📚 Energy Levels and Spectra | 能级与光谱
In A-Level Physics, understanding energy levels and spectra is essential for explaining how atoms interact with light. Electrons in atoms can only occupy discrete energy states, and transitions between these levels produce or absorb photons of specific wavelengths. The resulting line spectra provide a unique fingerprint for each element and underpin techniques from flame tests to astrophysical analysis. This revision guide covers all the key concepts for WJEC, from photon energy calculations to the hydrogen spectrum and fluorescence.
在 A-Level 物理中，理解能级与光谱对于解释原子如何与光相互作用至关重要。原子中的电子只能占据分立的能态，电子在这些能级之间的跃迁会发射或吸收特定波长的光子。所产生的线状光谱为每种元素提供了独一无二的“指纹”，并支撑着从焰色反应到天体物理分析的多种技术。本复习指南涵盖 WJEC 考试的所有核心概念，包括光子能量计算、氢光谱以及荧光等。
1. Discrete Energy Levels | 分立的能级
Atoms possess discrete energy levels for their electrons. According to quantum theory, an electron can only exist in certain allowed orbits or states, each with a definite energy. The lowest energy state is called the ground state. Higher energy states are referred to as excited states. An electron cannot have an energy value between these levels; it must ‘jump’ from one level to another.
原子中的电子具有分立的能级。根据量子理论，电子只能存在于某些特定的允许轨道或状态中，每个状态都有确定的能量。能量最低的状态称为基态，能量较高的状态则称为激发态。电子不能拥有介于这些能级之间的能量值，它必须从一个能级“跃迁”到另一个能级。
In a simple atomic model, the energy levels are often depicted as horizontal lines on an energy-level diagram. The spacing between lines decreases as the energy increases, reflecting that higher levels are closer together. The energy is typically measured in electronvolts (eV), with the ground state usually assigned a negative value, where zero energy corresponds to a free electron at rest infinitely far from the nucleus.
在简单的原子模型中，能级常用能级图上的水平线来表示。随着能量增加，线之间的间距变小，反映出较高能级靠得更近。能量通常以电子伏特 (eV) 为单位，基态一般被赋予一个负值，而零能量对应于一个静止在无穷远处的自由电子。
2. Photon Energy and Electronic Transitions | 光子能量与电子跃迁
When an electron moves from a higher energy level E₂ to a lower energy level E₁, the energy difference ΔE is released as a single photon. The photon energy is given by Planck’s equation:
当电子从较高能级 E₂ 跃迁到较低能级 E₁ 时，能量差 ΔE 以单个光子的形式释放。光子能量由普朗克方程给出：
ΔE = E₂ – E₁ = hf
where h is the Planck constant (6.63 × 10⁻³⁴ J s) and f is the frequency of the emitted photon. Since c = fλ, the relationship can also be written as
其中 h 为普朗克常数 (6.63 × 10⁻³⁴ J s)，f 为所发射光子的频率。因为 c = fλ，该关系式也可写成
ΔE = hc / λ
where c is the speed of light (3.00 × 10⁸ m s⁻¹) and λ is the wavelength. To make calculations easier, it is often useful to remember that a photon energy of 1 eV corresponds to a wavelength of about 1240 nm, using the conversion 1 eV = 1.60 × 10⁻¹⁹ J.
其中 c 为光速 (3.00 × 10⁸ m s⁻¹)，λ 为波长。为了方便计算，记住 1 eV 的光子能量对应约 1240 nm 的波长是很有用的，利用换算关系 1 eV = 1.60 × 10⁻¹⁹ J。
Conversely, an electron can absorb a photon and jump to a higher energy level only if the photon energy exactly matches the gap between two permitted levels. If the incoming photon energy is insufficient or excessive, no absorption occurs. This explains the dark lines seen in absorption spectra.
相反，只有当入射光子的能量恰好等于两个允许能级之间的能量间隙时，电子才能吸收光子并跃迁到较高能级。如果光子能量不足或过量，就不会发生吸收。这就解释了吸收光谱中观测到的暗线。
3. Emission Spectra | 发射光谱
An emission spectrum is produced when excited atoms return to lower energy states. The atoms can be excited by heating, electrical discharge, or bombardment with electrons. As electrons de-excite, they emit photons of specific energies, resulting in a pattern of bright lines on a dark background. Each line corresponds to a particular electronic transition.
