含义Meaning
一切运动的粒子都具有波动性（1924），波长与动量成反比。1927 年戴维森–革末和 G. P. 汤姆孙的电子衍射实验证实了这一点。
Every moving particle has a wavelength inversely proportional to its momentum (1924), confirmed by electron diffraction (Davisson–Germer and G. P. Thomson, 1927).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(p\) | 动量momentum | kg·m·s⁻¹ |
| \(E_k\) | 动能kinetic energy | J |
| \(U\) | 加速电压accelerating voltage | V |
推导Derivation
光子 \(p = E/c = h\nu/c = h/\lambda\)。
For photons \(p = h/\lambda\).
德布罗意假设同样关系适用于实物粒子：\(\lambda = h/p\)。
de Broglie postulated the same for matter.
电子经电压 U 加速 \(E_k = eU\)，代入得数值式；玻尔量子化条件 \(2\pi r = n\lambda\) 即“驻波”。
With \(E_k = eU\) for electrons; Bohr’s condition \(2\pi r = n\lambda\) is a standing wave.
适用条件与局限Assumptions & limitations
高能电子（> 约 50 kV）需相对论修正。
Fast electrons (> ~50 kV) need relativistic correction.
宏观物体波长极小，不可观测（1 g 小球以 1 m/s 运动 λ ≈ 6.6×10⁻³¹ m）。
Macroscopic objects have unobservably small wavelengths.
例题Worked example
100 V 加速的电子：λ = 0.123 nm，与晶体原子间距相当，所以能发生衍射。
An electron accelerated through 100 V has λ = 0.123 nm — comparable to atomic spacings.