A new method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods have been discussed critically.
We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
Une nouvelle méthode numérique pour ressoudre l'équation de Schrödinger dépendente du temps est proposée. Méthode consiste en factorisaion des opérateurs exponentielles. On démontre que cette méthode fourni une précison de troisième ordre par rapport à l'incrément temporel. La phase de la solution est obtenue avec une précision exceptionelle. On démontre que la nouvelle méthode est plus efficace que la méthode standard de deuxième ordre pour des potentiels non-dépendants ainsi que dépendant du temps. On présente la solution numérique des deux équations de Schrödinger couplées à l'aide de cette méthode. Abstract: A new method of splitting exponential operators is proposed in exponential form for the operator solution of the time dependent Schrödinger equation. The method is shown to be third order accurate in the time increment. In particular, the phase of the weavefunction is shown to be exceptionally accurate. The new method is shown to be more efficient that the standard second order evolution operator algorithms for time independent and dependent potentials. The numerical solution of two coupled Schrödinger equations also shwon in the new method.