📚 Quantum Physics: Photons, Wave-Particle Duality and Energy Levels | 量子物理：光子、波粒二象性与能级
Quantum physics revolutionised our understanding of the microscopic world, introducing the idea that energy and matter exhibit both particle and wave characteristics. This article reviews the key concepts required for CIE A-Level Physics, including photons, the photoelectric effect, wave-particle duality, de Broglie waves, atomic energy levels, electron diffraction and the uncertainty principle. Each section pairs the physics with the standard examination calculations and the language examiners expect to see.
量子物理学彻底改变了我们对微观世界的认识，提出了能量和物质同时具有粒子性与波动性的观念。本文回顾 CIE A-Level 物理课程中的关键概念，包括光子、光电效应、波粒二象性、德布罗意波、原子能级、电子衍射以及不确定原理。每一节都把物理原理与考试中常见的计算、以及考官期望看到的表述方式结合起来。
1. Photons and Energy Quantisation | 光子与能量量子化
In quantum physics, electromagnetic radiation is emitted, transmitted and absorbed in discrete packets of energy called photons. The energy carried by a single photon depends only on its frequency, not on the intensity of the radiation. A photon has no rest mass, is electrically neutral, and always travels at the speed of light in a vacuum. A beam of light of higher intensity simply contains more photons per second, not more energetic ones.
在量子物理中，电磁辐射以不连续的能量包形式被发射、传播和吸收，这些能量包称为光子。单个光子携带的能量只取决于它的频率，而与辐射强度无关。光子没有静止质量、不带电，并且在真空中始终以光速传播。强度更高的光束只是每秒包含更多的光子，而不是每个光子的能量更大。
- Photon energy: E = hf, measured in joules (J). | 光子能量：E = hf，单位为焦耳（J）。
- Planck constant: h = 6.63 × 10⁻³⁴ J s. | 普朗克常数：h = 6.63 × 10⁻³⁴ J s。
- Speed of light in vacuum: c = 3.00 × 10⁸ m s⁻¹. | 真空中光速：c = 3.00 × 10⁸ m s⁻¹。
- Since c = fλ, the photon energy can be written in terms of wavelength. | 由于 c = fλ，光子能量也可以用波长表示。
E = hf = hc / λ
Because h and c are both constants, photon energy is inversely proportional to wavelength. Ultraviolet photons are therefore far more energetic than infrared or visible photons, which is exactly why ultraviolet light can damage living tissue while radio waves pass through it harmlessly.
由于 h 和 c 都是常数，光子能量与波长成反比。因此紫外光子的能量远高于红外光或可见光子，这正是紫外光会损伤生物组织、而无线电波可以安全穿过人体的原因。
2. The Photoelectric Effect: Experimental Observations | 光电效应：实验现象
The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency is incident on it. The released electrons are called photoelectrons. The effect was one of the first pieces of evidence that light behaves as a stream of particles rather than as a continuous wave.
光电效应是指当频率足够高的电磁辐射照射到金属表面时，金属表面发射出电子的现象。被释放出来的电子称为光电子。这一效应是光表现为粒子流而非连续波的最早证据之一。
- Threshold frequency: below a certain frequency f₀, no electrons are emitted at all, however intense the radiation. | 阈值频率：低于某一频率 f₀ 时，无论辐射多强，都不会有电子逸出。
- Instantaneous emission: photoelectrons appear immediately (within about 10⁻⁹ s), even for very weak radiation. | 瞬时发射：即使辐射非常微弱，光电子也几乎立刻出现（约 10⁻⁹ s 内）。
- Maximum kinetic energy: the maximum kinetic energy of the photoelectrons increases with frequency but is independent of intensity. | 最大动能：光电子的最大动能随频率升高而增大，但与光强无关。
- Intensity: increasing the intensity increases the number of photoelectrons emitted per second, not their energy. | 强度：增大光强会增加每秒逸出的光电子数目，但不会提高它们的能量。
Classical wave theory predicted the opposite in every case: it expected energy to accumulate gradually, so that any frequency would eventually release electrons, and it expected brighter light to give faster electrons. The experimental results therefore forced a new model of light.
