查看“︁普朗克黑体辐射定律”︁的源代码

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[[File:Wiens law.svg|thumb|300px|普朗克定律描述的黑体辐射在不同温度下的频谱]]

在[[物理学]]中，'''普朗克黑体辐射定律'''（{{lang-en|Planck's law}}，{{lang|en|Blackbody radiation law}}，也简称作'''普朗克定律'''或'''[[黑体辐射]]定律'''）是指在任意[[温度]]<math>T\,</math>下，从一个[[黑体 (热力学)|黑体]]中发射出的[[电磁辐射]]的[[辐射率]]与[[頻率 (物理學)|频率]]彼此之間的关系。在这裏，辐射率是频率<math>\nu</math>的函数<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>：

:<math>I_{\nu}(\nu,T) =\frac{2  h\nu^{3}}{c^2}\frac{1}{e^{\frac{h\nu}{kT}} - 1}</math>，

如果写成[[波长]]的函数，則辐射率为<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>
:<math>I_{\lambda}(\lambda,T) =\frac{2 hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}} - 1}</math>；

其中，<math>I_{\nu}</math>或<math>I_{\lambda}</math>是[[輻射率]]，<math>\nu \,</math>是[[頻率 (物理學)|频率]]，<math>\lambda \,</math>是[[波长]]，<math>T \,</math>是黑体的[[温度]]，<math>h \,</math>是[[普朗克常数]]，<math>c \, </math>是[[光速]]，<math>k \, </math>是[[玻尔兹曼常数]]。

注意这两个函数具有不同的单位：第一个函数是描述单位频率间隔内的辐射率，而第二个则是单位波长间隔内的辐射率。因而<math>I_{\nu}(\nu,T)</math>和<math>I_{\lambda}(\lambda,T)</math>并不等价。它们之间存在有如下关系：

:<math>I_{\nu}(\nu,T)\,d\nu= - I_{\lambda}(\lambda,T)\,d\lambda</math>。

通过单位频率间隔和单位波长间隔之间的关系，这两个函数可以相互转换：

:<math>d\nu=d\left(\frac{c}{\lambda}\right)=c\,d\left(\frac{1}{\lambda}\right)= - \frac{c}{\lambda ^2}\,d\lambda</math>。

在低頻率極限，普朗克定律趨於[[瑞利-金斯定律]]，而在高頻率極限，普朗克定律趨於[[維恩近似]]。

[[馬克斯·普朗克]]於1900年發展出普朗克定律，並從實驗結果計算出所涉及的常數。後來，他又展示，當表達為能量分布時，該分布是電磁輻射在[[熱力學平衡]]下的唯一穩定分布。<ref name="Planck 1914 42">{{harvnb|Planck|1914|p=42}}</ref>當表達為能量分布時，該分布是熱力學平衡分布家族的成員之一，其它成員為[[玻色–爱因斯坦统计|玻色–愛因斯坦分布]]、[[费米-狄拉克统计|費米–狄拉克分布]]、[[麦克斯韦-玻尔兹曼分布]]等等。

== 概述 ==
[[File:Max planck.jpg|thumb|150px|[[德国]]物理学家[[马克斯·普朗克]]]]

电磁波[[波长]]和[[頻率 (物理學)|频率]]的关系为<ref>{{harv|Rybicki|Lightman|1979|p=1}}</ref>

:<math>\lambda = { c \over \nu }.</math>

普朗克定律有时写做[[能量密度]]频谱的形式<ref name="BrehmMullin">Brehm, J.J. and Mullin, W.J., "Introduction to the Structure of Matter: A Course in Modern Physics," (Wiley, New York, 1989) ISBN 0-471-60531-X.</ref>：

:<math>u_{\nu}(\nu,T) = {  4 \pi \over c }   I_{\nu}(\nu,T) = \frac{8\pi h\nu^3 }{c^3}~\frac{1}{e^{\frac{h\nu}{kT}} - 1},</math>

这是指单位频率在单位[[体积]]内的能量，单位是焦耳／（立方米·赫兹）。对全频域积分可得到与频率无关的能量密度。一个黑体的辐射场可以被看作是[[光子气体]]，此时的能量密度可由气体的[[热力学]]参数决定。

能量密度频谱也可写成波长的函数
:<math>u_{\lambda}(\lambda,T) = {8\pi h c\over \lambda^5}{1\over e^{\frac{h c}{\lambda kT}} - 1},</math>

下表中给出了函数中每一个物理量的意义和单位：

:{| class="wikitable"
|-
!物理量
!含义
![[国际单位制]]
![[厘米-克-秒制]]
|-
|<math>I \,</math>
| 辐射率，在单位时间内从单位表面积和单位立体角内以单位频率间隔或单位波长间隔辐射出的[[能量]]
|[[焦耳]]·秒<sup>-1</sup>·米<sup>-2</sup>·[[球面度]]<sup> -1</sup>·赫兹<sup>-1</sup>，或焦耳·秒<sup>-1</sup>·米<sup>-2</sup>·球面度<sup>- 1</sup>·米<sup>-1</sup><!--不要将米这一量纲写在一起-->
|[[尔格]]·秒<sup>-1</sup>·厘米<sup>-2</sup>·赫兹<sup>-1</sup>·球面度<sup>-1</sup>
|-
|<math>\nu \,</math>
|[[頻率 (物理學)|频率]]
|[[赫兹]] (Hz)
|赫兹
|-
|<math>\lambda \,</math>
|[[波长]]
|[[米 (單位)|米]] (m)
|厘米（cm）
|-
|<math>T \,</math>
|黑体的[[温度]]
| [[开尔文]] (K)
|开尔文
|-
|<math>h \,</math>
|[[普朗克常数]]
|焦耳·秒 (J·s)
|尔格·秒（erg·s）
|-
|<math>c \, </math>
|[[光速]]
|[[米/秒]] (m/s)
|厘米／秒（cm/s）
|-
|<math>e \,</math>
|[[自然对数的底]]，2.718281...
|[[无量纲|1]]
|1
|-
|<math>k \, </math>
|[[玻尔兹曼常数]]
|焦耳／开尔文 (J/K)
|尔格／开尔文 (erg/K)
|}

== 推导 ==
下面的推导并非普朗克的原始推导（来源<ref name="BrehmMullin"/>），需要用到[[电动力学]]、[[量子力学]]和[[统计力学]]的有关概念。

考虑一个充满了电磁辐射的边长为<math>L\,</math>的[[立方体]]：根据经典电动力学，在立方体壁表面的边界条件为[[电场]]的平行分量和[[磁场]]的垂直分量都为零。类似于处于束缚态的粒子的[[波函数]]，立方体内部的电磁场也是满足边界条件的周期性[[本征函数]]的线性叠加，在垂直于立方体壁表面的三个方向上各个本征函数的波长分别为<math>\lambda_{1},</math> <math>\lambda_{2}</math>和<math>\lambda_{3},</math>

:<math>\lambda_{i} = \frac{2L}{n_{i}},</math>

这里<math>n_{i}</math>是非负整数。对于每一组<math>n_{i}</math>值都有两个线性无关的解（两种不同的模）。根据量子力学中的[[谐振子]]理论，任意模式下的系统能级为

:<math>E_{n_{1},n_{2},n_{3}}\left(r\right)=\left(r+\frac{1}{2}\right)\frac{hc}{2L}\sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}. \qquad \mbox{(1)}</math>

这里量子数<math>r\,</math>可看作是立方体中的光子数，而两种不同模式对应的是光子的两种[[偏振]]态。注意到当光子数为零时能级不为零，这种电磁场的真空能量是一种量子效应，是产生[[卡西米尔效应]]的原因。下面我们计算在温度<math>T\,</math>下光子数为零时系统处于真空状态下的[[内能]]。

