A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde’s usually results in system of ode’s of the form ut −Gu= s (0.1) where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modifi...

A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular exam...

We prove an error estimate for a Lie–Trotter splitting operator associated with the Schrödinger–Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible Euler–Poisson equation is smooth, the error between the numerical solution and the exact solution is controlled in Sobolev spaces, in a suitable phase/ampli...

---------------------------------------------------------------------***--------------------------------------------------------------------Abstract In the field of information processing like data mining, information retrieval, natural language processing and machine learning, Quantum Computing play vital role for extracting the implicit, potentially useful and previously unknown information f...

Ehrenfest, Born-Oppenheimer, Langevin and Smoluchowski dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system avoiding caustics, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger...

The time-dependent Schrödinger equation for two quite general types of perturbation has been solved by introducing the Laplace transforms to eliminate the time variable. The resulting time-independent differential equation can then be solved by the perturbation method, the variation method, the variation-perturbation method, and other methods.

Let H(t) ≥ 1 be a time-dependent self-adjoint operator on a Hilbert space H with quadratic form domain Q(H(t)). If Q(H(t)) is independent of t, along with other suitable conditions, we construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation. We apply this extension of Yosida’s Theorem to study the time evolution of a density matrix in a quantum me...