环上粒子 Particle on a ring — 公式推导、符号与局限

\[E_{m_l} = \frac{m_l^2\hbar^2}{2mR^2},\qquad \psi_{m_l}(\phi) = \frac{1}{\sqrt{2\pi}}e^{im_l\phi},\quad m_l = 0, \pm1, \pm2, \ldots\]

含义Meaning

粒子在半径 R 的圆环上运动。除 \(m_l = 0\) 外每个能级二重简并；可解释苯等环状共轭体系的 π 能级和芳香性（4n+2）。

A particle moving on a ring of radius R. All levels except \(m_l = 0\) are doubly degenerate — a model for cyclic π systems and the 4n + 2 rule.

符号Symbol	含义 Meaning	单位 Unit
\(m_l\)	磁量子数magnetic quantum number	1
\(R\)	环半径ring radius	m
\(I = mR^2\)	转动惯量moment of inertia	kg·m²

极坐标中 \(-\frac{\hbar^2}{2mR^2}\frac{d^2\psi}{d\phi^2} = E\psi\)，解为 \(e^{im_l\phi}\)。
In polar form \(-\frac{\hbar^2}{2mR^2}\psi'' = E\psi\), solved by \(e^{im_l\phi}\).
周期性边界 \(\psi(\phi + 2\pi) = \psi(\phi)\) 要求 \(m_l\) 为整数。
Periodicity requires integer \(m_l\).
\(m_l = 0\) 允许（无零点能），\(\pm m_l\) 能量相同（顺、逆时针运动）。
\(m_l = 0\) is allowed; \(\pm m_l\) are degenerate.
填充：最低 2 个电子，之后每层 4 个 → 闭壳层电子数 2, 6, 10, …（4n+2）。
Filling 2 then 4 per level gives closed shells at 2, 6, 10, … electrons.

忽略环的离散原子结构；休克尔方法给出更真实的能级（弗罗斯特圆）。
Ignores discrete atoms; Hückel theory (Frost circle) is more realistic.