📚 E.2 Quantum Physics (HL only) | E.2 量子物理（仅限HL）
Quantum physics emerged from the failure of classical physics to explain phenomena at the atomic scale. Planck’s quantisation of energy and Einstein’s photon model provided a radical new framework. In IB Physics HL topic E.2, we explore the photoelectric effect, photon momentum, Compton scattering, matter waves, electron diffraction, wave–particle duality and the uncertainty principle.
量子物理诞生于经典物理无法解释原子尺度现象的困境。普朗克的能量量子化与爱因斯坦的光子模型提供了一个革命性的新框架。在 IB 物理高水平 E.2 中，我们将探讨光电效应、光子动量、康普顿散射、物质波、电子衍射、波粒二象性以及不确定性原理。
1. The Photoelectric Effect and Photons | 光电效应与光子
The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it. Classical wave theory predicted that any frequency could eject electrons if the intensity was high enough, but experiments showed a threshold frequency below which no electrons are emitted, regardless of intensity.
光电效应是当频率足够高的电磁辐射照射到金属表面时，电子从表面逸出的现象。经典波动理论预言，只要强度足够，任何频率的光都能打出电子，但实验表明存在一个截止频率，低于该频率无论强度多大都不能打出电子。
onstant and f is the frequency. A single photon interacts with a single electron, and electron emission occurs only if the photon energy exceeds the work function Φ of the metal.
爱因斯坦通过提出光由分离的量子（后来称为光子）组成解决了这个问题，每个光子携带能量 E = hf，其中 h 是普朗克常数，f 是频率。单个光子与单个电子相互作用，只有当光子能量超过金属的功函数 Φ 时，电子才会发射。
E = h f
2. Einstein’s Photoelectric Equation | 爱因斯坦光电方程
equation states that the maximum kinetic energy of the emitted photoelectron is: Kₘₐₓ = hf – Φ.
电子若要被打出，光子必须提供功函数 Φ，多余的能量转化为电子的动能。爱因斯坦光电方程指出，发射出的光电子最大动能为：Kₘₐₓ = hf – Φ。
Kₘₐₓ = h f – Φ
frequency f₀ = Φ/h, the independence of Kₘₐₓ on intensity, and the instantaneous emission – all classical wave theory could not account for.
该方程成功解释了截止频率 f₀ = Φ/h、最大动能与光强无关以及瞬时发射——这些都是经典波动理论无法解释的。
3. Stopping Potential and Threshold Frequency | 阻止电势与截止频率
the maximum kinetic energy by eVₛ = Kₘₐₓ, where e is the elementary charge. Thus eVₛ = hf – Φ.
阻止电势 Vₛ 是阻止最具能量的光电子到达集电极所需的反向电势。它通过 eVₛ = Kₘₐₓ 与最大动能关联，其中 e 是元电荷电量。因此 eVₛ = hf – Φ。
e Vₛ = h f – Φ
By plotting Vₛ against frequency f, the gradient gives h/e and the x-intercept yields the threshold frequency f₀. Below f₀, no photocurrent flows no matter how intense the illumination.
通过绘制 Vₛ 对频率 f 的图像，斜率给出 h/e，与横轴交点给出截止频率 f₀。低于 f₀，无论照度多强都不会有光电流。
4. Photon Momentum and Compton Scattering | 光子动量与康普顿散射
s was confirmed by Compton scattering, in which an X-ray photon collides with a stationary electron, causing the photon to lose energy and its wavelength to increase.
虽然光子静止质量为零，但它们携带动量 p = E/c = h/λ。这一点由康普顿散射所证实，即一个 X 射线光子与静止电子碰撞，导致光子损失能量，其波长变长。
p = h / λ
Treating the interaction as a relativistic collision between a particle‑like photon and an electron, energy and momentum conservation lead to a shift Δλ that depends on the scattering angle θ. This confirmed the particle nature of light beyond the photoelectric effect.
将该相互作用视为类粒子光子与电子的相对论碰撞，能量与动量守恒导致一个与散射角 θ 有关的偏移 Δλ。这进一步在光电效应之外证实了光的粒子性。
5. The Compton Wavelength Shift | 康普顿波长偏移
The Compton shift formula is Δλ = (h/(mₑ c)) (1 – ing angle. The constant h/(mₑ c) ≈ 2.43×10⁻¹² m is the Compton wavelength of the electron.
康普顿偏移公式为 Δλ = (h/(mₑ c)) (1 – cos θ)，其中 mₑ 是电子静止质量，θ 是光子散射角。常数 h/(mₑ c) ≈ 2.43×10⁻¹² m 是电子的康普顿波长。
Δλ = (h / (mₑ c)) (1 – cos θ)
ing) it is twice that value. The formula shows that the shift is independent of the incident wavelength – a direct signature of photon‑electron interaction modelled as particle collisions.
当 θ = 90° 时，偏移等于康普顿波长；当 θ = 180°（背散射）时是其两倍。该公式表明偏移与入射波长无关——这是按照粒子碰撞模型处理光子-电子相互作用的直接印记。
6. Matter Waves and de Broglie’s Hypothesis | 物质波与德布罗意假说
wavelength. The de Broglie wavelength is λ = h/p, where p is the momentum of the particle.
