📚 Modelling with Particles and Waves | 粒子与波建模
In CIE A-Level Physics, the way we interpret light, electrons and other quantum objects depends on the model we choose. A particle model treats energy and momentum as localised bundles, while a wave model treats disturbances as spreading through space and showing interference. This article explains when each model works, the key experimental evidence and the mathematical links behind wave-particle duality.
在 CIE A-Level 物理中，我们如何解释光、电子等量子对象取决于所选模型。粒子模型把能量和动量视为局域束，而波动模型把扰动视为在空间中传播并产生干涉。本文解释两种模型各自适用的情境、关键实验证据以及波粒二象性背后的数学联系。
1. Why Models Matter in Physics | 为什么物理需要模型
A model in physics is a simplified representation used to predict observations. No single classical model is perfect for quantum objects; light sometimes behaves like a stream of photons, and electrons sometimes behave like waves.
物理模型是一种用于预测观测结果的简化表示。对量子物体而言，没有单一经典模型是完美的；光有时表现为光子流，电子有时表现为波。
Using the wrong model can lead to incorrect predictions. For example, Maxwell’s wave model explains interference but cannot explain the photoelectric effect without quantisation of light.
使用错误模型会导致错误预测。例如，麦克斯韦波动模型能解释干涉，但若不引入光的量子化，就无法解释光电效应。
2. The Particle Model of Light | 光的粒子模型
In the particle model, light consists of discrete packets of electromagnetic energy called photons. Each photon carries energy E = hf, where h is the Planck constant and f is the frequency.
在粒子模型中，光由离散的电磁能量包组成，称为光子。每个光子携带能量 E = hf，其中 h 是普朗克常量，f 是频率。
E = hf
The photon model was introduced by Einstein in 1905 to explain the photoelectric effect. It treats photons as having no rest mass but carrying momentum p = h/λ.
光子模型由爱因斯坦于 1905 年提出，用于解释光电效应。该模型认为光子没有静止质量，但具有动量 p = h/λ。
p = h / λ
3. The Wave Model of Light | 光的波动模型
The wave model describes light as a transverse electromagnetic wave. Electric and magnetic fields oscillate perpendicular to the direction of energy transfer.
波动模型将光描述为横电磁波。电场和磁场垂直于能量传递方向振荡。
Young’s double-slit experiment provides strong evidence for the wave nature of light. Monochromatic light passing through two slits creates bright and dark fringes due to constructive and destructive interference.
杨氏双缝实验为光的波动性提供了有力证据。单色光通过双缝后因相长干涉和相消干涉产生明暗条纹。
The fringe spacing Δx is given by:
条纹间距 Δx 由以下关系给出：
Δx = λD / d
where λ is the wavelength, D is the distance from slits to screen, and d is the slit separation.
其中 λ 是波长，D 是双缝到屏的距离，d 是双缝间距。
4. Photoelectric Effect: Evidence for Photons | 光电效应：光子的证据
When ultraviolet light shines on a clean metal surface, electrons are emitted. The wave model predicts that brighter light should give electrons more kinetic energy, but experiments show otherwise.
当紫外光照射洁净金属表面时，会发射电子。波动模型预测光越强，电子动能越大，但实验结果并非如此。
Key observations are: emission is instantaneous above a threshold frequency f₀; maximum electron kinetic energy depends only on frequency, not intensity; intensity affects only the number of emitted electrons.
关键观察是：高于阈值频率 f₀ 时发射立即发生；电子最大动能只取决于频率而非光强；光强只影响发射电子的数量。
Eₖ,ₘₐₓ = hf – Φ
Here Φ is the work function of the metal, the minimum energy needed to remove an electron. The threshold frequency is f₀ = Φ / h.
这里 Φ 是金属的功函数，即移除一个电子所需的最小能量。阈值频率为 f₀ = Φ / h。
5. Electron Diffraction: Evidence for Matter Waves | 电子衍射：物质波的证据
Electrons accelerated through a potential difference V gain kinetic energy Eₖ = eV. When they pass through a thin graphite film, a diffraction pattern of concentric rings appears on a fluorescent screen.
电子经电势差 V 加速后获得动能 Eₖ = eV。当它们穿过薄石墨膜时，荧光屏上出现同心环衍射图样。
This pattern is evidence that electrons behave as waves. Increasing the accelerating voltage reduces the ring spacing, showing that the wavelength decreases as electron speed increases.