经典波动理论在每一点上都给出了相反的预言：它认为能量会逐渐积累，因此任何频率最终都能释放电子；它还认为光越强，电子速度越快。这些实验结果因此迫使人们建立关于光的全新模型。
3. Einstein's Photoelectric Equation | 爱因斯坦光电方程
Einstein explained the photoelectric effect by treating light as photons. A single photon is absorbed by a single electron in the metal. The electron must do a minimum amount of work to escape from the surface; this is called the work function Φ, and different metals have different values. Any remaining energy becomes kinetic energy of the emitted electron.
爱因斯坦把光看作光子来解释光电效应。单个光子被金属中的单个电子吸收。电子要逸出表面必须做一定的最小功，这称为逸出功 Φ，不同金属的逸出功不同。剩余的能量则成为逸出电子的动能。
hf = Φ + KEmax
Rearranging gives KEmax c/λ₀. If the photon energy hf is less than Φ, the electron cannot escape at all, which explains the threshold frequency directly from the model.
整理后得到 KEmax = hf − Φ。逸出功与阈值频率的关系是 Φ = hf₀，与阈值波长的关系是 Φ = hc/λ₀。如果光子能量 hf 小于 Φ，电子根本无法逸出，这就从模型本身直接解释了阈值频率。
| Observation | 实验现象 | Wave theory prediction | 波动理论预言 | Photon explanation | 光子解释 |
|---|---|---|
| Threshold frequency exists | 存在阈值频率 | No threshold expected | 不应有阈值 | hf must exceed Φ | 必须满足 hf 大于 Φ |
| Emission is instantaneous | 发射是瞬时的 | Energy must build up | 能量需要积累 | One photon, one electron | 一个光子对应一个电子 |
| KEmax depends on f, not intensity | 最大动能取决于频率而非光强 | Brighter light gives faster electrons | 光越强电子越快 | KEmax = hf − Φ | 最大动能等于 hf 减 Φ |
| Intensity sets electron number | 光强决定电子数目 | Not predicted | 未预言 | More photons, more electrons | 光子越多，电子越多 |
4. Stopping Potential and Maximum Kinetic Energy | 遏止电压与最大动能
The maximum kinetic energy of the photoelectrons can be measured using a vacuum photocell and a variable opposing voltage. When the voltage is just large enough to stop even the fastest electrons from reaching the anode, the photocurrent falls to zero. This voltage is called the stopping potential Vs.
光电子的最大动能可以用真空光电管和可调的反向电压来测量。当电压刚好大到足以阻止最快电子到达阳极时，光电流降为零。这个电压称为遏止电压 Vs。
KEmax = eVs = hf − Φ
al against frequency is a straight line whose gradient is h/e and whose x-intercept is the threshold frequency f₀. This is the standard experimental method for determining the Planck constant.
这里 e 是元电荷，等于 1.60 × 10⁻¹⁹ C。以遏止电压对频率作图，得到的是一条直线，其斜率为 h/e，与横轴的交点即为阈值频率 f₀。这是测定普朗克常数的标准实验方法。
Emitted electrons have a range of kinetic energies because electrons deeper within the metal need more energy to escape, and because some energy may be lost in collisions. The value KEmax corresponds to electrons released from the very surface with no internal energy loss.
逸出电子的动能有一个分布范围，因为金属内部较深处的电子需要更多能量才能逸出，而且有些能量可能在碰撞中损失。KEmax 对应的是从最表层逸出、且没有任何内部能量损失的电子。
5. Wave-Particle Duality | 波粒二象性
Light and matter both show behaviour that can only be described by waves in some experiments and by particles in others. This is called wave-particle duality. The two descriptions are complementary rather than contradictory: a single experiment reveals one aspect, and the full picture requires both.