路易·德布罗意提出，如果光可以同时表现出波动性和粒子性，那么电子等实物粒子也应具有相应的波长。德布罗意波长为 λ = h/p，其中 p 是粒子的动量。
λ = h / p
For a non‑ h a potential difference V, the kinetic energy is eV = p²/(2m), giving a wavelength λ = h/√2 m e V. Typical accelerating voltages of ~100 V produce wavelengths of the order of 0.1 nm, comparable to atomic spacing.
对于通过电势差 V 加速的非相对论电子，动能 eV = p²/(2m)，由此得到波长 λ = h/√2 m e V。大约 100 V 的典型加速电压产生约 0.1 nm 量级的波长，与原子间距相当。
7. Electron Diffraction: Davisson–Germer Experiment | 电子衍射：戴维森–革末实验
The wave nature of electrons was experimentally confirmed by Davisson and Germer in 1927. They directed a beam of electrons at a nickel crystal and observed intensity peaks at specific angles, exactly like X‑ray diffraction patterns.
电子的波动性由戴维森和革末在 1927 年通过实验证实。他们将电子束射向镍晶体，并在特定角度观察到强度峰值，与 X 射线衍射图样完全类似。
lerating voltage, using Bragg’s law nλ = 2d sin θ. This proved that electrons exhibit wave‑like interference, a demonstration of matter waves.
测得的衍射角与根据加速电压预测的德布罗意波长相符，利用了布拉格定律 nλ = 2d sin θ。这证明电子表现出类波干涉现象，是物质波的有力演示。
Electron diffraction is now routinely used in electron microscopes and surface analysis, confirming that all microscopic particles possess wave properties.
如今电子衍射已常规用于电子显微镜和表面分析，证实了所有微观粒子都具有波动性。
8. Wave–Particle Duality and Complementarity | 波粒二象性与互补原理
Wave–particle duality is the concept that quantum entities exhibit both wave‑like and particle‑like properties, but never both simultaneously in the same measurement. Bohr’s complementarity principle states that which aspect we observe depends on the experimental arrangement.
波粒二象性是指量子实体既表现出波动性又表现出粒子性，但在同一次测量中从不同时显现两者。玻尔的互补原理指出，我们观察到哪一方面的性质取决于实验装置。
For example, a double‑slit experiment with electrons builds up an interference pattern (wave behaviour) when which‑path information is not observed, but shows particle‑like clicks at the screen. If the path is determined, the interference disappears.
例如，电子双缝实验中，若未观测路径信息，则会累积成干涉图样（波动行为），而在屏幕上显示为粒子般的点击。一旦路径被确定，干涉立即消失。
This duality is not a contradiction but a fundamental feature of quantum mechanics: the wave function encodes probabilities, and measurement yields definite particle‑like outcomes.
这种二象性并非矛盾，而是量子力学的基本特征：波函数编码概率，测量则产生确定的粒子性结果。
9. The Heisenberg Uncertainty Principle | 海森伯不确定性原理
known simultaneously. For position Δx and momentum Δp, it is expressed as Δx Δp ≥ h/(4π).
不确定性原理为某些可观测量对（如位置与动量）的同时精确度设定了基本限度。对于位置 Δx 和动量 Δp，它表述为 Δx Δp ≥ h/(4π)。
Δx Δp ≥ h / (4π)
olds for energy and time: ΔE Δt ≥ h/(4π), limiting the lifetime and energy spread of short‑lived states.
这并不是源于测量缺陷，而是量子描述的内禀属性。能量与时间之间存在类似关系：ΔE Δt ≥ h/(4π)，这限制了短寿命态的寿命与能量展宽。
The uncertainty principle explains why a confined electron cannot have zero kinetic energy and why orbitals in atoms have finite widths. It reinforces that we cannot assign a precise trajectory to a quantum particle.
不确定性原理解释了为什么被束缚的电子不可能动能为零以及原子轨道为什么具有有限宽度。它进一步表明我们不能给量子粒子指定精确的轨迹。
10. Quantum Interpretation and Probability | 量子诠释与概率波
In quantum theory, the state of a particle is described by a wave function ψ. The square of its magnitude, |ψ|², gives the probability density of finding the particle at a point. Max Born’s probability interpretation bridges the wave description and the measured particle‑like events.
在量子理论中，粒子的状态由波函数 ψ 描述。其幅值的平方 |ψ|² 表示在一点找到粒子的概率密度。玻恩的概率诠释在波动描述与测量的粒子性事件之间架起了桥梁。
When a measurement is made, the wave function ‘collapses’ to a specific outcome. This probabilistic nature is a key departure from deterministic classical physics and is central to understanding phenomena like electron diffraction and quantum tunnelling.
进行测量时，波函数会“坍缩”到一个特定结果。这种概率性质是与确定性的经典物理学的关键分野，也是理解电子衍射和量子隧穿等现象的核心。
Quantum physics has led to technologies such as semiconductors, lasers, and quantum computing, underscoring that its counter‑intuitive rules work with remarkable accuracy.
量子物理已催生出半导体、激光器和量子计算等技术，凸显了这些违反直觉的规则以惊人的精确性运作着。
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