Wave packet time-independent

Kroes G J and Neuhauser D 1996 Performance of a time-independent scattering wave packet technique using real operators and wave functions J. Chem. Phys. 105 8690... 

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

By way of contrast, recall that in treating the free particle as a wave packet in Chapter 1, we required that the weighting factor A(p) be independent of time and we needed to specify a functional form for A(p) in order to study some of the properties of the wave packet. 

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrodinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. 

Further Analysis of Solutions to the Time-Independent Wave Packet Equations of Quantum Dynamics II. Scattering as a Continuous Function of Energy Using Finite, Discrete Approximate Hamiltonians. 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

Time-Independent Scattering Wave Packet Technique Using Real Operators and Wave Functions. 

In the previous sections, we have seen that a resonance can be described through both a time-dependent approach and a time-independent approach. The goal of this section is to create a unified picture that joins both methodologies and explain how general wave packets evolve into pseudostationary decaying states. 

In the control scheme [13,17] that we have focused on, the time evolution of the interference terms plays an important role. We have already discussed more explicit forms of Eq. (7.75). One example is the Franck-Condon wave packet considered in Section 7.2.2 another example, which we considered above, is the oscillating Gaussian wave packet created in a harmonic oscillator by an (intense) IR-pulse. Note that the interference term in Eq. (7.76) becomes independent of time when the two states are degenerate, that is, AE = 0. The magnitude of the interference term still depends, however, on the phase S. This observation is used in another important scheme for coherent control [14]. 

Coherent control Control of the motion of a microscopic object by using the coherent properties of an electromagnetic held. Coherent phase control uses a pair of lasers with long pulse durations and a well-defined relative phase to excite the target by two independent paths. Wave packet control uses tailored ultrashort pulses to prepare a wave packet at a desired position at a given time. 

Wave packet A localized wave function, consisting of a non-stationary superposition of eigenfunctions of the time-independent Schrodinger equation. 

The nonlinear relationship due to the final term causes the wave packet to spread as it propagates. Dropping it assumes that W is so small that the detector can be placed close enough to the scattering target to neglect the spread. Note that only for a photon wave packet is E strictly proportional to k E = tick. The physical situation that we will ultimately consider is that W tends to zero. In section 3.2.2 we showed that the absence of time resolution in an experiment results in the experiment being equivalent to an incoherent superposition of independent experiments, each with an incident plane wave, i.e. an incident wave packet of zero width. 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

Huang, Y., Iyengar, S.S., Kouri, D.J. and Hoffman, D.K. (1996) Further analysis of solutions to the time-independent wave packet equations of quantum dynamics. 2. Scattering as a continuous function of energy using finite, discrete approximate Hamiltonians, J. Chem. Phys. 105, 927-939. 

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