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On Application of Unbounded Hilbert Linear Operators in Quantum Mechanics

Eziokwu C. Emmanuel, Bright O. Osu, Iwuanyanwu Vitalis

Published Nov 18, 2020• 0 citations
OpenAlex

Abstract

This research work presents an important Banach space in functional analysis which is known and calledHilbert space. We verified the crucial operations in this space and their applications in physics, particularlyin quantum mechanics. The operations are restricted to the unbounded linear operators densely definedin Hilbert space which is the case of prime interest in physics, precisely in quantum machines. Precisely,we discuss the role of unbounded linear operators in quantum mechanics, particularly, in the study ofHeisenberg uncertainty principle, time-independent Schrodinger equation, Harmonic oscillation, andfinally, the application of Hamilton operator. To make these analyses fruitful, the knowledge of Hilbertspaces was first investigated followed by the spectral theory of unbounded operators, which are claimedto be densely defined in Hilbert space. Consequently, the theory of probability is also employed to studysome systems since the operators used in studying these systems are only dense in H (i.e., they must (orprobably) be in the domain of H defined by L2 ( ) −∞,+∞ ).

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This research work presents an important Banach space in functional analysis which is known and calledHilbert space. We verified the crucial operations in this space and their applications in physics, particularlyin quantum mechanics. The operations are restricted to the unbounded linear operators densely definedin Hilbert space which is the case of prime interest in physics, precisely in quantum machines. Precisely,we discuss the role of unbounded linear operators in quantum mechanics, particularly, in the study ofHeisenberg uncertainty principle, time-independent Schrodinger equation, Harmonic oscillation, andfinally, the application of Hamilton operator. To make these analyses fruitful, the knowledge of Hilbertspaces was first investigated followed by the spectral theory of unbounded operators, which are claimedto be densely defined in Hilbert space. Consequently, the theory of probability is also employed to studysome systems since the operators used in studying these systems are only dense in H (i.e., they must (orprobably) be in the domain of H defined by L2 ( ) −∞,+∞ ).
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.
The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models. The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.
This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X\mathbb{X}X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X\mathbb{X}X under the semi-inner product, and then the state space S(X)\mathcal{S} (\mathbb{X})S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X\mathbb{X}X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X)ω\in \mathcal{S}(\mathbb{X})ω∈S(X) to be a projection PPP (bounded operator such that P2=PP^2 =PP2=P) in X\mathbb{X}X satisfying the positivity condition 0≤ω(P)≤1,0 \le ω(P) \le 1,0≤ω(P)≤1, and a physical quantity at a quantum state ωωω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω.ω.ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
This study investigates the foundational and applied roles of linear algebra and functional analysis in quantum mechanics, employing a mixed-methods design that integrates qualitative conceptual analysis with quantitative simulations. The methodology combined literature synthesis of recent works (2018–2022), mathematical modeling of eigenvalue problems, and computational simulations within Hilbert spaces to validate theoretical constructs. The results demonstrated that linear algebra provides the structural basis for representing quantum states, operator norms, and matrix formulations, while functional analysis extends these principles to infinite-dimensional spaces, spectral decompositions, and unbounded operators. Tables presented comprehensive datasets, including eigenvalue distributions, operator convergence, and density matrix approximations, whereas figures offered multidimensional visualizations through line, bar, scatter, and pie charts. Together, these outputs revealed that functional analytic methods are indispensable in understanding spectral behavior and operator stability, while linear algebra underpins computational frameworks for quantum algorithms and quantum information processing. The discussion established that this synergy not only consolidates the mathematical rigor of quantum mechanics but also advances practical innovations such as quantum error correction, quantum machine learning, and operator-based quantum algorithms. The study concludes that linear algebra and functional analysis are complementary frameworks that enable both deeper theoretical insights and application-driven progress in quantum mechanics, underscoring their continuing importance in research, pedagogy, and technology development.
Quantum theory and functional analysis were created and put into essentially their final form during similar periods ending around 1930. Each was also a key outcome of the major revolutions that both physics and mathematics as a whole underwent at the time. This paper studies their interaction in this light, emphasizing the leading roles played by Hilbert in preparing the ground and by von Neumann in bringing them together during the crucial year of 1927, when he gave the modern, abstract definition of a Hilbert space and applied this concept to quantum mechanics (consolidated in his famous monograph from 1932). Subsequently, I give a very brief overview of three areas of functional analysis that have had fruitful interactions with quantum theory since 1932, namely unbounded operators, operator algebras, and distributions. The paper closes with some musings about the role of functional analysis in actual physics.