根据统计力学，在特定模式下不同能级的概率分布由下式给出

:<math>P_{r}=\frac{e^{ - \beta E\left(r\right)}}{Z\left(\beta\right)}.</math>

这里
:<math>\beta\ \stackrel{\mathrm{def}}{=}\   1/\left(kT\right).</math> 

分母<math>Z(\beta)\,</math>是系统在特定模式下的[[配分函数]]，它能够使[[概率]]分布<math>P_r\,</math>归一化。对[[正则系综]]有
:<math>Z\left(\beta\right)=\sum_{r=0}^{\infty}e^{-\beta E\left(r\right)}=\frac{e^{-\frac{\beta\varepsilon}{2}}}{1 - e^{ - \beta\varepsilon}}.</math>
这里我们定义单个光子的能量为
:<math>\varepsilon\ \stackrel{\mathrm{def}}{=}\  \frac{hc}{2L}\sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}},</math>

系统的平均能量和配分函数的关系为
:<math>\left\langle E\right\rangle=-\frac{d\ln Z}{d\beta}=\frac{\varepsilon}{2}+\frac{\varepsilon}{e^{\beta\varepsilon} - 1}.</math>

这个公式是[[玻色-爱因斯坦统计]]的一个特例。由于光子是[[玻色子]]，任一能级对光子的数量没有限制，系统的[[化学势]]为零。

系统的总能量是平均能量<math>\left\langle E\right\rangle</math>对所有可能的单光子态求和。考虑在热力学极限下，立方体边长<math>L\,</math>趋于无穷大，这时单光子能量<math>\varepsilon</math>近似成为连续值，我们将平均能量<math>\left\langle E\right\rangle</math>对单光子的连续能量积分就可以得到系统的总能量，这就需要我们首先确定在任意给定的能量范围内具有多少个光子态。假设处于能级<math>\varepsilon</math>和<math>\varepsilon + d\varepsilon</math>的单光子态总数为<math>g(\varepsilon)\,d\varepsilon</math>（这里<math>g(\varepsilon)</math>是所谓光子的[[能态密度]]，其具体表达式还需另行计算），则系统的总能量为

:<math>U = \int_{0}^{\infty}\frac{\varepsilon}{e^{\beta\varepsilon} - 1}g(\varepsilon)\,d\varepsilon. \qquad \mbox{(2)}</math>

为计算光子能态密度的表达式，我们将（1）式重写成

:<math>\varepsilon\ \stackrel{\mathrm{def}}{=}\  \frac{hc}{2L}n,</math>

这里<math>n\,</math>是[[矢量]]<math>\vec{n}=\left(n_{1},n_{2},n_{3}\right)</math>的模

:<math>n=\sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}.</math>

每一个矢量都对应有两个光子态，换句话说，在给定的一个由矢量<math>\vec{n}=\left(n_{1},n_{2},n_{3}\right)</math>构成的[[希尔伯特空间]]中的光子态总数是这个空间体积的2倍。一个微小的能量区间<math>d\varepsilon</math>对应着这个希尔伯特空间中一个薄球壳的厚度<math>dn= (2L/hc) d\varepsilon</math>。由于矢量<math>\vec{n}</math>的分量不能为负值，能量区间实际上只能对应整个薄球壳总体积的1/8（这是因为矢量有三个分量，每一个分量都为正数时的概率为1/8）。因而在能量区间<math>d\varepsilon</math>上光子态总数<math>g(\varepsilon)\,d\varepsilon</math>为

:<math>g(\varepsilon)\,d\varepsilon=2\frac{1}{8}4\pi n^{2}\,dn=\frac{8\pi L^{3}}{h^{3}c^{3}}\varepsilon^{2}\,d\varepsilon.</math>

将这个表达式代入（2）式，得到

:<math>U =L^{3}\frac{8\pi}{h^{3}c^{3}}\int_{0}^{\infty}\frac{\varepsilon^{3}}{e^{\beta\varepsilon} - 1}\,d\varepsilon. \qquad \mbox{(3)}</math>

注意到<math>L\,</math>的三次方是立方体体积，因此可直接得到能量密度的表达式，将它写成频率的频谱函数<math>u(\nu,T)</math>

:<math>\frac{U}{L^3} = \int_0^{\infty}u(\nu,T)\, d\nu,</math>

其中

:<math>u(\nu,T) = {8\pi h\nu^3\over c^3}{1\over e^{h\nu/kT} - 1}.</math>

这里<math>u(\nu,T)</math>即是黑体辐射的能量频谱密度，其意义为单位频率在单位体积内的能量。

如果写成波长的函数，

:<math>\frac{U}{L^3} = \int_0^\infty u(\lambda,T)\, d\lambda,</math>

其中
:<math>u(\lambda,T) = {8\pi h c\over \lambda^5}{1\over e^{h c/\lambda kT} - 1}.</math>

这是黑体辐射的能量密度频谱的另一种形式，其意义为单位波长在单位体积内的能量。在玻色或费米气体情形下对这一函数积分需要用到[[多对数函数]]展开。但这里可以用初等函数的办法得到一个近似形式，数学上做代换

:<math>\varepsilon = kTx,</math>

积分变量从而可写成如下形式

:<math>u(T) =\frac{8\pi (kT)^{4}}{(hc)^{3}} J,</math>

其中<math>J</math>的表达式为

:<math>J=\int_{0}^{\infty}\frac{x^{3}}{e^{x} - 1}\,dx = \frac{\pi^{4}}{15}.</math>

这一积分结果将后文附录中做说明。因而得到立方体中电磁场的总能量为

:<math>{U\over V} = \frac{8\pi^5(kT)^4}{15 (hc)^3},</math>

其中<math>V=L^3\,</math>是立方体体积（注意：这个表达式'''不是'''[[斯特藩-玻尔兹曼定律]]，它的含义并不是理想黑体在单位时间内从'''单位表面积'''辐射出的总能量，参见[[斯特藩-玻尔兹曼定律]]条目）。由于辐射[[各向同性]]，并且以[[光速]]传播，能量的辐射率（单位时间单位立体角所对应辐射行进截面积及单位频率下辐射的能量）为

:<math>I(\nu,T) = \frac{u(\nu,T)\,c}{4\pi},</math>

从而得到普朗克黑体辐射定律

:<math>I(\nu,T) = \frac{2 h\nu^3 }{c^2}~\frac{1}{e^{h\nu/kT} - 1}.</math>

== 历史 ==
{{See also|光子|能量均分定理|紫外灾变}}

[[File:RWP-comparison.svg|thumb|300px|普朗克定律（绿）、维恩近似（蓝）和瑞利-金斯定律（红）在频域下的比较，可见维恩近似在高频区域和普朗克定律相符，瑞利-金斯定律在低频区域和普朗克定律相符。]]
[[马克斯·普朗克]]于1900年建立了黑体辐射定律的公式，并于1901年发表<ref name="planck">
Planck, Max, "[http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html On the Law of Distribution of Energy in the Normal Spectrum] {{webarchive|url=https://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |date=2008-04-18 }}". [[Annalen der Physik]], vol. 4, p. 553 ff (1901)</ref>。其目的是改进由[[威廉·维恩]]提出的[[维恩近似]]（至于描述黑体辐射的另一公式：由[[瑞利勋爵]]和[[詹姆士·金斯|金斯爵士]]提出的[[瑞利-金斯定律]]，其建立时间要稍晚于普朗克定律。由此可见瑞利-金斯公式所导致的“[[紫外灾变]]”并不是普朗克建立黑体辐射定律的动机，参见后文叙述）。维恩近似在短波范围内和实验数据相当符合，但在长波范围内偏差较大；而瑞利-金斯公式则正好相反。普朗克得到的公式则在全波段范围内都和实验结果符合得相当好。在推导过程中，普朗克考虑将[[电磁场]]的能量按照物质中带电振子的不同振动模式分布。得到普朗克公式的前提假设是这些振子的能量只能取某些基本能量单位的整数倍，这些基本能量单位只与电磁波的频率<math>\nu\,</math>有关，并且和频率<math>\nu\,</math>成正比。

