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Journal of Heihe University

Discussion on the Application of Representation in Quantum Mechanics

Xiuping Zhang

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Representation is an abstract concept in quantum mechanics.Rationally,choosing a representation,which is similar to the coordinate system in geometry,can make the problem simply.To solve the same question,different representation in quantum system of micro-world is chosen,but its conclusion will not be changed.

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The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
Quantum states can be represented mathematically in different ways, depending on the chosen basis states. For example, position-based states have a definite position but an indefinite momentum, while momentum-based states have a definite momentum but an indefinite position.The Fourier transform is a mathematical operation that converts a quantum state from its position-space representation to its momentum-space representation, and vice versa. This transformation allows a function to be expressed in a different domain.This paper examines the concept and characteristics of the momentum-space representation of a quantum state and compares it to the position-space representation. Additionally, we investigate the applications and benefits of using the momentum-space representation to solve certain issues in quantum mechanics.
In this paper, the method how to further study the representational and the representational transformation is introduced by geometry coordinates, linear algebra and physical concept. According to this representational theory which can be simply compared with coordinate theory in geometry, two simple memory methods have been provided. Firstly, the transformation formula can be written as b =S+a from representation A to representation B, in which the matrix element of the transformation matrix is <ψ|φ>, which can be regarded as the inner product between the basis vector of representation A and representation B. At the same time, this formula can be thought of as the expression of the representation B of the state vector equals the transformation matrix acting on the expression of the representation A. Secondly, the transformation formula can be written as F' =S+FS for the mechanical operator from representation A to representation B, in which the operator F and operator F' are the expression of representations A and representations B, respectively, which this formula can be thought of as the expression of the representation B of the mechanical operator equals the conjugate matrix of transformation matrix and the expression of the representation A of the mechanical operator acting on transformation matrix. Finally, an example is given to prove the superiority of representation transformation. Meanwhile, the intrinsic function under two representations is deduced and the transformation between them is realized by the Fourier transform, thus deepening the understanding of representation transformation.
What is quantum mechanics and what does it do ? In very general terms, the basic problem that both classical Newtonian mechanics and quantum mechanics seek to address can be stated very simply: if the state of a dynamic system is known initially and something is done to it, how will the state of the system change with time in response ? In this chapter, we will give a brief overview of, first, how Newtonian mechanics goes about solving the problem for systems in the macroscopic world and, then, how quantum mechanics does it for systems on the atomic and subatomic scale. We will see qualitatively what the differences and similarities of the two schemes are and what the domain of applicability of each is. Brief overview of classical mechanics To answer the question posed above systematically, we must first give a more rigorous formulation of the problem and introduce the special language and terminology (in double quotation marks) that will be used in subsequent discussions. For the macroscopic world, common sense tells us that, to begin with, we should identify the “system” that we are dealing with in terms of a set of “static properties” that do not change with time in the context of the problem. For example, the mass of an object might be a static property. The change in the “state” of the system is characterized by a set of “dynamic variables.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
Quantum mechanics is traditionally taught in a specific representation—the position-space representation—because the time-independent Schrödinger equation is simplest to formulate there. But this leads to a number of conceptual issues both associated with language and with the formalism, especially if one starts more formally and then emphasizes representations for calculations. In this work, we describe a number of these issues and argue why restricting to a representation-independent formalism (that is, working with abstract operators and state vectors independent of any specific representation) fixes many of these issues. Then representations can be introduced later when needed for specific applications (but we find the need to do this can be sharply reduced from current paradigms).