一个量子力学阱和一个$π^2 $公式的推导
A quantum mechanical well and a derivation of a $π^2 $ formula
摘要：
束缚在无限大一维方势阱中的量子粒子是量子力学中的一个问题，大多数教科书都是从这个问题出发的。在那里，计算任意波函数的允许能谱通常涉及黎曼zeta函数，导致$π$级数。在这项工作中，两个“$π$公式”时，计算一个粒子的动量测量的可能结果的频谱限制在这样一个好，系列，$\frac{π^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$，和积分$\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =π$。动量算符的谱似乎只在量子数为偶数的状态下在经典允许的动量值上达到峰值。本文受\cite{wallys}中$π$公式的另一种量子力学推导的启发。
英文摘要：
Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a $π$ series. In this work, two "$π$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{π^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$, and the integral $\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =π$. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of $π$ formula in \cite{wallys}.
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