---
title: "黑体辐射：证据是一整条曲线"
description: "长波段的测量，怎样迫使热辐射定律改变？"
canonical_url: "https://sandboxphysics.com/zh/stories/quantum-evidence/blackbody-spectrum-planck-wien/"
language: "zh-CN"
page_type: "story"
alternate_languages:
  en: "https://sandboxphysics.com/stories/quantum-evidence/blackbody-spectrum-planck-wien/"
last_modified: "2026-10-15"
---

[量子物理：读懂证据](https://sandboxphysics.com/zh/stories/quantum-evidence/)

诺奖 A · 证据与操控

# 黑体辐射：证据是一整条曲线

Sandbox Physics · 2026-10-15

长波段的测量，怎样迫使热辐射定律改变？

## 实验测的是分布

等温腔体上开一个小孔，就近似得到一个理想辐射源：平衡辐射只由温度决定，与某块发光表面的细节无关。分光测量问的是每个波段到达多少辐射，并不逐次记录单次发射的能量。

装置仍然重要。要把探测器的电信号换算成辐亮度，得考虑孔径几何、光谱带宽、温度计标定、杂散辐射和探测器响应。这些量控制好以后，数据才能和普适定律比较；只凭热物体看上去的颜色不行。

证据阅读：M. Planck (1901), On the Law of Distribution of Energy in the Normal Spectrum — English transcription NIST SP 250-43, Radiance Temperature Calibrations — spectral radiance and instrument response

## Wien 的两条定律说的是两回事

位移定律讲光谱怎样随温度移动；Wien 的分布律说得更多，要给出整条曲线上的强度。分布律在某个波段失效，不等于所有冠以 Wien 之名的结果都跟着失效。

推动改变的是测量。Planck 在 1901 年论文开头明确提到，Lummer 与 Pringsheim，尤其是 Rubens 与 Kurlbaum 的长波测量，说明原有分布律并不普遍成立。他后来的诺奖演讲也回忆了红外测量带来的压力。当时的问题，是候选曲线和实测辐射对不上，不只是后来被称为“灾难”的那个预言。

证据阅读：M. Planck (1901), On the Law of Distribution of Energy in the Normal Spectrum — English transcription M. Planck (1920), The Genesis and Present State of Development of the Quantum Theory — Nobel lecture

## 比较峰值前，先说清是哪种光谱

用现代记号，下式给出理想平衡黑体在真空中按单位频率计的光谱辐亮度。频率用赫兹，温度用开尔文；辐亮度的单位是瓦每平方米每球面度每赫兹。式中常数分别是普朗克常数、真空光速和玻尔兹曼常数。

按单位波长计的光谱，是按波长区间来数辐射的。只换横轴的刻度，纵轴的密度并不会跟着换。所以，按波长画的曲线，峰值一般不在频率曲线峰值所对应的波长上。说明了用哪种约定，峰值才有意义。

$$
B_\nu(T)=\frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/(k_{\mathrm B}T)}-1}
$$

证据阅读：NIST SP 250-43, Radiance Temperature Calibrations — spectral radiance and instrument response

## 极限情形显出几条定律的差别

频率远低于热能除以普朗克常数时，把指数展开，就得到 Rayleigh–Jeans 极限；高频时，略去分母里减去的那个一，就得到 Wien 近似。两者都是同一个表达式的极限，不是凭眼睛挑出的两条无关曲线。

经典均分谱在高频端一路上升，延伸到所有频率，总量就会发散；Planck 的表达式把这段尾巴压了下去。但 Wien 近似在短波段本来就成功，所以检验必须进入几条候选定律真正分开的波段。在它们重合的极限里拟合得再好，也分不出谁对。

证据阅读：M. Planck (1901), On the Law of Distribution of Energy in the Normal Spectrum — English transcription NIST SP 250-43, Radiance Temperature Calibrations — spectral radiance and instrument response

## 能量元只是光子故事的开头

Planck 的计数论证，是把有限的能量元分给各个谐振子。这一步还不包含后来散射实验里自由光量子的图像。Planck 在回顾演讲中自己也说，光量子怎样传播，在当时还是没解决的概念问题。

Compton 那篇讲的是另一种装置中的能量和动量转移。平衡光谱的成功和散射波长的变化，检验的是理论的不同方面。本文梳理历史证据；黑体 Lab 是另行开发的模拟器，本文既不新增它，也不为它作验收。

证据阅读：M. Planck (1901), On the Law of Distribution of Energy in the Normal Spectrum — English transcription M. Planck (1920), The Genesis and Present State of Development of the Quantum Theory — Nobel lecture

## 接着操作模型

下面这些教学 Lab 产生的是模拟数据，和上文讨论的历史观测不是一回事。

- [黑体辐射观测台 · 模拟光谱读数](https://sandboxphysics.com/zh/labs/blackbody-radiation/)

## 来源与阅读说明

Planck 1901 年的论文读的是英文转录本，另参照他的回顾演讲；现代辐射测量约定取自 NIST。公式用现行记号。本文没有复制或拟合历史辐射计曲线。

1. [M. Planck (1901), On the Law of Distribution of Energy in the Normal Spectrum — English transcription](https://www.informationphilosopher.com/solutions/scientists/planck/Planck_1901a.pdf)
2. [M. Planck (1920), The Genesis and Present State of Development of the Quantum Theory — Nobel lecture](https://www.nobelprize.org/prizes/physics/1918/planck/lecture/)
3. [NIST SP 250-43, Radiance Temperature Calibrations — spectral radiance and instrument response](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication250-43.pdf)

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