Truncated quantum observables and their semiclassical limit

2024/04/24 by Fabio Deelan Cunden, Marilena Ligabò, Cunden, Fabio Deelan +3
Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2404.15863

openalex publication_date 2024/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For quantum observables H truncated on the range of orthogonal projections Π_N of rank N, we study the corresponding Weyl symbol in the phase space in the semiclassical limit of vanishing Planck constant ℏ→0 and large quantum number N→∞, with ℏ N fixed. Under certain assumptions, we prove the L²- convergence of the Weyl symbols to a symbol truncated (hence, in general discontinuous) on the classically allowed region in phase space. As an illustration of the general theorems we analyse truncated observables for the harmonic oscillator and for a free particle in a one-dimensional box. In the latter case, we also compute the microscopic pointwise limit of the symbols near the boundary of the classically allowed region.

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