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College Physics

Relationship between the coordinate representation and discretional mechanical parameter representation

Wei Wei

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

By discussing the relationship between the coordinate representation and the discretional mechanical parameter representation,this paper analyses the origin of the δ function normalization of eigenfunction with continuous eigenvalue.A conclusion is obtained that the eigenfunction of coordinate in the discretional mechanical parameter representation and the eigenfunction of discretional mechanical parameter in the coordinate representation are conjugated complex number of each other.Using this conclusion we give a new physical explanation of the eigenfunction closure.

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By discussing the relationship between the coordinate representation and the discretional mechanical parameter representation,this paper analyses the origin of the δ function normalization of eigenfunction with continuous eigenvalue.A conclusion is obtained that the eigenfunction of coordinate in the discretional mechanical parameter representation and the eigenfunction of discretional mechanical parameter in the coordinate representation are conjugated complex number of each other.Using this conclusion we give a new physical explanation of the eigenfunction closure.
On the basis of our previous papers~[1,2], we discuss the representations in quantum mechanics with negative metric. With the aid of the properties of the coherent state of the abnormal boson (the boson with negative metric), in addition to the properties of the normal product, we derive the ordered product form of the metric operator, and them discuss the eigenstates of the coordinate and the momentum with the imaginary eigenvalues. With the coherent-state method we prove that these eigenstates cannot be normalized, but we can find metric-like operators which can normalize the eigenstates as the δ function. We also provide the completeness condition for these eigenstates with imaginary eigenvalues and their relation to the eigenstates with real eigenvalues, as well as the physical vacuum state's new representations.
The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.
We delineate the scope of research on the completeness of eigenstates in quantum mechanics. Based on the limit of the potential function at infinity, the proof of completeness is divided into eight cases, and theoretical proofs or numerical simulations are provided for each case. We present the definition of orthonormalization for general free states and the solution to the normalization coefficients, as well as a general set of initial states, which simplifies and concretizes the proof of completeness. Additionally, we define the spectral function for continuous energy eigenvalues. By taking the spectral function as the original integral variable of the expansion function, the relationship between the measured probability amplitude and the expansion function is endowed with the physical meaning of coordinate-momentum transformation.