Discrete Models

Michele Cini

Timothy B. Boykin

Discrete versions of continuous models are central to numerical calculations in physics and engineering. A very common problem in setting up a discrete model is how to handle derivatives. There are, for example, three common approximations for the first derivative, and each embeds different properties in the discrete model. Discretizing continuous expressions simplified using rules of calculus is especially problematic, since many different discretizations can stand for the same continuous expression depending on the stage of simplification at which the discretization is carried out. The problems are resolved by requiring that the discrete model satisfies discrete versions of the properties satisfied by the continuous original. We illustrate by using some examples from undergraduate-level one-dimensional quantum mechanics.

Michael Stone

Yutaka Shikano, Tatsuaki Wada, Junsei Horikawa

Constructing a discrete model like a cellular automaton is a powerful method for understanding various dynamical systems. However, the relationship between the discrete model and its continuous analogue is, in general, nontrivial. As a quantum-mechanical cellular automaton, a discrete-time quantum walk is defined to include various quantum dynamical behavior. Here we generalize a discrete-time quantum walk on a line into the feed-forward quantum coin model, which depends on the coin state of the previous step. We show that our proposed model has an anomalous slow diffusion characterized by the porous-medium equation, while the conventional discrete-time quantum walk model shows ballistic transport.

Miguel Lorente

The Kravchuk and Meixner polynomials of discrete variable are introduced for the discrete models of the harmonic oscillator and hydrogen atom. Starting from Rodrigues formula we construct raising and lowering operators, commutation and anticommutation relations. The physical properties of discrete models are figured out through the equivalence with the continuous models obtained by limit process.

Rainer Dick