当受激原子回到较低能态时，就会产生发射光谱。原子可以通过加热、放电或电子轰击等方式被激发。电子在退激时会发射特定能量的光子，从而在暗背景上形成一系列亮线。每一条谱线都对应着一个特定的电子跃迁。
Because the set of energy levels is unique to each element, the emission spectrum serves as an atomic fingerprint. In a flame test, for example, sodium gives a strong yellow doublet near 589 nm, while lithium produces a red line. This principle is widely used in chemical analysis and in studying the composition of stars.
由于每种元素都有一套独特的能级，发射光谱便成为原子的“指纹”。例如，在焰色反应中，钠会在 589 nm 附近产生很强的黄色双线，而锂则产生一条红线。这一原理被广泛应用于化学分析以及恒星成分的研究。
4. Absorption Spectra | 吸收光谱
An absorption spectrum is observed when white light passes through a cool gas. Photons whose energies exactly match the gaps between the gas atoms’ energy levels are absorbed, lifting electrons to higher states. The transmitted light then shows a continuous spectrum crossed by dark absorption lines at those specific wavelengths.
当白光通过一种冷气体时，就会观察到吸收光谱。能量恰好等于气体原子能级间隙的光子会被吸收，使电子跃迁到较高能态。于是，透射光就会呈现出被若干暗吸收线切割的连续光谱，而这些暗线正位于那些特定波长处。
The dark lines appear at exactly the same wavelengths as the bright lines in the emission spectrum of the same element. This is because the energy gaps are identical. The classic example is the Fraunhofer lines in the solar spectrum, which reveal the elements present in the Sun’s cooler outer atmosphere.
这些暗线所在的波长与同一元素发射光谱中的亮线波长完全一致，因为能级间隙是相同的。最经典的例子是太阳光谱中的夫琅禾费线，它们揭示了太阳较冷的外层大气中所含的元素。
5. The Hydrogen Spectrum and the Bohr Model | 氢光谱与玻尔模型
The hydrogen atom has the simplest spectrum, and its regular pattern of lines provided key evidence for the Bohr model. In the visible region, the Balmer series consists of transitions from higher levels (n ≥ 3) down to n = 2. The wavelengths are given by the Balmer formula:
氢原子具有最简单的光谱，其规则的谱线排列为玻尔模型提供了关键证据。在可见光区域，巴尔末系由从较高能级 (n ≥ 3) 跃迁到 n = 2 的跃迁组成。其波长由巴尔末公式给出：
1/λ = R (1/2² – 1/n²)
where R is the Rydberg constant (1.097 × 10⁷ m⁻¹) and n = 3, 4, 5, … for the visible lines (Hα at 656 nm, Hβ at 486 nm, Hγ at 434 nm, etc.). Other series exist: the Lyman series (n = 1) in the ultraviolet, and the Paschen series (n = 3) in the infrared.
其中 R 为里德伯常数 (1.097 × 10⁷ m⁻¹)，对于可见谱线 n = 3, 4, 5, …（Hα 位于 656 nm，Hβ 位于 486 nm，Hγ 位于 434 nm，等等）。还有其他线系：紫外区的莱曼系 (n = 1) 和红外区的帕邢系 (n = 3)。
Bohr’s model postulates that electrons orbit the nucleus in circular paths without radiating energy, and that angular momentum is quantised: mvr = nh/2π. Although this model is now superseded by quantum mechanics, it successfully predicts the energy levels of hydrogen: Eₙ = –13.6 eV / n². The negative sign indicates that the electron is bound to the nucleus; 13.6 eV is the ionisation energy for hydrogen from the ground state.
玻尔模型假设电子在圆形轨道上绕核运动而不辐射能量，并且角动量是量子化的：mvr = nh/2π。尽管这一模型已被量子力学所取代，但它成功地预测了氢原子的能级：Eₙ = –13.6 eV / n²。负号表示电子被束缚在原子核附近；13.6 eV 就是氢原子从基态电离所需的能量。
6. Energy Level Diagrams | 能级图
An energy level diagram is a vertical scale of allowed energies, usually with the ground state at the bottom and the ionisation level at E = 0. Arrows pointing downwards represent emission transitions, and their lengths are proportional to the photon energies. Upward arrows indicate absorption or excitation. In WJEC exams, you may be asked to draw such diagrams and label the transitions corresponding to given spectral lines.
