Schrodinger equation time-dependent ground state

This mixed classical/quantum expression is valid for classical nuclear behavior and, strictly speaking, for the case of direct two-site interaction rather than superexchange, as the Landau-Zener expression was derived from the time-dependent Schrodinger equation assuming a two-state (reactant/product) electronic system with direct coupling. Nevertheless, it becomes clear on physical grounds that the form of Eqs. 4-5 can serve to define an effective A in the superexchange case in terms of the Rabi precession frequency characteristic of the two trap sites embedded in the complex system wherein 2A/h would be computed from this net effective Rabi precession frequency. 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

When the frequency of the field is lower, the dynamics will be confined to the electronic ground state. This is, typically, dynamics induced by fields in the infrared (IR) spectral region. In this case, the time-dependent Schrodinger equation for the atomic nuclei takes the form... 

The time-dependent Schrodinger equation (2.30) for the particle in a box has an infinite set of solutions xpn(x) given by equation (2.40). The first four wave functions xpn(x) for n = 1, 2, 3, and 4 and their corresponding probability densities ij)n(x) 2 are shown in Figure 2.2. The wave function xp (x) corresponding to the lowest energy level E is called the ground state. The other wave functions are called excited states. 

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 QMC method generates a many-body ground state wavefunction by propagating the time-dependent Schrodinger equation along the imaginary time axis. When the time variable is transformed as T = -it/h, the equation becomes iso-morphous to a diffusion equation that includes source and sink terms ... 

Selloni et al. [48] were the first to simulate adiabatic ground state quantum dynamics of a solvated electron. The system consisted of the electron, 32 K+ ions, and 31 Cl ions, with electron-ion interactions given by a pseudopotential. These simulations were unusual in that what has become the standard simulating annealing molecular dynamics scheme, described in the previous section, was not used. Rather, the wave function of the solvated electron was propagated forward in time with the time-dependent Schrodinger equation,... 

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