:<math>E=h\nu.\,</math>

这即是普朗克的能量量子化假说，这一假说的提出比[[爱因斯坦]]为解释[[光电效应]]而提出的[[光子]]概念还要至少早五年。然而普朗克并没有像爱因斯坦那样假设[[电磁波]]本身即是具有分立能量的量子化的波束，他认为这种量子化只不过是对于处在封闭区域所形成的腔（也就是构成物质的[[原子]]）内的微小振子而言的，用半经典的语言来说就是[[束缚态]]必然导出量子化。普朗克没能为这一量子化假设给出更多的物理解释，他只是相信这是一种数学上的推导手段，从而能够使理论和经验上的实验数据在全波段范围内符合。不过最终普朗克的量子化假说和爱因斯坦的光子假说都成为了[[量子力学]]的基石。

很多有关量子理论的大众科普读物，甚至某些物理学课本，在讨论普朗克黑体辐射定律的历史时都犯了严重的错误。尽管这些错误概念在四十多年前就已经被物理学史的研究者们指出，事实证明它们依然难以被消除。部分原因可能在于，普朗克最初量子化能量的动机并不是能用三言两语就能够道清的，这里面的原因在现代人看来相当复杂，因而不易被外人所理解<ref>关于究竟是什么动机致使普朗克建立了量子化的能量这一历史争论，请参看<br />{{lang|en|{{cite book | authorlink=Thomas Kuhn |last=Kuhn |first=Thomas | title=[[Black-Body Theory and the Quantum Discontinuity]]: 1894-1912 | publisher=Clarendon Press, Oxford | year=1978 | id=ISBN 0-226-45800-8}}<br />{{cite journal | first=Peter | last=Galison | authorlink=Peter Galison | title=Kuhn and the Quantum Controversy | journal=British Journal for the Philosophy of Science | volume=32 | issue=1 | year=1981 | pages=71–85}}}}</ref>。[[丹麦]]物理学家{{lang|en|Helge Kragh}}曾发表过一篇文章清晰地阐述了这种错误是如何发生的<ref name="helge">Kragh, Helge [http://www.physicsweb.org/articles/world/13/12/8/1 Max Planck: The reluctant revolutionary] {{Wayback|url=http://www.physicsweb.org/articles/world/13/12/8/1 |date=20070702004530 }} Physics World, December 2000.</ref>。

[[File:Blackbody-lg.png|thumb|300px|“紫外灾变”：在经典统计理论中，[[能量均分定理]]预言黑体辐射的强度在紫外区域会发散至无穷大，这和事实严重违背]]
首先是尽管普朗克给出了量子化的电磁波能量表达式，普朗克并没有将电磁波量子化，这在他1901年的论文以及这篇论文对他早先文献的引用中就可以看到<ref name="planck"/>。他还在他的著作《热辐射理论》（''{{lang|en|Theory of Heat Radiation}}''）中平淡无奇地解释说量子化公式中的普朗克常数（现代量子力学中的基本常数）只是一个适用于[[赫兹振荡器]]的普通常数。真正从理论上提出光量子的第一人是于1905年成功解释光电效应的爱因斯坦，他假设电磁波本身就带有量子化的能量，携带这些量子化的能量的最小单位叫[[光量子]]。1924年[[萨特延德拉·纳特·玻色]]发展了光子的统计力学，从而在理论上推导了普朗克定律的表达式。

另一错误概念是，普朗克发展这一定律的动机并不是试图解决“[[紫外灾变]]”。“紫外灾变”这一名称是[[保罗·埃伦费斯特]]于1911年提出的，从时间上看这比普朗克定律的提出要晚十年之久。紫外灾变是指将经典统计力学的[[能量均分定理]]应用于一个空腔中的黑体辐射（又叫做空室辐射或具空腔辐射）时，系统的总能量在紫外区域将变得发散并趋于无穷大，这显然与实际不符。普朗克本人从未认为[[能量均分定理]]永远成立，从而他根本没有觉察到在黑体辐射中有任何“灾变”存在——不过仅仅过了五年之后，这一问题随着爱因斯坦、瑞利勋爵和金斯爵士的发现而就变得尖锐起来。

== 附录 ==
{{See also|黎曼ζ函数|Γ函数}}

有一个简便方法计算下面的积分

:<math>J=\int_{0}^{\infty}\frac{x^3}{e^x - 1}\,dx</math>

我们可以首先用<math>x^n\,</math>替换式中的<math>x^3\,</math>，计算一般形式下的积分

:<math>\int_{0}^{\infty}\frac{x^n}{e^x - 1}\,dx = \int_{0}^{\infty}\frac{x^n e^{ - x}}{1 - e^{ - x}}\,dx</math>

由于分母总是小于1，我们可以将它按<math>e^{-x}</math>展开写成收敛的[[几何级数]]

:<math> \frac{1}{1 - e^{ - x}} = \sum_{k=0}^{\infty} e^{-kx}.</math>

这就是几何级数的求和公式。等号左边的表达式正是右边的<math>1 + e^{ - x} + e^{ - 2x} + e^{ - 3x} + \cdots.</math>求和结果，右边的几何级数公比为<math>e^{-x}</math>.

从而得到

:<math>\int_{0}^{\infty}x^{n} e^{ - x} \sum_{k=0}^{\infty} e^{ - kx}\,dx.</math>

表达式乘以<math>e^{-x}</math>后相当于将<math>e^{ - x} + e^{ - 2x} + e^{ - 3x} + \cdots</math>变成<math>e^{ - 2x} + e^{ - 3x} + e^{ - 4x} + \cdots</math>，因而我们将求和符号中的序号加1，并消去原先的<math>e^{ - x}</math>:

:<math>\int_{0}^{\infty}x^{n}  \sum_{k=1}^{\infty} e^{ - kx}\,dx.</math>

通过变量替换<math>u = kx</math>，我们得到<math>x^n = \frac{u^n}{k^n}</math>以及<math>dx = \frac{du}{k}</math>，积分式进一步写成

:<math>\int_{0}^{\infty}\frac{u^n}{k^n} \sum_{k=1}^{\infty} e^{-u}\frac{du}{k}</math>

即

:<math>\int_{0}^{\infty}u^n \sum_{k=1}^{\infty}\frac{1}{k^{n + 1}} e^{ - u}du.</math>

形如上式的积分是收敛的，我们将求和的部分移到积分之外：

:<math>\sum_{k=1}^{\infty} \frac{1}{k^{n+1}} \int_{0}^{\infty}u^{n} e^{ - u}\,du.</math>

前面的求和系数正是[[黎曼ζ函数]]<math>\zeta(n+1)</math>，而后面的积分正是[[Γ函数]]<math> \Gamma(n+1) </math>。从而我们得到一个一般的关系式：

:<math>\int_{0}^{\infty}\frac{x^{n}}{e^x - 1}\,dx = \zeta(n+1) \Gamma{\left(n+1\right)}.</math>

或等价为

:<math>\int_{0}^{\infty}\frac{x^{n - 1}}{e^x - 1}\,dx = \zeta{\left(n\right)} \Gamma{\left(n\right)}.</math>

对于我们所需要的积分，积分式的分子为<math>x^3</math>，因此代入上面等式中得到

:<math>J=\zeta{\left(4\right)} \Gamma{\left(4\right)} = \frac{\pi^{4}}{90} \times 6 = \frac{\pi^4}{15}.</math>

这里我们用到了

:<math>\zeta{\left(4\right)} = \sum_{n=1}^{\infty}\frac{1}{n^{4}}=\pi^{4}/90</math>

