📚 Line Spectra and Energy Level Transitions | 线状光谱的起源与能级跃迁
The study of line spectra has been central to modern physics. When atoms are excited, they emit light only at specific wavelengths, producing a pattern of discrete lines. This observation challenged classical physics and led directly to the development of quantum theory.
线状光谱的研究是现代物理学的核心内容之一。当原子被激发时，它们只在特定波长处发光，形成一条条分立的谱线。这一观测事实挑战了经典物理学，并直接推动了量子理论的发展。
1. The Bohr Model and the Origin of Line Spectra | 玻尔模型与线状光谱的起源
In 1913, Niels Bohr proposed a semiclassical model of the hydrogen atom to explain why its spectrum is discrete, not continuous. He postulated that electrons orbit the nucleus only in certain allowed circular orbits without radiating energy.
1913年，尼尔斯·玻尔提出了一个半经典的氢原子模型，用以解释为什么氢原子光谱是分立的而不是连续的。他假设电子只能在某些允许的圆形轨道上绕核运动，并且在轨道上运动时不辐射能量。
Each allowed orbit corresponds to a specific total energy Eₙ, where n (the principal quantum number) can take positive integer values 1, 2, 3, … The energy of the hydrogen atom is quantised:
每一条允许的轨道都对应一个特定的总能量 Eₙ，其中主量子数 n 可取正整数值 1、2、3…… 氢原子的能量是量子化的：
Eₙ = −13.6 eV / n² (n = 1, 2, 3, …)
The negative sign indicates that the electron is bound to the nucleus. The lowest energy state, n = 1, is called the ground state; states with n > 1 are excited states.
负号表示电子被束缚在原子核周围。能量最低的状态 n = 1 称为基态；n > 1 的状态则称为激发态。
2. Energy Levels and Quantisation | 能级与量子化
The energy level diagram of the hydrogen atom resembles a ladder that becomes progressively closer together as n increases. The energy spacing between adjacent levels decreases with n, and the levels converge towards zero energy as n → ∞, where the electron is just free from the nucleus.
氢原子的能级图就像一把阶梯，随着 n 的增大，相邻能级之间的间距越来越小。随着 n → ∞，能级逐渐收敛于零能量，此时电子恰好脱离原子核的束缚。
To ionise a hydrogen atom from its ground state, an external photon or collision must supply at least 13.6 eV. This quantity is the ionisation energy of hydrogen. CIE exam questions frequently ask candidates to calculate this value from the energy level diagram.
要将氢原子从基态电离，外界必须通过光子或碰撞提供至少 13.6 eV 的能量。这个数值就是氢原子的电离能。CIE 考试中经常要求考生根据能级图计算出该值。
The quantisation of energy means that the atom cannot exist with an energy between two allowed levels. This is a key departure from classical physics, where any energy would be possible.
能量的量子化意味着原子不可能处于两个允许能级之间的一种能量状态。这是与经典物理学的重大区别，在经典理论中任何能量值都是可能的。
3. Emission and Absorption Transitions | 发射与吸收跃迁
When an electron jumps from a higher energy level Eᵢ to a lower level E_f, the atom emits a photon of energy equal to the difference between the two levels:
当电子从较高能级 Eᵢ 跃迁到较低能级 E_f 时，原子会发射一个光子，其能量等于两个能级之差：
hf = Eᵢ − E_f = ΔE ⇒ λ = hc / ΔE
where h is the Planck constant, f is the photon frequency, λ is the wavelength, and c is the speed of light. Because the energy levels are discrete, the emitted wavelengths are also discrete, producing a line spectrum rather than a continuous one.
其中 h 是普朗克常量，f 是光子频率，λ 是波长，c 是光速。因为能级是分立的，所以发射光的波长也是分立的，形成线状光谱而不是连续光谱。
Conversely, when a photon of exactly the right energy passes through a gas of atoms, it can be absorbed, causing an electron to jump from a lower to a higher level. This produces an absorption spectrum, which consists of dark lines superimposed on a continuous spectrum. The dark lines occur at exactly the same wavelengths as the emission lines for the same transition.
反过来，当一束能量恰好匹配的光子穿过原子气体时，它可能被吸收，使电子从较低能级跃迁到较高能级。这就产生了吸收光谱，它是在连续光谱上叠加的一系列暗线。这些暗线出现的波长与同一跃迁对应的发射线波长完全相同。
4. The Rydberg Formula and Spectral Series | 里德伯公式与谱线系
All the spectral lines of the hydrogen atom can be described by a single empirical equation known as the Rydberg formula:
氢原子的所有谱线都可以用一个统一的经验公式来描述，称为里德伯公式：
1/λ = R_H (1/n_f² − 1/n_i²)
where R_H is the Rydberg constant, equal to 1.097 × 10⁷ m⁻¹, n_f is the final level and n_i is the initial level, with n_i > n_f. A table of the main spectral series is given below.
