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Some impressive properties of unbounded operators in quantum mechanics

Niloofar Parviz, Dolati, Maedeh, Nojan Behzadi Alam

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

In this paper we have presented the necessity of defining the domain of the operators in quantum mechanics. We have investigated some controversial issues concerning algebra of unbounded operators. Our main focus has been the position and momentum operators. We have also shown the necessity of the presence Schwartz functions in quantum mechanics. The original aim of this paper is to present some mathematical issues of quantum mechanics in an understandable form for physics students.

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In this paper we have presented the necessity of defining the domain of the operators in quantum mechanics. We have investigated some controversial issues concerning algebra of unbounded operators. Our main focus has been the position and momentum operators. We have also shown the necessity of the presence Schwartz functions in quantum mechanics. The original aim of this paper is to present some mathematical issues of quantum mechanics in an understandable form for physics students.
During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain almost no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographes [32] and [2]. Some physics appeared first in [28] and [31], in the attempt to describe systems with a very large (10 24 ) number of degrees of freedom, following some general ideas originally proposed in the famous Haag and Kastler’s paper, [27], on QM ∞ .
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.
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 ( ) −∞,+∞ ).
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.
According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.