光和物质在某些实验中只能用波来描述，在另一些实验中只能用粒子来描述，这就是波粒二象性。这两种描述是互补的，而不是互相矛盾的：单个实验只显示其中一个方面，完整的图像需要两者兼备。
- Evidence for wave behaviour in light: interference (Young's double-slit experiment) and diffraction at a grating or slit. | 光具有波动性的证据：干涉（杨氏双缝实验）以及光栅或单缝衍射。
- Evidence for particle behaviour in light: the photoelectric effect and Compton scattering. | 光具有粒子性的证据：光电效应和康普顿散射。
- Evidence for wave behaviour in electrons: electron diffraction through a thin graphite film. | 电子具有波动性的证据：电子穿过薄石墨膜的衍射。
- Practical consequence: scanning electron microscopes exploit the very short de Broglie wavelength of electrons to resolve detail far smaller than visible light allows. | 实际应用：扫描电子显微镜利用电子极短的德布罗意波长，分辨出远小于可见光极限的细节。
A useful working rule is that wave behaviour dominates when the wavelength is comparable to the size of the obstacles or apertures involved, while particle behaviour dominates in interactions with individual atoms and electrons.
一个实用的判断准则是：当波长与障碍物或孔径的尺寸相当时，波动性占主导；而在与单个原子、电子发生相互作用时，粒子性占主导。
6. The de Broglie Wavelength | 德布罗意波长
In 1924 de Broglie proposed that every particle has an associated wavelength, given by the Planck constant divided by its momentum. This idea extended wave-particle duality from light to matter and was confirmed experimentally within a few years.
1924 年德布罗意提出，任何粒子都有一个与之对应的波长，其大小等于普朗克常数除以其动量。这一思想把波粒二象性从光推广到物质，并在几年内就被实验证实。
λ = h / p = h / (mv)
at 30 m s⁻¹ has a de Broglie wavelength of about 1.4 × 10⁻³⁴ m, which is many orders of magnitude smaller than an atomic nucleus, so no diffraction can be observed.
由于普朗克常数极其微小，宏观物体的波长小到无法探测。一个质量为 0.16 kg、以 30 m s⁻¹ 运动的板球，其德布罗意波长约为 1.4 × 10⁻³⁴ m，比原子核还要小好几个数量级，因此观察不到任何衍射现象。
from ½mv² = eV. Combining this with the de Broglie relation gives a very convenient form.
对于从静止开始经电势差 V 加速的电子，获得的动能为 eV，因此可由 ½mv² = eV 求出动量。把它与德布罗意关系式结合，可得一个非常实用的形式。
λ = h / √2meV
o typical atomic spacings in a crystal. This is why crystals, rather than mechanical slits, are used to diffract electrons.
例如，经 100 V 电压加速的电子，其波长约为 1.2 × 10⁻¹⁰ m，与晶体中典型的原子间距相当。这正是用晶体而不是机械狭缝来使电子发生衍射的原因。
7. Electron Diffraction: Evidence for Matter Waves | 电子衍射：物质波的证据
In the Davisson-Germer experiment, a beam of electrons was fired at a thin polycrystalline graphite film and the transmitted electrons were detected on a fluorescent screen. The result was a pattern of concentric rings, exactly the kind of pattern produced by X-ray diffraction through the same material. This could not be explained by electrons travelling as simple particles in straight lines.
在戴维孙-革末实验中，一束电子射向薄的多晶石墨膜，透射电子被荧光屏探测。结果出现了同心圆环图样，与 X 射线穿过同样材料产生的衍射图样完全同类。这无法用电子沿直线运动的简单粒子图像来解释。
- Ring formation: electrons diffract at the regularly spaced atomic planes, and constructive interference occurs at specific angles. | 圆环的形成：电子在规则排列的原子平面处发生衍射，在特定角度上产生相长干涉。
- Changing the voltage: increasing the accelerating voltage decreases the de Broglie wavelength, so the diffraction rings become smaller in diameter. | 改变电压：提高加速电压会减小德布罗意波长，因此衍射环的直径变小。
- Key conclusion: electrons have a wavelength given by λ = h/p, confirming de Broglie's hypothesis. | 关键结论：电子的波长满足 λ = h/p，从而证实了德布罗意的假设。
- Modern use: electron diffraction is used to study crystal structures, thin films and surfaces. | 现代应用：电子衍射用于研究晶体结构、薄膜和表面。
Neutrons and even large molecules such as C₆₀ have since been diffracted, so wave behaviour is a general property of matter, not a special feature of electrons.