其中 R_H 是里德伯常量，等于 1.097 × 10⁷ m⁻¹，n_f 是末能级，n_i 是初能级，且 n_i > n_f。下列表格列出了主要的谱线系。
| Series | n_f | Spectral Region | Key Wavelengths |
| Lyman | 1 | Ultraviolet | 121.6 nm (Hα), 102.6 nm, 97.3 nm |
| Balmer | 2 | Visible | 656.3 nm, 486.1 nm, 434.1 nm |
| Paschen | 3 | Infrared | 1875 nm, 1282 nm, 1094 nm |
| Brackett | 4 | Far infrared | 4052 nm, 2625 nm |
| Pfund | 5 | Far infrared | 7460 nm, 4654 nm |
The Balmer series, in which electrons fall to the n = 2 level, is the only series with wavelengths in the visible region. The first line of this series, known as Hα, corresponds to the 3 → 2 transition at 656.3 nm and is responsible for the characteristic red glow of hydrogen discharge tubes.
巴耳末系是电子跃迁到 n = 2 能级的谱线系，也是唯一一个波长落在可见光区的谱线系。该线系的第一条谱线称为 Hα，对应 3 → 2 跃迁，波长 656.3 nm，正是氢放电管发出特征红光的原因。
5. Quantum Numbers and Atomic States | 量子数与原子状态
Beyond the Bohr model, quantum mechanics describes the electron state by four quantum numbers. The principal quantum number n determines the main energy level. The orbital angular momentum quantum number l ranges from 0 to n − 1 and determines the shape of the orbital. The magnetic quantum number mₗ ranges from −l to +l and determines the orientation of the orbital in space. The spin quantum number mₛ is either +½ or −½, describing the intrinsic spin of the electron.
在玻尔模型之外，量子力学用四个量子数来描述电子的状态。主量子数 n 决定主能级。轨道角动量量子数 l 取值从 0 到 n − 1，决定轨道形状。磁量子数 mₗ 取值从 −l 到 +l，决定轨道在空间中的取向。自旋量子数 mₛ 为 +½ 或 −½，描述电子的内禀自旋。
For the hydrogen atom, the energy depends only on n, so states with the same n but different l and mₗ are degenerate, meaning they have the same energy. In multi-electron atoms, however, electron-electron repulsion lifts this degeneracy, and energy depends on both n and l.
对于氢原子，能量只取决于 n，因此相同 n 但不同 l 和 mₗ 的态是简并的，即能量相同。然而在多电子原子中，电子与电子之间的排斥作用会解除这种简并，能量同时取决于 n 和 l。
6. Selection Rules and Allowed Transitions | 选择定则与允许跃迁
Not every transition between two energy levels is equally likely. Quantum mechanics imposes selection rules that govern which transitions are allowed. For electric dipole transitions, the orbital angular momentum quantum number l must change by exactly ±1:
并非任何两个能级之间的跃迁都是同等可能发生的。量子力学给出了选择定则，它规定了哪些跃迁是允许的。对于电偶极跃迁，轨道角动量量子数 l 的改变量必须恰好为 ±1：
Δl = ±1 and Δmₗ = 0, ±1
This rule arises from the conservation of angular momentum between the atom and the emitted or absorbed photon, which carries one unit of angular momentum.
这一规则源于原子与所发射或吸收的光子之间角动量的守恒，因为光子携带一个单位的角动量。
Transitions that violate this rule are called forbidden transitions. They can still occur, but with much lower probability, leading to very weak spectral lines or long-lived metastable states. A good example is the 2s → 1s transition in hydrogen, which is forbidden by the Δl = ±1 rule and has a lifetime of about 0.12 seconds, far longer than allowed transitions which occur on the nanosecond timescale.
违反该定则的跃迁称为禁戒跃迁。这类跃迁仍然可能发生，但概率低得多，导致谱线非常微弱，或产生寿命很长的亚稳态。一个典型的例子是氢原子中的 2s → 1s 跃迁，它被 Δl = ±1 定则所禁止，其寿命约为 0.12 秒，远长于发生在纳秒时间尺度上的允许跃迁。
7. Line Intensity and Transition Probability | 谱线强度与跃迁概率
The intensity of a spectral line is not simply proportional to the energy difference of the transition. It depends on the number of atoms in the initial state and the transition probability per unit time, known as the Einstein A coefficient.
谱线的强度并不简单地与跃迁的能量差成正比。它取决于处于初态的原子数目以及单位时间内的跃迁概率，即爱因斯坦 A 系数。
At thermal equilibrium, the population of an excited state N_i relative to the ground state N₁ follows the Boltzmann distribution:
在热平衡条件下，激发态布居数 Nᵢ 相对于基态布居数 N₁ 服从玻尔兹曼分布：
Nᵢ / N₁ = (gᵢ / g₁) exp(−ΔE / kT)
where gᵢ and g₁ are the degeneracies (numbers of states with the same energy), ΔE is the energy separation, k is the Boltzmann constant, and T is the absolute temperature. Higher temperature populates the excited states more heavily, thus increasing the intensity of emission lines originating from them.
