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양자력학(量子力學)의 Operator의 섭동리론(攝動理論)에 대(對)한 연구(硏究) -수리물리학(數理物理學)의 입장(立場)에서-

주영흠

Published Jan 1, 1975• 0 citations
OpenAlex

Abstract

This research deals with the general perturbation theory of linear operators which are applied to Quantum Mechanics in a view-point of Mathematical Physics. The principal theoretical instrument in the formation and development of the Quantum Mechanics is the theory of linear operators. The general theory of linear operators is presented, and the perturbation theories of closed operator, eigenvalue, eigenvector and continuous spectrum on Banach and Hilbert spaces are discussed using the method of functional analysis.

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This research deals with the general perturbation theory of linear operators which are applied to Quantum Mechanics in a view-point of Mathematical Physics. The principal theoretical instrument in the formation and development of the Quantum Mechanics is the theory of linear operators. The general theory of linear operators is presented, and the perturbation theories of closed operator, eigenvalue, eigenvector and continuous spectrum on Banach and Hilbert spaces are discussed using the method of functional analysis.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
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.
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.