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Journal of Hebei Polytechnic University

The Sufficiency and Necessity of Hermitian Operator

Yang Han-song

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Hermitianity is a very important physical character to operators in quantum mechanics.It will play an important role to study and research quantum mechanics for us if we can acknowledge the condition to determine the Hermitianity of operator.This articles takes P^ for example to give the concrete illumination and reasoning for the conditions of sufficiency and necessity to the Hermitianity of operator.It constructs the academic foundation to acknowledge it completely and correctly for us.

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Operators

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Quantum mechanics has brought revolutionary changes in the conceptual foundations of physics and continues to shape the modern world. In this book, the author's emphasis is on helping students comprehend the significance of the underlying principles and understand the ways the new concepts were introduced. Including many worked examples and problems, this book will be an invaluable resource for students in physics, chemistry and electrical engineering needing a clear and rigorous introduction to quantum mechanics.

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Hermitianity is a very important physical character to operators in quantum mechanics.It will play an important role to study and research quantum mechanics for us if we can acknowledge the condition to determine the Hermitianity of operator.This articles takes P^ for example to give the concrete illumination and reasoning for the conditions of sufficiency and necessity to the Hermitianity of operator.It constructs the academic foundation to acknowledge it completely and correctly for us.
We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.
This chapter considers the idea of representing physical as well as nonphysical quantities by operators and collect together some frequently used operator techniques. It postulates that any quantity in quantum physics is specified by a linear operator. In a manipulation with operators, it is very important to maintain the order of the operators because operator algebra is not in general commutative for multiplication. An important concept in quantum physics is a commutator. The chapter focuses on to define a special class of operators called Hermitian operators. They are distinguished by the relationship of  and † The Hermitian operators are of great importance in quantum physics, as they represent physical (measurable) quantities. The chapter defines a very important property of wave functions: orthogonality. This property, for example, allows to identify whether the given wave functions belong to the same operator or not. The orthogonality condition of two functions is analogous to the orthogonality condition of two vectors.
Quantum mechanics has brought revolutionary changes in the conceptual foundations of physics and continues to shape the modern world. In this book, the author's emphasis is on helping students comprehend the significance of the underlying principles and understand the ways the new concepts were introduced. Including many worked examples and problems, this book will be an invaluable resource for students in physics, chemistry and electrical engineering needing a clear and rigorous introduction to quantum mechanics.