其中 gᵢ 和 g₁ 是简并度（具有相同能量的状态数），ΔE 是能量间隔，k 是玻尔兹曼常量，T 是绝对温度。温度越高，激发态的布居数越多，因而从这些激发态出发的发射谱线强度也越大。
In discharge tubes and flames, collisions between atoms can also populate excited states. This is why a sodium flame test produces a strong yellow emission at 589 nm, corresponding to the 3p → 3s transition.
在放电管和火焰中，原子之间的碰撞也可以布居激发态。这就是钠的焰色反应产生 589 nm 强黄色发射线的原因，它对应 3p → 3s 跃迁。
8. Fine Structure and Spin-Orbit Coupling | 精细结构与自旋轨道耦合
When observed with high-resolution spectrometers, many spectral lines are found to consist of several closely spaced lines. This phenomenon is called fine structure. The most important cause in hydrogen and alkali atoms is spin-orbit coupling: the electron’s spin interacts with the magnetic field produced by its own orbital motion around the nucleus.
用高分辨率光谱仪观察时会发现，许多谱线其实由若干相距很近的谱线组成。这一现象称为精细结构。在氢原子和碱金属原子中，最重要的原因是自旋-轨道耦合：电子的自旋与其绕核轨道运动所产生的磁场发生相互作用。
The spin-orbit interaction splits an energy level with given n and l into two levels with slightly different energies, corresponding to the two possible orientations of the total angular momentum. For hydrogen, this splitting is of the order of 10⁻⁵ eV, far smaller than the separation between different principal levels. The 2p level, for instance, splits into 2p₁/₂ and 2p₃/₂, giving rise to the two closely spaced lines of the hydrogen Hα transition.
自旋-轨道相互作用将一个给定的 n、l 能级分裂成两个能量略有不同的能级，对应于总角动量的两种可能取向。对于氢原子，这种分裂的能级间距约为 10⁻⁵ eV，远小于不同主能级之间的间隔。例如 2p 能级分裂为 2p₁/₂ 和 2p₃/₂，使氢的 Hα 跃迁产生两条相距很近的谱线。
Relativistic corrections to the electron’s kinetic energy contribute an additional small shift to the levels. The combination of spin-orbit coupling and relativistic effects is fully described by the Dirac equation, whose predictions for hydrogen are confirmed experimentally to extraordinary precision.
电子动能的相对论修正会对能级产生额外的微小移动。自旋-轨道耦合与相对论效应的总和由狄拉克方程完整描述，该方程对氢原子的预言已被实验以极高的精度证实。
9. Multi-Electron Atoms and the X-ray Spectrum | 多电子原子与X射线谱
In atoms with more than one electron, the energy levels depend not only on n but also on l. The shell structure leads to characteristic spectra that are unique to each element. Inner-shell transitions produce X-ray lines, which are particularly important in applications such as medical imaging and materials analysis.
在含多个电子的原子中，能级不仅取决于 n，还取决于 l。壳层结构使得每种元素都有其独特的特征光谱。内壳层跃迁产生 X 射线谱线，在医学成像和材料分析等应用中特别重要。
If a high-energy electron or photon knocks out an electron from the innermost K-shell (n = 1), an electron from the L-shell (n = 2) may fall down to fill the vacancy, emitting an X-ray photon. These transitions are called Kα lines. Similarly, M-shell to K-shell transitions produce Kβ lines, and transitions from M-shell to L-shell produce Lα lines. Because these energies depend strongly on the nuclear charge, each element has a distinctive X-ray fingerprint.
如果高能电子或光子将最内层 K 壳层（n = 1）的电子击出，L 壳层（n = 2）的电子就可能落入空位，同时发射一个 X 射线光子。这类跃迁称为 Kα 线。类似地，M 壳层到 K 壳层的跃迁产生 Kβ 线，M 壳层到 L 壳层的跃迁产生 Lα 线。由于这些能量强烈依赖于核电荷，每一种元素都有独特的 X 射线指纹。
Moseley’s law relates the frequency of the Kα X-ray line to the atomic number Z:
莫塞莱定律将 Kα X 射线谱线的频率与原子序数 Z 联系起来：
f^(½) = a(Z − b)
where a and b are constants. This relationship was historically used to order the elements in the periodic table and to discover new elements.
