It is a classical derivation that the Wigner equation, derived from the Schrödinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant \(\varepsilon \rightarrow 0\). Since the latter presents the Newton’s second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schrödinger equation. More specifically, we show that using the initial condition and final state of the Schrödinger equation to reconstruct the potential term, in the classical regime with \(\varepsilon \rightarrow 0\), becomes using the initial and final state to reconstruct the potential term in the Liouville equation. This formally bridges an inverse problem in quantum mechanics with an inverse problem in classical mechanics.

这是一个经典的推导，从包含量子信息的薛定谔方程导出的 Wigner 方程在重新调整普朗克常数\(\varepsilon \rightarrow 0\)时收敛到 Liouville 方程。由于后者提出了牛顿第二定律，因此该过程通​​常被称为（半）经典极限。在本文中，我们研究了薛定谔方程逆问题的经典极限。更具体地说，我们展示了使用薛定谔方程的初始条件和最终状态来重建势项，在经典的\(\varepsilon \rightarrow 0\), 变成使用初始和最终状态来重建 Liouville 方程中的潜在项。这在形式上将量子力学中的逆问题与经典力学中的逆问题联系起来。