这一图样证明电子表现出波动行为。提高加速电压会减小环间距，表明电子速度增加时波长减小。
The electron wavelength is given by:
电子波长由下式给出：
λ = h / p = h / √(2meV)
where m is the electron mass and e is the elementary charge.
其中 m 是电子质量，e 是元电荷。
6. De Broglie Wavelength | 德布罗意波长
Louis de Broglie proposed that all particles have a wavelength related to their momentum. This unified the particle and wave descriptions of matter.
德布罗意提出所有粒子都具有与其动量相关的波长。这统一了物质的粒子描述和波动描述。
The de Broglie wavelength is:
德布罗意波长为：
λ = h / p = h / mv
For macroscopic objects, the wavelength is extremely small, so wave behaviour is not observed. For electrons and other small particles, the wavelength can be comparable to atomic spacing, producing observable diffraction.
对于宏观物体，波长极小，因此观察不到波动行为。对于电子和其他小粒子，波长可与原子间距相当，从而产生可观察的衍射。
Example: An electron accelerated through 54 V has a de Broglie wavelength of about 1.67 × 10⁻¹⁰ m, similar to atomic spacing. This is why electron diffraction is useful for studying crystal structure.
例题：经 54 V 加速的电子德布罗意波长约为 1.67 × 10⁻¹⁰ m，与原子间距相近。这就是电子衍射可用于研究晶体结构的原因。
7. Wave-Particle Duality | 波粒二象性
Wave-particle duality states that quantum objects have both wave-like and particle-like properties. Neither classical model alone gives a complete description.
波粒二象性指出量子物体同时具有波动性和粒子性。单独使用任一经典模型都无法给出完整描述。
Light shows particle behaviour in the photoelectric effect and wave behaviour in interference. Electrons show particle behaviour in collisions and wave behaviour in diffraction.
光在光电效应中表现出粒子性，在干涉中表现出波动性。电子在碰撞中表现出粒子性，在衍射中表现出波动性。
Which property is observed depends on the experiment. A single quantum object interacts at a point like a particle, but the probability distribution of many events shows a wave-like interference pattern.
观察到哪种性质取决于实验。单个量子物体像一个粒子那样在一点发生相互作用，但大量事件的概率分布却显示出类波的干涉图样。
8. Atomic Spectra and Energy Quantisation | 原子光谱与能量量子化
Atoms emit or absorb light at discrete wavelengths, producing line spectra. These spectra can only be explained if electron energy levels are quantised.
原子以离散波长发射或吸收光，产生线光谱。只有电子能级是量子化的，才能解释这些光谱。
When an electron drops from a higher energy level E₂ to a lower level E₁, it emits a photon of energy:
当电子从较高能级 E₂ 跃迁到较低能级 E₁ 时，会发射一个光子，能量为：
hf = E₂ – E₁
The hydrogen spectrum is a classic example. The Balmer series lies in the visible region and corresponds to transitions ending at n = 2.
氢光谱是一个经典例子。巴耳末系位于可见光区，对应于终止在 n = 2 的跃迁。
This quantisation supports the particle model of light, as photons carry fixed energy differences, while the wavelengths of spectral lines link back to the wave model through c = fλ.
这种量子化支持了光的粒子模型，因为光子携带固定的能量差，而谱线波长又通过 c = fλ 与波动模型联系起来。
| Spectral series | 光谱系 | Region | 区域 | Ending level | 终止能级 |
| Lyman | 莱曼系 | Ultraviolet | 紫外 | n = 1 |
| Balmer | 巴耳末系 | Visible | 可见光 | n = 2 |
| Paschen | 帕邢系 | Infrared | 红外 | n = 3 |
9. Probabilistic Interpretation of Waves | 波的概率诠释
In quantum physics, a particle’s wave function describes the probability amplitude for finding the particle at a location. The square of the wave function gives the probability density.
在量子物理中，粒子的波函数描述在某位置找到粒子的概率幅。波函数的平方给出概率密度。
This means a wave-like diffraction pattern does not show a particle spreading out; it shows the distribution of possible detection positions over many identical trials.
这意味着类波衍射图样并不表示粒子分散开来，而是显示在多次相同实验中可能被探测到的位置分布。
The probabilistic model is important for A-Level when understanding why individual electrons arrive at the screen one by one, yet build up an interference pattern over time.
概率模型在 A-Level 中很重要，可帮助理解为什么单个电子一个接一个到达屏幕，但随时间积累却能形成干涉图样。
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