其中 a 和 b 是常数。这一关系在历史上被用来确定元素周期表的排列顺序，并发现了新的元素。
10. Applications in Astronomy and Astrophysics | 在天文学与天体物理学中的应用
Spectroscopy is the most powerful tool available to astrophysicists for studying the composition, temperature, and motion of distant objects. The absorption lines in the solar spectrum, known as Fraunhofer lines, reveal the presence of hydrogen, helium, calcium, iron, and many other elements in the Sun’s atmosphere.
光谱分析是天体物理学家研究遥远天体的成分、温度和运动的最强大工具。太阳光谱中的吸收线，即夫琅禾费线，揭示了太阳大气中存在氢、氦、钙、铁以及许多其他元素。
Because each element produces a unique pattern of lines, astronomers can identify elements in stars that are thousands of light-years away. The ratio of line intensities gives an estimate of temperature and density. The Doppler shift of spectral lines reveals the radial velocity of a star or galaxy; the systematic redshift of galaxies was essential evidence for the expanding universe.
因为每一种元素都产生独特的谱线图样，天文学家可以识别数千光年之外的恒星中所含有的元素。谱线强度之比可用于估算温度和密度。谱线的多普勒频移揭示恒星或星系在视线方向上的运动速度；星系谱线的系统性红移是宇宙膨胀的重要证据。
Gas nebulae, such as the Orion Nebula, exhibit bright emission-line spectra produced by atoms excited by ultraviolet radiation from nearby hot stars. Many of these lines, including the famous green line of doubly ionised oxygen at 500.7 nm, are forbidden transitions that would be extremely weak in laboratory conditions but are prominent in the rarefied gas of space.
气体星云（如猎户座星云）显示出明亮的发射线光谱，这些谱线是由附近炽热恒星发出的紫外辐射激发原子所产生的。其中许多谱线，包括著名的双电离氧在 500.7 nm 处的绿色谱线，属于禁戒跃迁，在实验室条件下极弱，但在太空中极其稀薄的气体中却非常显著。
11. Worked Example: Calculating Wavelength | 例题：计算谱线波长
Let us calculate the wavelength of the photon emitted when a hydrogen atom makes the transition from n = 3 to n = 2. The energy levels are:
让我们计算氢原子从 n = 3 跃迁到 n = 2 时发射光子的波长。能级值为：
E₂ = −13.6/4 eV = −3.40 eV
E₃ = −13.6/9 eV = −1.51 eV
The energy difference is:
能量差为：
ΔE = E₃ − E₂ = −1.51 − (−3.40) = 1.89 eV
Converting to joules: ΔE = 1.89 eV × 1.60 × 10⁻¹⁹ J eV⁻¹ = 3.02 × 10⁻¹⁹ J. Then:
换算为焦耳：ΔE = 1.89 eV × 1.60 × 10⁻¹⁹ J eV⁻¹ = 3.02 × 10⁻¹⁹ J。于是：
λ = hc / ΔE = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / (3.02 × 10⁻¹⁹) m = 6.59 × 10⁻⁷ m = 659 nm
This corresponds to the red Hα line of the Balmer series. Note that no energy level between these two values is permitted, so intermediate wavelengths cannot be emitted by this atom.
这对应巴耳末系中的红色 Hα 线。注意，这两个能级之间的任何中间能量值都是不允许的，因此该原子不可能发射介于两者之间的波长。
12. Key Points for CIE Examination Success | CIE考试要点提示
- Line spectra arise from quantised energy levels: electrons emit or absorb photons only when moving between allowed levels.
- 线状光谱源于量子化的能级：电子只在允许能级之间跃迁时发射或吸收光子。
- The energy of the photon equals the magnitude of the energy difference: E = hf = ΔE.
- 光子的能量等于能级差的绝对值：E = hf = ΔE。
- Emission spectra show bright lines; absorption spectra show dark lines at the same wavelengths.
- 发射光谱显示亮线；吸收光谱在相同波长处显示暗线。
- Energy levels must be drawn with correct spacing: converging as n increases.
- 画能级图时必须注意间距正确：随着 n 增大能级间距逐渐减小。
- Know the ground state energy of hydrogen (−13.6 eV) and be able to calculate ionisation energy.
- 记住氢原子基态能量（−13.6 eV），并能计算电离能。
- Use the Rydberg formula to find wavelengths; use 1 eV = 1.60 × 10⁻¹⁹ J in energy conversions.
- 使用里德伯公式计算波长；能量单位换算时注意 1 eV = 1.60 × 10⁻¹⁹ J。
Finally, always check the units in your calculation. Wavelength should be expressed in metres, nanometres, or ångströms, with 1 nm = 10⁻⁹ m. A common examiner’s trap is to give energy in electronvolts but the Planck constant in joule-seconds. Convert carefully.
最后，计算中一定要注意单位。波长应当以米、纳米或埃为单位，且 1 nm = 10⁻⁹ m。考官常见的陷阱是能量给电子伏特量级，而普朗克常量用焦耳秒量级。换算时要格外小心。
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