题型概述：近代物理与综合短题Question types: modern physics and mixed short questions
Round 2 常把陌生模型写在题干中，要求从基础定律推导，再用数量级判断结果。本页选取一组综合短题和两道近代物理长题，每题均保留全部小问；顶部统计数值核对完成度，不代表官方得分。常见失分在于 eV 与 J 混用、遗漏几何因子、把估算前提当成已知，以及过早舍入。Round 2 often introduces an unfamiliar model and asks you to derive consequences from basic laws and assess their scale. This page contains one mixed set and two modern-physics questions, retaining all parts. The header counts numerical checks, not official marks. Typical losses come from mixing eV and J, missing geometric factors, hiding estimate assumptions and rounding too early.
1能量与光子数分开算Separate energy from photon count
同样功率下，单个光子能量越低，光子数越多。量子产率是光子数比，荧光能量效率还要乘波长比。At fixed power, lower photon energy means more photons. Quantum yield is a photon-number ratio; energy efficiency also includes the wavelength ratio.
波长用 m，能量用 J；1 eV = 1.60 × 10⁻¹⁹ J。Use metres and joules; 1 eV = 1.60 × 10⁻¹⁹ J.
2闭合条件筛选能级Closure selects energy levels
经典力学允许连续半径；波的单值条件筛选离散半径。双体系统要围绕共同质心处理，两种同位素都要使用约化质量。Classical mechanics allows continuous radii; wave closure selects discrete ones. Treat a two-body atom about its common centre of mass and use reduced mass for both isotopes.
不要把波形起伏当作电子真的沿波浪线飞行。Do not mistake wave ripples for a literal electron trajectory.
3比值、守恒和适用范围Ratios, conservation and validity
指数关系取比值可以消去未知比例常数。电容题先守恒节点电荷，再比较能量；守恒量的选择比代数更重要。Ratios eliminate unknown prefactors in exponential laws. For capacitors, conserve node charge before comparing energies; choosing the right conservation law precedes algebra.
以下小工具只演示能量与波长的对应，真题交互在各题页。The small tool below maps wavelength to energy; question-specific interactions appear in their tabs.
| 子题型Sub-type | 识别特征Recognition | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 综合短题Mixed short questions | 多个独立情境Independent scenarios | 各自建模，检查单位Model separately; check units | 2024 Q1 |
| 能级与同位素Levels and isotopes | 闭合波与质心Wave closure and centre of mass | 库仑力 + 量子化Coulomb force + quantisation | 2023 Q2 |
| 荧光测量Fluorescence measurement | 光子数、增益和指数衰减Photon counts, gain and decay | 逐级计数，再取比值Count stages, then take ratios | 2024 Q4 |
全页采用试卷舍入常数：h = 6.63×10⁻³⁴ J s，c = 3.00×10⁸ m/s，e = 1.60×10⁻¹⁹ C，ε₀ = 8.85×10⁻¹² F/m，me = 9.11×10⁻³¹ kg，mp = 1.67×10⁻²⁷ kg，G = 6.67×10⁻¹¹，kB = 1.38×10⁻²³ J/K。默认数值容差 2%。Paper constants throughout: h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m/s, e = 1.60×10⁻¹⁹ C, ε₀ = 8.85×10⁻¹² F/m, me = 9.11×10⁻³¹ kg, mp = 1.67×10⁻²⁷ kg, G = 6.67×10⁻¹¹, kB = 1.38×10⁻²³ J/K. Default numerical tolerance: 2%.