Truncated quantum observables and their semiclassical limit

TL;DR

This paper investigates the semiclassical behavior of truncated quantum observables, showing their Weyl symbols converge to a truncated classical symbol in phase space as Planck's constant approaches zero.

Contribution

It establishes the $L^2$-convergence of Weyl symbols of truncated observables to a discontinuous classical symbol in the semiclassical limit, with applications to harmonic oscillator and particle in a box.

Findings

Weyl symbols converge in $L^2$ to a truncated classical symbol

Explicit analysis of symbols for harmonic oscillator and particle in a box

Computed microscopic limits near the boundary of the classically allowed region

Abstract

For quantum observables $H$ truncated on the range of orthogonal projections $\Pi_N$ of rank $N$, we study the corresponding Weyl symbol in the phase space in the semiclassical limit of vanishing Planck constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ fixed. Under certain assumptions, we prove the $L^2$- 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.