能级图是一个垂直排列的允许能量标尺，通常基态位于最底部，而电离能级位于 E = 0 处。向下的箭头代表发射跃迁，其长度正比于光子能量。向上的箭头则表示吸收或激发。在 WJEC 考试中，你可能会被要求绘制这种图，并标出与给定谱线对应的跃迁。
For a hydrogen atom, the diagram shows levels converging towards 0 eV. The spacing between levels decreases as n increases. The transitions that produce the Balmer lines all end at n = 2. You should be able to calculate the photon energy, frequency, and wavelength for any arrow on the diagram using ΔE = hf.
对于氢原子，能级图显示出各能级向 0 eV 收敛。随着 n 增大，能级间距逐渐减小。产生巴尔末线的跃迁全部终止于 n = 2。你应当能够利用 ΔE = hf 计算图上任何箭头所对应的光子能量、频率和波长。
The ionisation energy is the minimum energy required to remove an electron completely from the atom, i.e. to take it from the ground state to E = 0. In an energy level diagram, this is the vertical distance from the ground level to the 0 eV line. For hydrogen, this is 13.6 eV; for other elements, it can be obtained from the convergence limit of a spectral series.
电离能是将电子完全移离原子所需的最小能量，即将其从基态提升到 E = 0 处。在能级图上，这就是从基态能级到 0 eV 线的垂直距离。对氢而言，这个值是 13.6 eV；对其他元素，可以从某一谱线系的收敛极限求得。
7. Excitation and Ionisation by Electron Impact | 电子碰撞激发与电离
Atoms can be excited not only by photons but also by collisions with free electrons, as in a gas discharge tube. A fast-moving electron can give part of its kinetic energy to an atomic electron, raising it to a higher level. The minimum energy needed for excitation is the energy gap between the ground state and the first excited state.
原子不仅可以被光子激发，还可以通过与自由电子碰撞而被激发，比如在气体放电管中。一个快速运动的电子可以将其一部分动能传递给原子中的电子，使之跃迁到较高能级。激发所需的最小能量就是基态与第一激发态之间的能量间隙。
If the incident electron’s kinetic energy is equal to or greater than the ionisation energy, it can knock an electron out of the atom entirely, creating a positive ion. This process is called ionisation by electron impact. In the lab, the ionisation energy can be measured using a gas-filled tube and observing the current flow as the accelerating voltage is increased; a sharp rise in current indicates the onset of ionisation.
如果入射电子的动能等于或大于电离能，它就可以把一个电子完全撞出原子，形成一个正离子，这一过程称为电子碰撞电离。在实验室中，可以利用充气管，在增加加速电压的同时观察电流的变化来测量电离能；电流的急剧上升标志着电离的开始。
8. Fluorescence and Phosphorescence | 荧光与磷光
Fluorescence occurs when a material absorbs high-energy (often ultraviolet) photons and then re-emits lower-energy (visible) photons. The incoming photon excites an electron to a high energy level; the electron may first drop through several smaller steps, losing energy as heat, before emitting the final visible photon. This is why fluorescent materials can glow under UV light.
当材料吸收高能（通常是紫外）光子，然后再发射出低能（可见）光子时，就会发生荧光。入射光子将电子激发到高能级；电子可能先通过若干个较小的步阶下降，以热能形式损失能量，然后才发射出最终的可见光子。这就是荧光材料在紫外光下能够发光的原因。
A related phenomenon is phosphorescence, where the re-emission is delayed. In phosphorescent materials, the excited electrons become trapped in metastable states and take seconds or longer to return to the ground state. This is used in glow-in-the-dark paints and safety signs. The energy level diagram for fluorescence includes intermediate levels that allow non-radiative transitions, converting some energy to heat.
与之相关的现象是磷光，其再发射是延迟的。在磷光材料中，受激电子被束缚在亚稳态，需要数秒或更长时间才能返回基态。这一特性被用于夜光涂料和安全标志中。荧光的能级图包含一些允许非辐射跃迁的中间能级，从而将一部分能量转化为热能。
9. Continuous and Line Spectra | 连续光谱与线状光谱
Not all light sources produce line spectra. A heated solid, liquid, or dense gas emits a continuous spectrum containing all wavelengths, because the atoms are closely packed and their energy levels are broadened into bands. A classic example is the filament of an incandescent light bulb, which gives a smooth rainbow of colours.