和<math>\Gamma(n+1) = n!</math>。（参见[[黎曼ζ函数]]和[[Γ函数]]的有关性质）。

== 参考文献 ==
{{reflist|colwidth=30em}}

==目錄==
{{refbegin}}
*{{cite book
 |last=Adkins |first=C. J.
 |year=1983
 |title=Equilibrium Thermodynamics
 |url=https://archive.org/details/equilibriumtherm0000adki_b6w0 |edition=3rd
 |publisher=Cambridge University Press |isbn=0-521-25445-0
 |ref=harv
}}
*{{cite journal
 |last1=Bohr
 |first1=N.
 |year=1913
 |title=On the constitution of atoms and molecules
 |url=http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Bohr_1913.pdf
 |journal=Philosophical Magazine
 |volume=26
 |issue=153
 |pages=1–25
 |doi=10.1080/14786441308634993
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2017-08-09
 |archive-url=https://web.archive.org/web/20170809013732/http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Bohr_1913.pdf
 |dead-url=no
 }}
*{{cite book
 |last1=Bohren |first1=C. F.
 |last2=Clothiaux |first2=E. E.
 |year=2006
 |title=Fundamentals of Atmospheric Radiation
 |publisher=Wiley-VCH
 |isbn=3-527-40503-8
 |ref=harv
}}
*{{cite journal
 |last1=Boltzmann |first1=L.
 |year=1878
 |title=Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht
 |journal=Sitzungsberichte Mathematisch-Naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften in Wien
 |volume=76
 |issue=2
 |pages=373–435
 |ref=harv
}}
*{{cite book
 |last1=Born |first1=M.
 |last2=Wolf |first2=E.
 |year=1999
 |title=Principles of Optics
 |url=https://archive.org/details/principlesofopti0006born |edition=7th
 |publisher=Cambridge University Press
 |isbn=0-521-64222-1
 |ref=harv
}}
*{{cite journal
 |last1=Born |first1=M.
 |last2=Jordan |first2=P.
 |year=1925
 |title=Zur Quantenmechanik
 |journal=Zeitschrift für Physik
 |volume=34
 |pages=858–888
 |bibcode=1925ZPhy...34..858B
 |doi=10.1007/BF01328531
 |ref=harv
}} Translated in part as "On quantum mechanics" in {{cite book
 |last1=van der Waerden |first1=B. L.
 |year=1967
 |title=Sources of Quantum Mechanics
 |pages=277–306
 |publisher=North-Holland Publishing
 |ref=harv
}}
*{{cite journal
 |last1=Bose |first1=Satyendra Nath
 |year=1924
 |title=Plancks Gesetz und Lichtquantenhypothese
 |language=de
 |journal=Zeitschrift für Physik
 |volume=26 |pages=178–181
 |bibcode=1924ZPhy...26..178B
 |doi=10.1007/BF01327326
 |ref=harv
}}
*{{cite book
 |last1=Brehm |first1=J. J.
 |last2=Mullin |first2=W. J.
 |year=1989
 |title=Introduction to the Structure of Matter
 |url=https://archive.org/details/introductiontost00breh |publisher=John Wiley & Sons
 |isbn=0-471-60531-X
 |ref=harv
}}
*{{cite book
 |last1=Brillouin |first1=L.
 |year=1970
 |title=Relativity Reexamined
 |url=https://archive.org/details/relativityreexam0000bril |publisher=Academic Press
 |isbn=978-0-12-134945-5
 |ref=harv
}}
*{{cite book
 |last1=Caniou |first1=J.
 |year=1999
 |title=Passive Infrared Detection: Theory and Applications
 |url=https://books.google.com/?id=X-aFGcf6pOEC&pg=116
 |publisher=Springer
 |isbn=978-0-7923-8532-5
 |ref=harv
}}
*{{cite book
 |last1=Chandrasekhar |first1=S.
 |year=1960
 |origyear=1950
 |title=Radiative Transfer
 |url=https://archive.org/details/radiativetransfe0000chan |edition=Revised reprint
 |publisher=Dover Publications
 |isbn=978-0-486-60590-6
 |ref=harv
}}
*{{cite journal
 |last1=Cotton |first1=A.
 |year=1899
 |title=The present status of Kirchhoff's law
 |url=https://archive.org/details/sim_astrophysical-journal_1899-04_9_4/page/237 |journal=The Astrophysical Journal
 |volume=9
 |pages=237–268
 |ref=harv
 |bibcode=1899ApJ.....9..237C
 |doi=10.1086/140585
}}
*{{cite journal
 |last1=Crova
 |first=A. P. P.
 |year=1880
 |title=Étude des radiations émises par les corps incandescents. Mesure optique des hautes températures
 |url=http://gallica.bnf.fr/ark:/12148/bpt6k34859d/f470.image.r=Annales%20de%20Chimie%20et%20de%20Physique.langEN
 |journal=Annales de chimie et de physique
 |series=Série 5
 |volume=19
 |pages=472–550
 |doi=
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2020-05-20
 |archive-url=https://web.archive.org/web/20200520133055/https://gallica.bnf.fr/ark:/12148/bpt6k34859d/f470.image.r=Annales%20de%20Chimie%20et%20de%20Physique.langEN
 |dead-url=no
 }}
*{{cite journal
 |last1=Dougal |first1=R. C.
 |date=1976
 |title=The presentation of the Planck radiation formula (tutorial)
 |journal=Physics Education
 |volume=11 |issue=6 |pages=438–443
 |bibcode=1976PhyEd..11..438D
 |doi=10.1088/0031-9120/11/6/008
 |ref=harv
}}
*{{cite journal
 |last1=Ehrenfest |first1=P.
 |year=1911
 |title=Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?
 |url=https://archive.org/details/sim_annalen-der-physik_1911_36_11/page/91 |journal=Annalen der Physik
 |volume=36 |issue=11
 |pages=91–118
 |bibcode=1911AnP...341...91E
 |doi=10.1002/andp.19113411106
 |ref=harv
}}
*{{cite journal
 |last1=Ehrenfest |first1=P.
 |last2=Kamerlingh Onnes |first2=H.
 |year=1914
 |title=Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory
 |journal=Proceedings of the Royal Dutch Academy of Sciences in Amsterdam
 |volume=17 |issue=2 |pages=870–873
 |bibcode=1914KNAB...17..870E
 |ref=harv
}}
*{{cite journal
 |last1=Einstein
 |first1=A.
 |year=1905
 |title=Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt
 |url=http://onlinelibrary.wiley.com/doi/10.1002/andp.19053220607/pdf
 |journal=Annalen der Physik
 |volume=17
 |issue=6
 |pages=132–148
 |bibcode=1905AnP...322..132E
 |doi=10.1002/andp.19053220607
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2017-07-22
 |archive-url=https://web.archive.org/web/20170722060318/http://onlinelibrary.wiley.com/doi/10.1002/andp.19053220607/pdf
 |dead-url=no
 }} Translated in {{cite journal
 |last1=Arons
 |first1=A. B.
 |last2=Peppard
 |first2=M. B.
 |year=1965
 |title=Einstein's proposal of the photon concept: A translation of the ''Annalen der Physik'' paper of 1905
 |url=http://astro1.panet.utoledo.edu/~ljc/PE_eng.pdf
 |journal=American Journal of Physics
 |volume=33
 |issue=5
 |page=367
 |bibcode=1965AmJPh..33..367A
 |doi=10.1119/1.1971542
 |author=
 |access-date=2018-06-12
 |archive-url=https://web.archive.org/web/20160304031335/http://astro1.panet.utoledo.edu/~ljc/PE_eng.pdf
 |archive-date=2016-03-04
 |dead-url=yes
 }}
*{{cite journal
 |last1=Einstein |first1=A.
 |year=1916
 |title=Zur Quantentheorie der Strahlung
