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The Physics Educator

The Discretized Momentum Operator

Timothy B. Boykin

Published Jun 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s2661339519200038

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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.
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.
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.
Abstract We use non-standard finite differences to propose a quantum momentum operator to be used when the spectrum of the operator is discrete. The defined discrete operator complies with the discrete versions of the properties that the continuous variable operator has. The discrete derivative is exact for its eigenfunction, that is, exponential functions. We obtain the discrete adjoint of the momentum operator. The canonical commutation relationship between conjugate operators for discrete variables is diagonal along a particular direction.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf