The Completeness of Eigenstates in Quantum Mechanics

TL;DR

This work addresses the completeness of eigenstates in one-dimensional quantum systems by classifying potentials according to their asymptotic limits into eight cases and providing rigorous proofs or numerical demonstrations for each. It develops a spectral-function framework with $\kappa(\varepsilon)$ as the momentum-like expansion variable and casts the expansion as a coordinate–momentum transformation, unifying the physical interpretation of measurement amplitudes with state expansion. Key contributions include a formal orthonormalization of free states via a generalized $I(\kappa-\kappa')$, a general set of initial states based on the infinite-square-well basis, and detailed completeness analyses for symmetric (infinite wells, cosine and Dirac comb) and asymmetric (unequal constant, one-sided open box, semi-infinite periodic) potentials, plus explicit counterexamples (periodic square) where completeness fails without physically motivated initial states. The results provide a practical, physically meaningful route to assess and achieve completeness across a broad class of one-dimensional quantum systems and hint at a unified framework for extending to higher dimensions.

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.

Figures (7)

• Figure 1: Comparison of initial states $\Psi_n(x)$ and expansion functions $f_n(x)$ of double-well eigenstates. The double-well is located in the region $0 < x < 2$. ($n=j$) Initial states completely within the double-well. ($n=i$) Initial states spanning both sides of the double-well.
• Figure 2: A cosine potential: (a)–(c) Band structure of the spectral function $\kappa (\varepsilon)$; (d) Wavefunction at $\varepsilon= -0.17,\kappa=0.44$; (e)–(j) Comparison between the expansion functions $f_j(x)$ and the initial states $\Psi_j(x)$.
• Figure 3: (a)Band structure of the spectral function $\kappa (\varepsilon)$ in periodic square potential; (b) Comparison of two initial states.
• Figure 4: Dirac comb potential: (a)–(b) Band structure of the spectral function $\kappa (\varepsilon)$; (c) Wavefunction at $k=1.5,\kappa=-1.01$; (d)–(f) Comparison between the expansion functions $f_j(x)$ and the initial states $\Psi_j(x)$.
• Figure 5: Unequal constant potentiaL: (a)–(b)for $V_0=2.645$ band structure of the spectral function $\kappa$;(c)the integration path $\Gamma$;(d)the calculation method of argument range.

Theorems & Definitions (2)

• Theorem 1
• Theorem 1