVR processes involves the explicit calculation of the time evolution of the initial state i) (75,76). We will begin by recalling that the solution to the Schrodinger time-dependent equation can be written in terms of the evolution operator (propagator)... 

Here x ( ) and p(t) are time-dependent parameters. This wave function is centered at x(t) and so, as time progresses, the wave function moves with its center along x(f), as shown in Figure 8.1. But the shape remains Gaussian. If the potential is harmonic, this wave function is an exact solution of the Schrodinger time-dependent equation and, and this is the point, the parameters x(t) and p(t) are the coordinate and momentum of a classical trajectory at the same energy as the (mean) energy of the quantum system." ... 

O. The quantum mechanical period of a vibrational motion of a diatomic molecule. Write the initial (time = 0) wave function as a linear superposition of two adjacent vibrational states, (a) Solve the Schrodinger time-dependent... 

The difTerential equation above is known as the time-dependent Schrodinger equation. There is an interesting and... 

From the fact that f/conmuites with the operators Pj) h is possible to show that the linear momentum of a molecule in free space must be conserved. First we note that the time-dependent wavefiinction V(t) of a molecule fulfills the time-dependent Schrodinger equation... 

We can describe the conservation of linear momenUim by noting the analogy between tire time-dependent Schrodinger equation, (equation A1.4.1 OS ), and (equation A1.4.991. For an isolated molecule, //does not depend explicitly on t and we can repeat the arguments expressed in (equation Al.4.98), (equation A1.4.99), (equation A1.4.1 OOl. (equation A 1.4.1011 and (equation A1.4.1021 with X replaced by t and Py replaced by // to show that... 

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

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

Balint-Kurti G G, Dixon R N and Marston C C 1992 Grid methods for solving the Schrodinger equation and time-dependent quantum dynamics of molecular photofragmentation and reactive scattering processes/of. Rev. Phys. Chem. 11 317—44... 

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... 

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

Neuhauser D and Baer M 1989 The time dependent Schrodinger equation application of absorbing boundary conditions J. Chem. Phys. 90 4351... 

Peskin U and Steinberg M 1998 A temperature-dependent Schrodinger equation based on a time-dependent self consistent field approximation J. Chem. Phys. 109 704... 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

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. 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

De Raedt, H. Product formula algorithms for solving the time-dependent Schrodinger equation. Comput. Phys. Rep. 7 (1987) 1-72. 

Inserting the separation ansatz, i.e., U , results in two nonlinearly coupled single particle Schrodinger equations, the so-called time dependent self-consistent field (TDSCF) equations ... 

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. 

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