并非所有光源都产生线状光谱。受热的固体、液体或稠密气体发射出包含所有波长的连续光谱，因为原子排列紧密，其能级被展宽为能带。一个典型的例子是白炽灯泡的灯丝，它会发出平滑的彩虹色光。
A low-pressure gas, on the other hand, produces a line emission spectrum if excited electrically. The distinction between continuous, emission, and absorption spectra is a common WJEC exam question. You must be able to identify the type of spectrum from its appearance: a full rainbow (continuous), a set of bright lines on a dark background (emission), or a rainbow crossed by dark lines (absorption).
另一方面，低压气体在被电激发时则产生线状发射光谱。连续光谱、发射光谱和吸收光谱之间的区别是 WJEC 考试中的常见考点。你必须能够根据谱线外观识别光谱类型：完整的彩虹（连续光谱）、暗背景上的一系列亮线（发射光谱），或是被暗线横穿的彩虹（吸收光谱）。
10. Applications of Spectra in Astrophysics | 光谱在天体物理中的应用
The analysis of spectra from stars and galaxies provides a wealth of information. The absorption lines in a star’s spectrum reveal its chemical composition. The Doppler shift of these lines tells us whether a star is moving towards or away from Earth and at what speed. This is how the expansion of the universe was first discovered.
对恒星和星系光谱的分析提供了丰富的信息。恒星光谱中的吸收线揭示了它的化学成分。这些谱线的多普勒频移告诉我们恒星是在靠近还是远离地球，以及其运动速度。宇宙的膨胀正是通过这种方法被首次发现的。
For WJEC, you may be asked to explain how the redshift of a known spectral line, such as a hydrogen line, can be used to calculate the recessional velocity using Δλ/λ ≈ v/c (for non-relativistic speeds). The broadening of spectral lines can also indicate the rotation of a star or the temperature and pressure in its atmosphere.
在 WJEC 考试中，你可能会被要求解释如何利用已知谱线（如氢线）的红移，通过公式 Δλ/λ ≈ v/c（适用于非相对论速度）来计算退行速度。谱线的展宽还能指示恒星的自转或其大气中的温度和压力。
11. Line Spectra as Evidence for Quantisation | 线状光谱作为量子化的证据
The very existence of discrete line spectra is direct evidence for the quantisation of energy in atoms. If atomic electrons could possess any energy, the emitted photons would have a continuous range of energies, producing a continuous spectrum. The fact that only certain wavelengths appear confirms that energy changes are restricted to fixed amounts.
分立线状光谱的存在本身就是原子能量量子化的直接证据。如果原子中的电子可以拥有任意能量，那么所发射的光子就会具有连续的能量范围，从而产生连续光谱。只有特定波长出现这一事实证实了能量变化被限制在固定的数值范围内。
In the early 20th century, the failure of classical physics to explain the hydrogen spectrum led to the development of quantum theory. The Bohr model was a stepping stone, and the full quantum mechanical model using wavefunctions and probability densities now provides a complete description. Nevertheless, the simple energy level picture remains a powerful tool for problem-solving.
在 20 世纪初，经典物理学无法解释氢光谱，这促成了量子理论的发展。玻尔模型是一块垫脚石，而如今，采用波函数和概率密度的量子力学模型提供了完整的描述。然而，简单的能级图像依然是解决问题的有力工具。
12. Summary and Key Equations | 总结与关键公式
The following table summarises the essential formulas and constants you must know for the WJEC examination:
下表总结了你在 WJEC 考试中必须掌握的关键公式和常数：
| Photon energy | E = hf = hc/λ |
| Planck constant | h = 6.63 × 10⁻³⁴ J s |
| Speed of light | c = 3.00 × 10⁸ m s⁻¹ |
| Electronvolt | 1 eV = 1.60 × 10⁻¹⁹ J |
| Rydberg formula | 1/λ = R (1/n₁² – 1/n₂²), R = 1.097 × 10⁷ m⁻¹ |
| Hydrogen energy levels | Eₙ = –13.6 eV / n² |
| Ionisation energy (H) | 13.6 eV |
Always check your units: convert all energies to joules when using h in J s, or work consistently in eV when using the 1240 nm eV rule. Practice drawing and interpreting energy level diagrams, and be ready to identify spectral series and perform Doppler shift estimates for stars.
务必检查单位：当使用以 J s 为单位的 h 时，要将所有能量转换为焦耳；或者当使用 1240 nm·eV 经验法则时，要始终使用 eV。多练习绘制和解读能级图，并准备好识别谱线系，以及对恒星进行多普勒频移估算。
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