The Completeness of Eigenstates in Quantum Mechanics

TL;DR

This paper systematically analyzes the completeness of eigenstates in quantum mechanics across various potential scenarios, providing theoretical proofs, numerical simulations, and clarifying the physical interpretation of spectral functions.

Contribution

It introduces a unified framework for proving completeness, defines orthonormalization for general free states, and clarifies the physical meaning of spectral functions in quantum expansions.

Findings

Completeness proof divided into eight cases based on potential limits.

Defined orthonormalization and normalization coefficients for general states.

Linked spectral functions with coordinate-momentum transformations.

Abstract

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.