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Journal of Mathematical Economics and Finance

Critical notes of Quantum Mechanics: Momentum Operators

David Carfì

Published Jun 30, 2019• 0 citations
OpenAlex

Abstract

In this note, we examine critically some basic features of momentum operators, both in the case of simple spatial context and in the case of spacetime. The natural time for the domain H of the momentum operator The Laurent Schwartz space Sn endowed with its own semimetric topology and with the Dirac inner product We emphasize that the right topology to consider is the standard Schwartz topology, not the topology induced by the above Dirac inner product which needs to provide probability issues and not continuity properties. The above choice for the space H satisfy at once some fundamental requirements of the theory:0. all the functions of the space H are both smooth and square integrable;1. the operator P reveals indeed a linear endomorphism everywhere defined uponH;2. the operator is continuous with respect to the Schwartz topology;3. the operator P is uniquely extensible to the entire space of tempered distributionsSn′ and this extension is continuous with respect to the standard topology of such distribution space.4. The above unique extension shows the so-called de Broglie waves as natural eigenvectors, although those de Broglie waves are not square integrable. The above third and fourth points induce us to consider that extension as the standard right choice for the momentum operator, not only because that extension contains in- side the previous operator defined on the Schwartz function space, but because - even more stringently - the space of temperate distributions allow us to define rigorously the fundamental concept of continuous quantum state basis and the concepts of Dirac orthogonality and Dirac normalizability for such type of bases. It happens, by the way, that some de Broglie families of quantum waves are indeed quantum basis of the space Sn′ and reveal Dirac orthogonal and Dirac normalizable.

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In this note, we examine critically some basic features of momentum operators, both in the case of simple spatial context and in the case of spacetime. The natural time for the domain H of the momentum operator The Laurent Schwartz space Sn endowed with its own semimetric topology and with the Dirac inner product
We emphasize that the right topology to consider is the standard Schwartz topology, not the topology induced by the above Dirac inner product which needs to provide probability issues and not continuity properties. The above choice for the space H satisfy at once some fundamental requirements of the theory:0. all the functions of the space H are both smooth and square integrable;1. the operator P reveals indeed a linear endomorphism everywhere defined uponH;2. the operator is continuous with respect to the Schwartz topology;3. the operator P is uniquely extensible to the entire space of tempered distributionsSn′ and this extension is continuous with respect to the standard topology of such distribution space.4. The above unique extension shows the so-called de Broglie waves as natural eigenvectors, although those de Broglie waves are not square integrable.
The above third and fourth points induce us to consider that extension as the standard right choice for the momentum operator, not only because that extension contains in- side the previous operator defined on the Schwartz function space, but because - even more stringently - the space of temperate distributions allow us to define rigorously the fundamental concept of continuous quantum state basis and the concepts of Dirac orthogonality and Dirac normalizability for such type of bases. It happens, by the way, that some de Broglie families of quantum waves are indeed quantum basis of the space Sn′ and reveal Dirac orthogonal and Dirac normalizable.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.
The aim of this chapter is to present (in a compact way) the fundamentals of conventional quantum theory. We select the for- mulation of QM as an axiomatic theory in the complete accordance with the Hilbert’s program on axiomatization of physical theories. We remark that the first step toward the present formulation of QM was done in the seminal work of Hilbert, Nordheim, and von Neumann [112]. This work was substantially extended, completed, and clarified in the book of von Neumann [296]. Another version of the rigorous mathematical approach to QM was presented in the book of Dirac [74]. From the mathematical viewpoint, the main difference between the approaches of von Neumann and Dirac is that the first one is based on the theory of self-adjoint operators in complex Hilbert space and the second one is based on the theory of self-adjoint operators acting (by using the modern mathematical terminology) in rigged Hilbert spaces. This theory operates with generalized eigenvectors belonging to the spaces of distributions. Although the von Neumann’s formalization of QM became dominant in quantum mathematical physics, the Dirac’s formalization is widely used in the works on the physical level of rigorousness. Moreover, the Dirac’s terminology, bra-and ket- vectors, became the standard for quantum information theory. However, since quantum information is based on the castrated version of QM, namely, QM in finite-dimensional Hilbert spaces, the so-called n-qubit spaces, the basic tool of the Dirac’s formalism, usage of generalized eigenvectors, is useless (since in the finite- dimensional case everything is reduced to manipulation with matrices). During the last 80 years, there were written hundreds of books on QM enlightening its foundational aspects. Unfortunately, the majority of these writings advertise exotic (and some of them really esoteric) views on QM and, in particular, on interpretation of the wave function. They are full of such terms as the collapse of the wave function, quantumnonlocality, manyworlds, thewave function of universe, the preparation of Schro¨dinger cats, influence of the future on the past, quantum teleportation. Of course, manipulations with such terminology create an atmosphere of mystery and a feeling that something unusual and impossible can happen. And this atmosphere plays a positive role in attracting young people to QM and especially to quantum foundations.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.
In this paper we start to construct some fundamental features of Dirac Calculus, specifically, we go inside the theory of Heisenberg continuous matrices, which, in our Schwartz Linear Algebra, are represented by Schwartz families. We distinguish the important subclass of transposable continuous matrices and give some basic and very important examples in Quantum Mechanics. So we define transposable Schwartz families and their transpose families, we prove the transposability of Dirac families and Fourier families. We find the transpose of regular-distribution families in a much general case. We define symmetric families, the analogous of symmetric ma- trices in the continuous case. We prove the symmetry of Dirac families and of Fourier families. We define Hermitian families, the analogous of Hermitian matrices in the continuous case. We prove the Hermitianity of Dirac families and of Fourier families. We define unitary families, the analogous of unitary matrices in the continuous case. We prove the unitarity of Dirac families and of the fundamental normalized de Broglie family. Then, we use the transpose of a family to find the components of the superpositions of transposable families, we give a general result and we apply this result to the Dirac families and the eigenfamilies of the vector-wave operator. We shall use the transposable families in next chapters to define the Dirac product in distribution spaces, basic product for the entire foundation of Dirac Calculus and Quantum Mechanics formalism.
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.