 |journal=Mitteilungen der Physikalischen Gesellschaft Zürich
 |volume=18 |pages=47–62
 |ref=harv
}} and a nearly identical version {{cite journal
 |last1=Einstein |first1=A.
 |year=1917
 |title=Zur Quantentheorie der Strahlung
 |journal=Physikalische Zeitschrift
 |volume=18 |pages=121–128
 |bibcode=1917PhyZ...18..121E
 |ref=harv
}} Translated in {{cite book
 |last1=ter Haar |first1=D.
 |year=1967
 |pages=[https://archive.org/details/oldquantumtheory00haar/page/167 167]–183
 |title=The Old Quantum Theory
 |url=https://archive.org/details/oldquantumtheory00haar |publisher=Pergamon Press
 |lccn=66029628
 |ref=harv
}} See also [https://web.archive.org/web/20110723012756/http://astro1.panet.utoledo.edu/~ljc/einstein_ab.pdf].
*{{cite book
 |last1=Einstein |first1=A.
 |others=English translation by Beck, A.
 |year=1993
 |title=The Collected Papers of Albert Einstein
 |publisher=Princeton University Press
 |isbn=0-691-10250-3
 |volume=3
 |ref=harv
}}
*{{cite book
 |last1=Feynman |first1=R. P.
 |last2=Leighton |first2=R. B.
 |last3=Sands |first3=M.
 |year=1963
 |title=The Feynman Lectures on Physics, Volume 1
 |publisher=Addison-Wesley
 |isbn=0-201-02010-6
 |ref=harv
}}
*{{cite web
 |last=Fischer
 |first=T.
 |date=1 November 2011
 |title=Topics: Derivation of Planck's Law
 |url=https://nanohub.org/wiki/DerivationofPlancksLaw
 |publisher=ThermalHUB
 |accessdate=2015-06-19
 |ref=harv
 |archive-date=2020-11-25
 |archive-url=https://web.archive.org/web/20201125081651/https://nanohub.org/wiki/DerivationofPlancksLaw
 |dead-url=no
 }}
*{{cite book
 |last1=Goody |first1=R. M.
 |last2=Yung |first2=Y. L.
 |year=1989
 |title=Atmospheric Radiation: Theoretical Basis
 |url=https://archive.org/details/atmosphericradia0000good_2nded |edition=2nd
 |publisher=Oxford University Press
 |isbn=978-0-19-510291-8
 |ref=harv
}}
*{{cite book
 |last1=Guggenheim |first1=E. A.
 |year=1967
 |title=Thermodynamics. An Advanced Treatment for Chemists and Physicists
 |publisher=North-Holland Publishing Company
 |edition=5th revised
 |ref=harv
}}
*{{cite book
 |last=Haken |first=H.
 |title=Light
 |url=https://archive.org/details/light0001hake |year=1981
 |publisher=North-Holland Publishing
 |location=Amsterdam
 |isbn=0-444-86020-7
 |edition=Reprint
 |ref=harv
}}
*{{cite book
 |last=Hapke |first=B.
 |title=Theory of Reflectance and Emittance Spectroscopy
 |year=1993
 |publisher=Cambridge University Press, Cambridge UK
 |isbn=0-521-30789-9
 |ref=harv
}}
*{{cite journal
 |last1=Heisenberg |first1=W.
 |year=1925
 |title=Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen
 |journal=Zeitschrift für Physik
 |volume=33
 |pages=879–893
 |bibcode=1925ZPhy...33..879H
 |doi=10.1007/BF01328377
 |ref=harv
}} Translated as "Quantum-theoretical Re-interpretation of kinematic and mechanical relations" in {{cite book
 |last1=van der Waerden |first1=B. L.
 |year=1967
 |title=Sources of Quantum Mechanics
 |pages=261–276
 |publisher=North-Holland Publishing
 |ref=harv
}}
*{{Cite book
  |last1=Heisenberg |first1=W.
  |others=Eckart, C.; Hoyt, F. C.  (transl.)
  |year=1930
  |title=The Physical Principles of the Quantum Theory
  |url=https://archive.org/details/in.ernet.dli.2015.166273 |publisher=University of Chicago Press
  |ref=harv
}}
*{{Cite book
  |last1=Hermann |first1=A.
  |others=Nash, C.W. (transl.)
  |year=1971
  |title=The Genesis of Quantum Theory
  |url=https://archive.org/details/genesisofquantum00herm |publisher=MIT Press
  |isbn=0-262-08047-8
  |ref=harv
}} a translation of ''Frühgeschichte der Quantentheorie (1899–1913)'', Physik Verlag, Mosbach/Baden, 1969.
*{{cite journal
 |last1=Hettner |first1=G.
 |year=1922
 |title=Die Bedeutung von Rubens Arbeiten für die Plancksche Strahlungsformel
 |journal=Naturwissenschaften
 |volume=10 |issue=48
 |pages=1033–1038
 |bibcode=1922NW.....10.1033H
 |doi=10.1007/BF01565205
 |ref=harv
}}
*{{Cite book
 |last1=Jammer |first1=M.
 |year=1989
 |title=The Conceptual Development of Quantum Mechanics
 |url=https://archive.org/details/conceptualdevelo0012jamm_t1q1 |edition=second
 |publisher=Tomash Publishers/American Institute of Physics
 |isbn=0-88318-617-9
 |ref=harv
}}
*{{Cite book
 |last1=Jauch |first1=J. M.
 |last2=Rohrlich |first2=F.
 |origyear=1955
 |year=1980
 |title=The Theory of Photons and Electrons. The Relativistic Quantum Field Theory of Charged Particles with Spin One-half
 |edition=second printing of second
 |publisher=pringer
 |isbn=0-387-07295-0
 |ref=harv
}}
*{{cite journal
 |last1=Jeans |first1=J. H.
 |year=1901
 |title=The Distribution of Molecular Energy
 |volume=196 |issue=274–286 |page=397
 |journal=Philosophical Transactions of the Royal Society A
 |bibcode=1901RSPTA.196..397J
 |doi=10.1098/rsta.1901.0008
 |jstor=90811
 |ref=harv
}}
*{{Cite journal
 |last1=Jeans |first1=J. H.
 |year=1905a
 |title=XI. On the partition of energy between matter and æther
 |journal=Philosophical Magazine
 |volume=10 |issue=55 |page=91
 |doi=10.1080/14786440509463348
 |ref=harv
}}
*{{Cite journal
 |last1=Jeans |first1=J. H.
 |year=1905b
 |title=On the Application of Statistical Mechanics to the General Dynamics of Matter and Ether
 |volume=76 |issue=510 |page=296
 |journal=Proceedings of the Royal Society Ａ
 |bibcode=1905RSPSA..76..296J
 |doi=10.1098/rspa.1905.0029
 |jstor=92714
 |ref=harv
}}
*{{Cite journal
 |last1=Jeans |first1=J. H.
 |year=1905c
 |title=A Comparison between Two Theories of Radiation
 |journal=Nature
 |volume=72 |issue=1865 |page=293
 |bibcode=1905Natur..72..293J
 |doi=10.1038/072293d0
 |ref=harv
}}
*{{Cite journal
 |last1=Jeans |first1=J. H.
 |year=1905d
 |title=On the Laws of Radiation
 |journal=Proceedings of the Royal Society A
 |volume=76|issue=513 |page=545
 |bibcode=1905RSPSA..76..545J
 |doi=10.1098/rspa.1905.0060
 |jstor=92704
 |ref=harv
}}
*{{cite book
 |last1=Jeffreys |first1=H.
 |year=1973
 |title=Scientific Inference
 |url=https://archive.org/details/scientificinfere0000jeff |edition=3rd
 |publisher=Cambridge University Press
 |isbn=978-0-521-08446-8
 |ref=harv
}}
*{{cite book
 |last1=Kangro |first1=H.
 |year=1976
 |title=Early History of Planck's Radiation Law
 |url=https://archive.org/details/earlyhistoryofpl0000kang |publisher=Taylor & Francis
 |isbn=0-85066-063-7
 |ref=harv
}}
*{{cite journal
