标题： 量子粒子如何出现在经典禁止区域中？ 显示英文标题

标题： How Can a Quantum Particle Be Found in a Classically Forbidden Region?

摘要： 量子力学的许多令人困惑的结果之一与基础物理中的一个结果相矛盾：即在经典物理学中，能量守恒会禁止出现的区域中，可能会找到一个量子粒子。 这种现象的一个特别有趣的例子，并且具有实际应用的是量子隧穿效应。 在这里，我们通过关注经典力学和量子力学中诸如动能和势能等量的数学表示差异，来研究这一令人困惑结果的原因。 在量子力学中，物理可观测量，如能量，是用算符而不是实数来表示的。 这种差异的后果将通过一个玩具模型进行明确说明，在该模型中，动能和势能算符由$2 \times 2$矩阵表示，这些矩阵不像它们的经典对应物那样对易。 然后，该模型将说明像量子粒子出现在禁区内这样的经典上令人困惑的结果是如何产生的。 显示英文摘要

摘要： Among the many perplexing results of quantum mechanics is one that contradicts a result from introductory physics: the possibility of finding a quantum particle in a region that would be forbidden classically by energy conservation. An especially interesting example of this phenomenon with practical applications is quantum tunneling. Here we investigate the reasons for this puzzling result by focusing on the difference between how quantities like kinetic and potential energy are represented mathematically in classical and quantum mechanics. In quantum mechanics, physical observables, like energy, are represented by operators rather than real numbers. The consequences of this difference will be illustrated explicitly using a toy model in which the kinetic and potential energy operators are represented by $2 \times 2$ matrices, which do not commute like their classical analogs. This model will then illustrate how classically perplexing results, like a quantum particle being found in the forbidden region, can arise.