此后中子、甚至 C₆₀ 这样的大分子也都被观察到衍射现象，因此波动性是物质的普遍性质，而不是电子独有的特征。
8. Atomic Energy Levels and Line Spectra | 原子能级与线状光谱
An atom can only exist in certain discrete energy states, called energy levels. The lowest level is the ground state, and all higher levels are excited states. An electron normally occupies the ground state; the atom can be excited by absorbing energy from a photon, from an electron collision, or from thermal energy.
原子只能处于某些特定的能量状态，这些状态称为能级。最低的能级是基态，所有更高的能级都是激发态。电子通常占据基态；原子可以通过吸收光子能量、电子碰撞能量或热能而被激发。
- Excitation: an electron moves to a higher level after absorbing exactly the right amount of energy. | 激发：电子吸收恰好合适的能量后跃迁到较高能级。
- Ionisation: the electron is removed from the atom completely. | 电离：电子被完全移出原子。
- Emission spectrum: bright lines on a dark background, produced when excited atoms emit photons. | 发射光谱：暗背景上的亮线，由激发态原子发射光子产生。
- Absorption spectrum: dark lines on a continuous bright background, produced when cold gas absorbs specific wavelengths. | 吸收光谱：连续亮背景上的暗线，由冷气体吸收特定波长的光产生。
For hydrogen, the energy levels can be calculated from En egative sign shows that the electron is bound to the nucleus and that energy must be supplied to remove it. The ionisation energy of hydrogen is therefore 13.6 eV.
对氢原子而言，能级可由 En = −13.6 / n² eV 计算，其中 n = 1, 2, 3……负号表示电子被原子核束缚，必须提供能量才能把它移走。因此氢原子的电离能为 13.6 eV。
9. Energy Level Diagrams and Photon Transitions | 能级图与光子跃迁
An energy level diagram shows the permitted energies as horizontal lines, with energy increasing upwards. When an electron moves from a higher level E₂ to a lower level E₁, the atom emits a single photon whose energy equals the difference between the two levels.
能级图用水平线表示允许的能量值，能量向上增加。当电子从较高能级 E₂ 跃迁到较低能级 E₁ 时，原子发出一个光子，其能量等于两个能级之差。
hf = E₂ − E₁
Because only certain energy differences are possible, only certain photon energies and therefore certain wavelengths are emitted. This produces a line spectrum rather than a continuous one, and it is the most direct evidence that atomic energy levels are quantised.
由于只有某些特定的能量差是可能的，因此只能发射特定能量的光子，即特定波长的光。这产生了线状光谱而不是连续光谱，也是原子能级量子化最直接的证据。
- An upward transition requires absorption of a photon of exactly the right energy. | 向上跃迁需要吸收能量恰好合适的光子。
- A downward transition emits a photon of exactly that energy difference. | 向下跃迁则发射能量恰好等于该能级差的光子。
- ding at n = 2 in hydrogen form the Balmer series, which lies partly in the visible region. | 终止于氢原子 n = 2 能级的跃迁构成巴尔末系，其中一部分位于可见光区。
- Transitions ending at n = 1 form the Lyman series in the ultraviolet. | 终止于 n = 1 能级的跃迁构成紫外区的莱曼系。
In an emission tube, a collision between a free electron and an atom can excite the atom, but the photon emitted later always has an energy equal to a level difference. An absorbed photon, however, must match a level difference exactly and cannot be partly absorbed.
在放电管中，自由电子与原子碰撞可以使原子激发，但随后发出的光子能量始终等于某个能级差。而被吸收的光子则必须与某个能级差精确匹配，不能被部分吸收。
10. The Uncertainty Principle | 不确定原理
Heisenberg's uncertainty principle states that position and momentum cannot both be known with unlimited precision at the same time. The more accurately the position of a particle is determined, the less accurately its momentum can be known, and vice versa. This is not a limitation of instruments but a fundamental property of nature.
海森堡不确定原理指出，位置和动量不能同时被无限精确地确定。粒子的位置测得越准确，其动量就越不准确，反之亦然。这不是仪器的局限，而是自然界的基本性质。
Δx Δp ≥ h / 4π
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