 |last=Karplus |first=R.
 |last2=Neuman |first2=M.
 |year=1951
 |title=The Scattering of Light by Light
 |journal=Physical Review
 |volume=83 |issue=4 |pages=776–784
 |bibcode=1951PhRv...83..776K
 |doi=10.1103/PhysRev.83.776
 |ref=harv
}}
*{{cite journal
 |last1=Kirchhoff |first1=G. R.
 |year=1860a
 |title=Über die Fraunhofer'schen Linien
 |journal=Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin
 |pages=662–665
 |doi=
 |ref=harv
}}
*{{cite journal
 |last1=Kirchhoff |first1=G. R.
 |date=1860b
 |title=Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme
 |journal=Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin
 |pages=783–787
 |doi=
 |ref=harv
}}
*{{cite journal
 |last1=Kirchhoff |first1=G. R.
 |year=1860c
 |title=Über das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht
 |journal=Annalen der Physik und Chemie
 |volume=109
 |issue=2
 |pages=275–301
 |doi=10.1002/andp.18601850205
 |ref=harv
|bibcode=1860AnP...185..275K }} Translated by Guthrie, F. as {{cite journal
 |last1=Kirchhoff |first1=G. R.
 |year=1860
 |title=On the relation between the radiating and absorbing powers of different bodies for light and heat
 |journal=Philosophical Magazine
 |series=Series 4
 |volume=20 |pages=1–21
}}
*{{Citation
 |last=Kirchhoff |first=G. R.
 |title=Gessamelte Abhandlungen
 |publisher=Johann Ambrosius Barth
 |year=1862
 |chapter=Über das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht
 |pages=571–598
 |ref=harv
}}
*{{cite book
 |last1=Kittel |first1=C.
 |last2=Kroemer |first2=H.
 |year=1980
 |title=Thermal Physics
 |url=https://archive.org/details/thermalphysics0000kitt |edition=2nd
 |publisher=W. H. Freeman
 |isbn=0-7167-1088-9
 |ref=harv
}}
*{{cite journal
 |last1=Klein |first1=M. J.
 |year=1962
 |title=Max Planck and the beginnings of the quantum theory
 |journal=Archive for History of Exact Sciences
 |volume=1 |issue=5 |pages=459–479
 |doi=10.1007/BF00327765
 |ref=harv
}}
*{{cite book
 |last1=Kragh |first1=H.
 |year=1999
 |title=Quantum Generations. A History of Physics in the Twentieth Century
 |url=https://archive.org/details/quantumgeneratio0000krag |publisher=Princeton University Press
 |isbn=0-691-01206-7
 |ref=harv
}}
*{{cite journal
 |last1=Kragh
 |first1=H.
 |date=December 2000
 |title=Max Planck: The reluctant revolutionary
 |url=http://physicsworld.com/cws/article/print/373
 |journal=Physics World
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2012-04-01
 |archive-url=https://web.archive.org/web/20120401221617/http://physicsworld.com/cws/article/print/373
 |dead-url=yes
 }}
*{{cite journal
 |last1=Kramm |first1=Gerhard
 |last2=Mölders |first2=N.
 |year=2009
 |title=Planck's Blackbody Radiation Law: Presentation in Different Domains and Determination of the Related Dimensional Constant
 |journal=Journal of the Calcutta Mathematical Society
 |volume=5 |issue=1–2 |pages=27–61
 |arxiv=0901.1863
 |bibcode=2009arXiv0901.1863K
 |ref=harv
}}
*{{cite book
 |last1=Kuhn |first1=T. S.
 |year=1978
 |title=Black–Body Theory and the Quantum Discontinuity
 |url=https://archive.org/details/blackbodytheoryq0000kuhn |publisher=Oxford University Press
 |isbn=0-19-502383-8
 |ref=harv
}}
*{{cite book
 |last1=Landsberg |first1=P. T.
 |year=1961
 |title=Thermodynamics with Quantum Statistical Illustrations
 |url=https://archive.org/details/thermodynamicswi0002land |publisher=Interscience Publishers
 |ref=harv
}}
*{{cite book
 |last1=Landsberg |first1=P. T.
 |year=1978
 |title=Thermodynamics and Statistical Mechanics
 |publisher=Oxford University Press
 |isbn=0-19-851142-6
 |ref=harv
}}
*{{Cite journal
 |last1=Lewis |first1=G. N.
 |year=1926
 |title=The Conservation of Photons
 |journal=Nature
 |volume=118 |issue=2981 |page=874
 |bibcode=1926Natur.118..874L
 |doi=10.1038/118874a0
 |ref=harv
}}
*{{cite book
 |last1=Loudon
 |first1=R.
 |year=2000
 |title=The Quantum Theory of Light
 |url=https://books.google.com/books?id=AEkfajgqldoC&pg=PA3
 |edition=3rd
 |publisher=Oxford University Press
 |isbn=0-19-850177-3
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2020-05-20
 |archive-url=https://web.archive.org/web/20200520133117/https://books.google.com/books?id=AEkfajgqldoC&pg=PA3
 |dead-url=no
 }}
*{{Cite journal
 |last1=Lowen |first1=A. N.
 |last2=Blanch |first2=G.
 |year=1940
 |title=Tables of Planck's radiation and photon functions
 |journal=Journal of the Optical Society of America
 |volume=30 |issue=2 |page=70
 |doi=10.1364/JOSA.30.000070
 |ref=harv
|bibcode=1940JOSA...30...70L
 }}
*{{cite journal
 |last1=Lummer |first1=O.
 |last2=Kurlbaum |first2=F.
 |year=1898
 |title=Der electrisch geglühte "absolut schwarze" Körper und seine Temperaturmessung
 |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft
 |volume=17
 |pages=106–111
 |ref=harv
}}
*{{cite journal
 |last1=Lummer |first1=O.
 |last2=Pringsheim |first2=E.
 |year=1899
 |title=1. Die Vertheilung der Energie in Spectrum des schwarzen Körpers und des blanken Platins; 2. Temperaturbestimmung fester glühender Körper
 |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft
 |volume=1
 |pages=215–235
 |ref=harv
}}
*{{cite journal
 |last1=Lummer |first1=O.
 |last2=Kurlbaum |first2=F.
 |year=1901
 |title=Der elektrisch geglühte "schwarze" Körper
 |journal=Annalen der Physik
 |volume=310
 |issue=8
 |pages=829–836
 |bibcode=1901AnP...310..829L
 |doi=10.1002/andp.19013100809
 |ref=harv
}}
*{{cite book
 |last1=Mandel |first1=L.
 |last2=Wolf |first2=E.
 |year=1995
 |title=Optical Coherence and Quantum Optics
 |url=https://archive.org/details/opticalcoherence0000mand |publisher=Cambridge University Press
 |isbn=0-521-41711-2
 |ref=harv
}}
*{{cite book
 |last1=Mehra |first1=J.
 |last2=Rechenberg |first2=H.
 |year=1982
 |title=The Historical Development of Quantum Theory
 |volume=1 |issue=1
 |publisher=Springer-Verlag
 |isbn=0-387-90642-8
 |ref=harv
}}
*{{cite book
 |last1=Messiah |first1=A.
 |others=Temmer, G. G. (transl.)
 |year=1958
 |title=Quantum Mechanics
 |publisher=John Wiley & Sons
 |ref=harv
}}
*{{cite journal
 |last1=Michelson |first1=V. A.
 |year=1888
 |title=Theoretical essay on the distribution of energy in the spectra of solids
 |journal=Philosophical Magazine
 |series=Series 5
 |volume=25 |issue=156
 |pages=425–435
 |doi=10.1080/14786448808628207
