We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger eq...

We explore a novel approach to the study of large-scale structure formation in which self-gravitating cold dark matter (CDM) is represented by a complex scalar field whose dynamics are governed by coupled Schrödinger and Poisson equations. We show that, in the quasi-linear regime, the Schrödinger equation can be reduced to the free-particle Schrödinger equation. We advocate using the free-parti...

We present some general results for the time-dependent mass Hamiltonian problem with H = − 2e∂xx + h(2)(t)e2ν(t)x2, where ν(t) is a continuous function of t. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P 2 and X2 terms. We give the specific transformations to a different quadratic Schrödinger(TQ) equation and to a diff...

We present some general results for the time-dependent mass Hamiltonian problem with H = − 2e∂xx + h(2)(t)e2νx2. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P 2 and X2 terms. We give the specific transformations to a different quantum Schrödinger(TQ) equation and to a different time-dependent oscillator (TO) equation. ...

In this paper, we attack the specific time-dependent Hamiltonian problem H = − 1 2 e Υ(t−to) ∂ xx + 1 2 ω 2 e −Υ(t−to) x 2. This corresponds to a time-dependent mass (T M) Schrödinger equation. We give the specific transformations to i) the more general quadratic (T Q) Schrödinger equation and to ii) a different time-dependent oscillator (T O) equation. For each Schrödinger system, we give the ...

We describe a parallel algorithm for solving the time-independent 3d Schrödinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ∝ (Nnodes). This makes it possible t...

In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrödinger equation with a smooth potential near the semiclassical limit is presented. We are interested computing arbitrarily accurate solutions when the non dimensional Planck’s constant, e, is small, but not negligible. The method is based on a time dependent transformation of the in...

It is shown that the eigenvalue problem for the Hamiltonians of the standard form, H = p/(2m) + V (x), is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the so...

In the present work, we improve a numerical method, developed to solve the Gross-Pitaevkii nonlinear Schrödinger equation. A particular scaling is used in the equation, which permits us to evaluate the wave-function normalization after the numerical solution. We have a two-point boundary value problem, where the second point is taken at infinity. The differential equation is solved using the sh...