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Journal of Yunnan Normal University

A hermitianity Discussion on the Momentum Operator

Zhang Shu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The measurable mechanical-quantity operator of quantum mechanics has to be hermite operator.Every momentum operator which shows hermite property must obey any rules,while it is not hermite for all wave functions.It will play an important role in the studying and the understanding quantum mechanics for us if we can clarify the condition for the hermitianity of momentum operator.In this paper,the hermitianity of momentum operator had been well discussed term by term by some suitable examples.

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The measurable mechanical-quantity operator of quantum mechanics has to be hermite operator.Every momentum operator which shows hermite property must obey any rules,while it is not hermite for all wave functions.It will play an important role in the studying and the understanding quantum mechanics for us if we can clarify the condition for the hermitianity of momentum operator.In this paper,the hermitianity of momentum operator had been well discussed term by term by some suitable examples.
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.
Hamiltonians in the geveralized Feshbach-Villars and Foldy-Wouthuysen representations describing an interaction of a scalar particle with electromagnetic fields in the Minkowski spacetime are self-adjoint and Hermitian (or pseudo-Hermitian) when they are presented in terms of operators of covariant derivatives. When one uses curvilinear coordinates in special relativity, the transition to the canonical momentum operator does not change these properties. When the curvilinear coordinates are applied in general relativity, the corresponding transition to the canonical momentum operator leads to the seeming non- Hermiticity of the Hamiltonians. Since the Hamiltonians remain in fact Hermitian, this seeming non-Hermiticity should not be eliminated by any nonunitary transformation.
In arXiv:0709.0483 Gunther and Samsonov outline a ``generalization'' of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be ``capable to produce effects beyond those of standard Hermitian quantum mechanics.'' We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding ``generalization of measurement theory'' suffers from a dynamical inconsistency and that it is by no means adequate to replace the standard measurement theory.
In the present chapter, before discussing the mechanical operators, operator has been lucidly defined and examples are cited for its clear understanding. It includes linear and non-linear operator, commutator and commutation, and Hermitian operator and its properties. It also consists of Schmidt orthogonalisation, del and del 2 operators, linear momentum operator and it applications, angular momentum vector and commutation, and angular momentum in spherical polar co-ordinates. This chapter also provides a good discussion regarding ladder operator, Hamiltonian operator and its commutation, and relation of angular momentum operators with Hamiltonian operators. It also discusses projection operator clearly and solution of problem related to projection operator. The properties of projection operator have also been given. Suitable references have also been provided for the consultation. A large number of problems and their solutions relating to different operators have also been cited. Questions on concepts have also been provided for exercise purposes. The readers are advised to solve these problems for better understanding.