 |ref=harv
}}
*{{cite book
 |last1=Mihalas |first1=D.
 |last2=Weibel-Mihalas |first2=B.
 |year=1984
 |title=Foundations of Radiation Hydrodynamics
 |publisher=Oxford University Press
 |isbn=0-19-503437-6
 |ref=harv
}}
*{{cite journal
 |last1=Milne |first1=E. A.
 |year=1930
 |title=Thermodynamics of the Stars
 |journal=Handbuch der Astrophysik
 |volume=3
 |issue=1
 |pages=63–255
 |doi=
 |ref=harv
}}
*{{cite journal
 |last1=Mohr
 |first1=P. J.
 |last2=Taylor
 |first2=B. N.
 |last3=Newell
 |first3=D. B.
 |year=2012
 |title=CODATA Recommended Values of the Fundamental Physical Constants: 2010
 |url=http://physics.nist.gov/cuu/Constants/codata.pdf
 |journal=Reviews of Modern Physics
 |volume=84
 |issue=4
 |pages=1527–1605
 |arxiv=1203.5425
 |bibcode=2012RvMP...84.1527M
 |doi=10.1103/RevModPhys.84.1527
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2018-06-12
 |archive-url=https://web.archive.org/web/20180612163158/https://physics.nist.gov/cuu/Constants/codata.pdf
 |dead-url=no
 }}
*{{cite book
 |last1=Paltridge |first1=G. W.
 |last2=Platt |first2=C. M. R.
 |year=1976
 |title=Radiative Processes in Meteorology and Climatology
 |url=https://archive.org/details/radiativeprocess0000palt |publisher=Elsevier
 |isbn=0-444-41444-4
 |ref=harv
}}
*{{cite journal
 |last1=Paschen |first1=F.
 |year=1895
 |title=Über Gesetzmäßigkeiten in den Spectren fester Körper und über ein neue Bestimmung der Sonnentemperatur
 |journal=Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Mathematisch-Physikalische Klasse)
 |pages=294–304
 |bibcode=
 |doi=
 |ref=harv
}}
*{{cite book
 |last1=Pauli |first1=W.
 |year=1973
 |editor-last=Enz |editor-first=C. P.
 |others=Margulies, S.; Lewis, H. R. (transl.)
 |title=Wave Mechanics
 |publisher=MIT Press
 |isbn=0-262-16050-1
 |ref=harv
}}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1900a
 |title=Über eine Verbesserung der Wien'schen Spectralgleichung
 |url=https://archive.org/stream/verhandlungende01goog#page/n212/mode/2up
 |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft
 |volume=2 |pages=202–204
 |ref=harv
}} Translated in {{cite book
 |last1=ter Haar
 |first1=D.
 |year=1967
 |chapter=On an Improvement of Wien's Equation for the Spectrum
 |url=http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20%281900%29,%20Improvement%20of%20Wien%27s.pdf
 |title=The Old Quantum Theory
 |publisher=Pergamon Press
 |pages=79–81
 |lccn=66029628
 |access-date=2018-06-12
 |archive-url=https://web.archive.org/web/20161010010043/http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20%281900%29,%20Improvement%20of%20Wien%27s.pdf
 |archive-date=2016-10-10
 |dead-url=yes
 }}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1900b
 |url=https://archive.org/stream/verhandlungende01goog#page/n247/mode/2up
 |title=Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum
 |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft
 |volume=2 |pages=237–245
 |ref=harv
}} Translated in {{cite journal
 |last1=ter Haar
 |first1=D.
 |year=1967
 |url=http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20%281900%29,%20Distribution%20Law.pdf
 |title=The Old Quantum Theory
 |publisher=Pergamon Press
 |page=82
 |lccn=66029628
 |journal=
 |author=
 |access-date=2014-02-08
 |archive-url=https://web.archive.org/web/20160920053757/http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20%281900%29,%20Distribution%20Law.pdf
 |archive-date=2016-09-20
 |dead-url=yes
 }}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1900c
 |title=Entropie und Temperatur strahlender Wärme
 |journal=Annalen der Physik
 |volume=306 |issue=4 |pages=719–737
 |bibcode=1900AnP...306..719P
 |doi=10.1002/andp.19003060410
 |ref=harv
}}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1900d
 |title=Über irreversible Strahlungsvorgänge
 |journal=Annalen der Physik
 |volume=306
 |issue=1
 |pages=69–122
 |bibcode=1900AnP...306...69P
 |doi=10.1002/andp.19003060105
 |ref=harv
}}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1901
 |title=Über das Gesetz der Energieverteilung im Normalspektrum
 |url=https://archive.org/details/sim_annalen-der-physik_1901_4_3/page/553 |journal=Annalen der Physik
 |volume=4 |issue=3
 |page=553
 |ref=harv
 |bibcode=1901AnP...309..553P
 |doi=10.1002/andp.19013090310
}} Translated in {{cite web
 |last1=Ando
 |first1=K.
 |title=On the Law of Distribution of Energy in the Normal Spectrum
 |url=http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf
 |accessdate=2011-10-13
 |deadurl=yes
 |archiveurl=https://web.archive.org/web/20111006162543/http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf
 |archivedate=6 October 2011
}}
*{{cite book
 |last1=Planck |first1=M.
 |year=1906
 |title=Vorlesungen über die Theorie der Wärmestrahlung
 |publisher=Johann Ambrosius Barth
 |ref=harv
 |lccn=07004527
}}
*{{cite book
 |last1=Planck |first1=M.
 |others=Masius, M. (transl.)
 |year=1914
 |title=The Theory of Heat Radiation
 |url=https://archive.org/details/theoryofheatradi0000plan |publisher=P. Blakiston's Son & Co
 |edition=2nd
 |ol=7154661M
 |ref=harv
}}
*{{cite book
 |last1=Planck |first1=M.
 |others=Wills, A. P. (transl.)
 |year=1915
 |title=Eight Lectures on Theoretical Physics
 |publisher=Dover Publications
 |isbn=0-486-69730-4
 |ref=harv
}}
*{{cite journal
 |last1=Planck |first1=M.
 |year=1943
 |title=Zur Geschichte der Auffindung des physikalischen Wirkungsquantums
 |journal=Naturwissenschaften
 |volume=31 |issue=14–15 |pages=153–159
 |bibcode=1943NW.....31..153P
 |doi=10.1007/BF01475738
 |ref=harv
}}
*{{Cite journal
 |last1=Rayleigh |first1=Lord
 |year=1900
 |title=LIII. Remarks upon the law of complete radiation
 |journal=Philosophical Magazine
 |series=Series 5 |volume=49 |issue=301 |page=539
 |doi=10.1080/14786440009463878
 |ref=harv
}}
*{{cite journal
 |last1=Rayleigh |first1=Lord
 |year=1905
 |title=The Dynamical Theory of Gases and of Radiation
 |journal=Nature
 |volume=72 |issue=1855 |pages=54–55
 |bibcode=1905Natur..72...54R
 |doi=10.1038/072054c0
 |ref=harv
}}
*{{cite book
 |last1=Razavy |first1=M.
 |year=2011
 |title=Heisenberg's Quantum Mechanics
 |publisher=World Scientific
 |isbn=978-981-4304-10-8
 |ref=harv
}}
*{{cite journal
 |last1=Rubens |first1=H.
 |last2=Kurlbaum |first2=F.
 |year=1900a
