含义Meaning
双原子分子振动的基本模型。能级等间距，间隔 hν；最低能量 \(\tfrac12h\nu\)（零点能）。选择定则 Δv = ±1。
The basic model of molecular vibration: evenly spaced levels hν apart, with zero-point energy ½hν; selection rule Δv = ±1.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(v\) | 振动量子数vibrational quantum number | 1 |
| \(k\) | 力常数force constant | N·m⁻¹ |
| \(\mu\) | 约化质量reduced mass | kg |
| \(H_v\) | 厄米多项式Hermite polynomial | 1 |
推导Derivation
势能 \(V = \tfrac12kx^2\)，方程 \(-\frac{\hbar^2}{2\mu}\psi'' + \tfrac12kx^2\psi = E\psi\)。
Potential \(V = \tfrac12kx^2\).
换元 \(y = x\sqrt{\mu\omega/\hbar}\)，\(\omega = \sqrt{k/\mu}\)：\(\psi'' + (\varepsilon - y^2)\psi = 0\)，\(\varepsilon = 2E/\hbar\omega\)。
Rescale to \(\psi'' + (\varepsilon - y^2)\psi = 0\).
大 y 时 \(\psi \sim e^{-y^2/2}\)；令 \(\psi = H(y)e^{-y^2/2}\) 得厄米方程，要求级数截断（ψ 可归一化）给出 \(\varepsilon = 2v + 1\)。
Write \(\psi = H(y)e^{-y^2/2}\); requiring the series to terminate gives \(\varepsilon = 2v + 1\).
所以 \(E_v = (v + \tfrac12)\hbar\omega\)。或用升降算符 \(\hat a^\dagger, \hat a\) 代数求解。
Hence \(E_v = (v + \tfrac12)\hbar\omega\); ladder operators give the same result.
跃迁矩 \(\langle v'|x|v\rangle\) 只在 \(v' = v \pm 1\) 时非零 → 选择定则。
\(\langle v'|x|v\rangle \neq 0\) only for \(v' = v \pm 1\) — the selection rule.
适用条件与局限Assumptions & limitations
真实键势不是抛物线：高 v 时能级间隔缩小，最终解离（用莫尔斯势）。
Real bonds are anharmonic: spacing shrinks and the bond dissociates (Morse).
非谐性使泛频（Δv = ±2, ±3）弱弱地出现。
Anharmonicity allows weak overtones.