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Time Dependent Supersymmetry in Quantum Mechanics  [PDF]
Vladislav G. Bagrov,Boris F. Samsonov
Physics , 1997,
Abstract: The well-known supersymmetric constructions such as Witten's supersymmetric quantum mechanics, Spiridonov-Rubakov parasupersymmetric quantum mechanics, and higher-derivative SUSY of Andrianov et al. are extended to the nonstationary Schr\"odinger equation. All these constructions are based on the time-dependent Darboux transformation. The superalgebra over the conventional Lie algebra is constructed. Examples of time-dependent exactly solvable potentials are given.
Time-dependent Perturbation Theory in Quantum Mechanics  [PDF]
C. Y. Chen
Physics , 1999,
Abstract: After revealing difficulties of the standard time-dependent perturbation theory in quantum mechanics mainly from the viewpoint of practical calculation, we propose a new quasi-canonical perturbation theory. In the new theory, the dynamics of physical observables, instead of that of coefficients of wave-function expansion, is formulated so that the gauge-invariance and correspondence principles are observed naturally.
Quantum mechanics with time-dependent parameters  [PDF]
G. Sardanashvily
Physics , 2000, DOI: 10.1063/1.533405
Abstract: Smooth composite bundles provide the adequate geometric description of classical mechanics with time-dependent parameters. We show that the Berry's phase phenomenon is described in terms of connections on composite Hilbert space bundles.
Inverse Time-Dependent Quantum Mechanics  [PDF]
J. C. Lemm
Physics , 2000,
Abstract: Using a new Bayesian method for solving inverse quantum problems, potentials of quantum systems are reconstructed from coordinate measurements in non-stationary states. The approach is based on two basic inputs: 1. a likelihood model, providing the probabilistic description of the measurement process as given by the axioms of quantum mechanics, and 2. additional "a priori" information implemented in form of stochastic processes over potentials.
Functional differentiability in time-dependent quantum mechanics  [PDF]
Markus Penz,Michael Ruggenthaler
Mathematics , 2014, DOI: 10.1063/1.4916390
Abstract: In this work we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions Fr\'echet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schr\"odinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fr\'echet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.
The effect of cusps in time-dependent quantum mechanics  [PDF]
Zeng-hui Yang,Neepa T. Maitra,Kieron Burke
Physics , 2011, DOI: 10.1103/PhysRevLett.108.063003
Abstract: Spatial cusps in initial wavefunctions can lead to non-analytic behavior in time. We suggest a method for calculating the short-time behavior in such situations. For these cases, the density does not match its Taylor-expansion in time, but the Runge-Gross proof of time-dependent density functional theory still holds, as it requires only the potential to be time-analytic.
Time dependent problems in relativistic quantum mechanics
Moshinsky, M.;Sadurní, E.;
Revista mexicana de física , 2006,
Abstract: the most effective procedure for dealing with time dependent problems is through the feynman propagator. in this note we indicate the explicit expression for this propagator in relativistic problems using spectral decomposition. we then take as an initial state one of a dirac oscillator and consider the behaviour of the wave function when the interaction is suddenly supressed.
Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics  [PDF]
Marek Czachor,Heinz-Dietrich Doebner,Monika Syty,Krzysztof Wasylka
Physics , 1999, DOI: 10.1103/PhysRevE.61.3325
Abstract: Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dot\rho(t)=[ h,\rho(t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i\dot\rho_1(t)=[h_1(t),\rho_1(t)]$ where $h_1(t)$ is time-dependent and not isospectral to $h$. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where $h$ corresponds to a 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pulse on a three-level system.
On the Mapping of Time-Dependent Densities onto Potentials in Quantum Mechanics  [PDF]
Roi Baer
Physics , 2007, DOI: 10.1063/1.2822124
Abstract: The mapping of time-dependent densities on potentials in quantum mechanics is critically examined. The issue is of significance ever since Runge and Gross (Phys. Rev. Lett. 52, 997 (1984)) established the uniqueness of the mapping, forming a theoretical basis for time-dependent density functional theory. We argue that besides existence (so called v-representability) and uniqueness there is an important question of stability and chaos. Studying a 2-level system we find innocent, almost constant densities that cannot be constructed from any potential (non-existence). We further show via a Lyapunov analysis that the mapping of densities on potentials has chaotic regions in this case. In real space the situation is more subtle. V-representability is formally assured but the mapping is often chaotic making the actual construction of the potential almost impossible. The chaotic nature of the mapping, studied for the first time here, has serious consequences regarding the possibility of using TDDFT in real-time settings.
Quantum mechanics of time-dependent systems. Construction of pure states  [PDF]
Victor Tapia
Physics , 1995,
Abstract: For time-dependent systems the wavefunction depends explicitly on time and it is not a pure state of the Hamiltonian. We construct operators for which the above wavefunction is a pure state. The method is based on the introduction of conserved quantities $Q$ and the pure states are defined by ${\hat Q}\psi=q\psi$. The conserved quantities are constructed using parametrised mechanics and the Noether theorem.
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