含义Meaning
能量低于势垒的粒子仍有一定概率穿过势垒。透射率随势垒宽度 a 和 \(\sqrt{m}\) 指数衰减，所以电子、质子隧穿显著，重原子几乎不隧穿。
A particle with E < V₀ can still pass a barrier. Transmission decays exponentially with width and √m, so electrons and protons tunnel appreciably while heavy atoms hardly do.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(T\) | 透射系数transmission coefficient | 1 |
| \(V_0\) | 势垒高度barrier height | J |
| \(a\) | 势垒宽度barrier width | m |
| \(\kappa\) | 衰减常数decay constant | m⁻¹ |
推导Derivation
势垒内 \(E < V_0\)，方程解为 \(Ce^{\kappa x} + De^{-\kappa x}\)（指数而非振荡）。
Inside the barrier the solutions are exponentials \(e^{\pm\kappa x}\).
三个区域在 x = 0、a 处 ψ 与 ψ′ 连续，求出透射振幅。
Match ψ and ψ′ at both edges to obtain the transmitted amplitude.
厚势垒（\(\kappa a \gg 1\)）时主导项为 \(e^{-2\kappa a}\)，得到近似式。
For thick barriers the \(e^{-2\kappa a}\) term dominates.
适用条件与局限Assumptions & limitations
矩形势垒是理想化；真实反应势垒用 WKB 近似 \(T \approx \exp(-2\int\kappa(x)dx)\) 或 Wigner/Bell 校正。
Rectangular barriers are idealised; use WKB or Wigner/Bell corrections for real barriers.
例题Worked example
扫描隧道显微镜：隧道电流随针尖–样品距离每增加 1 Å 下降约一个数量级。
STM current falls about tenfold per ångström of tip–sample separation.