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arXiv:0805.2017

Quantum mechanics and umbral calculus

E. Lopez-Sendino, J. Negro, M. A. del Olmo, Erik Salgado

Published May 14, 2008• 0 citations
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Abstract

In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.

DOI
10.48550/arxiv.0805.2017

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In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrödinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schrödinger equation with a potential depending only in the space variable is discussed.
The umbral calculus is an old mathematical tool that began its first steps in the XVII century [1]. Since the second half of XIX [2, 3] it was applied in a systematical way although until the second half of XX [4, 5] there was not a formal theory. Recently it has been used to provide discrete representations of canonical commutation relations [6]–[12]. In this approach it can be used to map equations and solutions from a continuous frame to a discrete one. The theory works very well for polynomials, however this is not the case for its extension to non-polynomial functions. A complicated, and then interesting, problem in this formalism is the harmonic oscillator. Due to the non preservation of the complete set of symmetries after discretization, some divergences appear. To second physical solutions of the discrete harmonic oscillator we use a non standard form of the umbral calculus discretization. Some surprising results are obtained.
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail.
Observables with a continuous spectrum are known to fundamentally differ from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. Here we show that it is possible to manipulate, detect, and classify continuous-variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define, and adapt, to arbitrary continuous quantum systems, quantum protocols, and algorithms developed for discrete systems.
It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and LLL contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length λPλ_PλP​, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator [p,x]q[p,x]^q[p,x]q under the assumption of quantized space-time and contrast it with the usual commutator [p,x][p,x][p,x].
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.