 |title=Über die Emission langer Wellen durch den schwarzen Körper
 |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft
 |volume=2 |page=181
 |ref=harv
}}
*{{cite journal
 |last1=Rubens |first1=H.
 |last2=Kurlbaum |first2=F.
 |year=1900b
 |title=Über die Emission langwelliger Wärmestrahlen durch den schwarzen Körper bei verschiedenen Temperaturen
 |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin
 |pages=929–941
 |ref=harv
}} Translated in {{cite journal
 |last1=Rubens |first1=H.
 |last2=Kurlbaum |first2=F.
 |year=1901
 |title=On the heat-radiation of long wave-length emitted by black bodies at different temperatures
 |url=https://archive.org/details/sim_astrophysical-journal_1901-12_14_5/page/335 |journal=The Astrophysical Journal
 |volume=14
 |pages=335–348
 |bibcode=1901ApJ....14..335R
 |doi=10.1086/140874
 |ref=harv
}}
*{{Cite book
 |last1=Rybicki
 |first1=G. B.
 |last2=Lightman
 |first2=A. P.
 |year=1979
 |title=Radiative Processes in Astrophysics
 |url=https://books.google.com/books?id=LtdEjNABMlsC&printsec=frontcover
 |publisher=John Wiley & Sons
 |isbn=0-471-82759-2
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2021-03-09
 |archive-url=https://web.archive.org/web/20210309111129/https://books.google.com/books?id=LtdEjNABMlsC&printsec=frontcover
 |dead-url=no
 }}
*{{cite book
 |last1=Sharkov
 |first1=E. A.
 |year=2003
 |chapter=Black-body radiation
 |title=Passive Microwave Remote Sensing of the Earth
 |url=http://www.iki.rssi.ru/asp/pub_sha1/Sharch06.pdf
 |publisher=Springer
 |isbn=978-3-540-43946-2
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2021-02-17
 |archive-url=https://web.archive.org/web/20210217100500/http://www.iki.rssi.ru/asp/pub_sha1/Sharch06.pdf
 |dead-url=no
 }}
*{{cite book
 |last1=Schiff|first1=L. I.
 |year=1949
 |title=Quantum Mechanics
 |url=https://archive.org/details/quantummechanics0000leon_g7q0|publisher=McGraw-Hill
 |ref=harv
}}
*{{cite book
 |last1=Schirrmacher |first1=A.
 |year=2001
 |title=Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck
 |publisher=Münchner Zentrum für Wissenschafts und Technikgeschichte
 |ref=harv
}}
*{{cite book
 |last1=Schwinger |first1=J.
 |year=2001
 |chatper=
 |editor-last=Englert |editor-first=B.-G.
 |title=Quantum Mechanics: Symbolism of Atomic Measurements
 |url=https://archive.org/details/quantummechanics0000schw_w7l9 |publisher=Springer
 |isbn=3-540-41408-8
 |ref=harv
}}
*{{cite book
 |last1=Scully |first1=M. O.
 |last2=Zubairy |first2=M. S.
 |year=1997
 |title=Quantum Optics
 |publisher=Cambridge University Press
 |isbn=0-521-43458-0
 |ref=harv
}}
*{{cite journal
 |last1=Siegel |first1=D. M.
 |year=1976
 |title=Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to "Kirchhoff's radiation law"
 |journal=Isis
 |volume=67 |issue=4
 |pages=565–600
 |doi=10.1086/351669
 |ref=harv
}}
*{{cite book
 |last1=Siegel
 |first1=R.
 |last2=Howell
 |first2=J. R.
 |year=2002
 |title=Thermal Radiation Heat Transfer, Volume 1
 |url=https://books.google.com/books?id=O389yQ0-fecC&pg=PA25#v=onepage&q&f=false
 |edition=4th
 |publisher=Taylor & Francis
 |isbn=978-1-56032-839-1
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2020-05-20
 |archive-url=https://web.archive.org/web/20200520133151/https://books.google.com/books?id=O389yQ0-fecC&pg=PA25#v=onepage&q&f=false
 |dead-url=no
 }}
*{{cite book
 |last1=Sommerfeld
 |first1=A.
 |others=Brose, H. L. (transl.)
 |year=1923
 |title=Atomic Structure and Spectral Lines
 |url=https://books.google.com/books/about/Atomic_structure_and_spectral_lines.html?id=u1UmAAAAMAAJ
 |edition=from 3rd German
 |publisher=Methuen Publishing
 |ref=harv
 |access-date=2018-06-12
 |archive-date=2020-05-20
 |archive-url=https://web.archive.org/web/20200520133156/https://books.google.com/books/about/Atomic_structure_and_spectral_lines.html?id=u1UmAAAAMAAJ
 |dead-url=no
 }}
*{{cite book
 |last1=Stehle |first1=P.
 |year=1994
 |title=Order, Chaos, Order. The Transition from Classical to Quantum Physics
 |publisher=Oxford University Press
 |isbn=0-19-507513-7
 |ref=harv
}}
*{{cite journal
 |last1=Stewart |first1=B.
 |year=1858
 |title=An account of some experiments on radiant heat
 |journal=Transactions of the Royal Society of Edinburgh
 |volume=22 |pages=1–20
 |doi=
 |ref=harv
}}
*{{cite book
 |last1=ter Haar |first1=D.
 |year=1967
 |title=The Old Quantum Theory
 |url=https://archive.org/details/oldquantumtheory00haar |publisher=Pergamon Press
 |lccn=66-029628
}}
*{{Cite book
 |last1=Thornton |first1=S. T.
 |last2=Rex |first2=A. F.
 |year=2002
 |title=Modern Physics
 |publisher=Thomson Learning
 |isbn=0-03-006049-4
 |ref=harv
}}
*{{Cite book
 |last1=Tisza |first1=L.
 |year=1966
 |title=Generalized Thermodynamics
 |publisher=MIT Press
 |ref=harv
}}
*{{cite journal
 |last1=Tommasini |first1=D.
 |last2=Ferrando |first2=F.
 |last3=Michinel |first3=H.
 |last4=Seco |first4=M.
 |year=2008
 |title=Detecting photon-photon scattering in vacuum at exawatt lasers
 |journal=Physical Review A
 |volume=77 |issue=|page=042101
 |arxiv=quant-ph/0703076
 |bibcode=2008PhRvA..77a2101M
 |doi=10.1103/PhysRevA.77.012101
 |ref=harv
}}
*{{cite journal
 |last1=Tyndall |first1=J.
 |year=1865a
 |title=Über leuchtende und dunkle Strahlung
 |journal=Annalen der Physik und Chemie
 |volume=200
 |pages=36–53
 |bibcode=1865AnP...200...36T
 |doi=10.1002/andp.18652000103
 |ref=harv
}}
*{{cite book
 |last1=Tyndall |first1=J.
 |year=1865b
 |title=Heat considered as a Mode of Motion
 |url=https://archive.org/details/heatconsideredas00tyndrich
 |publisher=D. Appleton & Company
 |ref=harv
}}
*{{cite journal
 |last1=Wien |first1=W.
 |year=1896
 |title=Über die Energievertheilung im Emissionsspectrum eines schwarzen Körpers
 |journal=Annalen der Physik und Chemie
 |volume=294
 |issue=8
 |pages=662–669
 |bibcode=1896AnP...294..662W
 |doi=10.1002/andp.18962940803
 |ref=harv
}}
*{{Cite book
 |last1=Wilson |first1=A. H.
 |year=1957
 |title=Thermodynamics and Statistical Mechanics
 |url=https://archive.org/details/thermodynamicsst0000wils |publisher=Cambridge University Press
 |ref=harv
}}
{{refend}}

== 参见 ==
* [[黑体 (热力学)]]
* [[维恩位移定律]]
* [[维恩近似]]
* [[瑞利-金斯定律]]
* [[盒中氣體]]

== 延伸阅读 ==
* {{lang|en|{{cite book  | author=Peter C. Milonni  | year=1994  | title=The Quantum Vacuum  | publisher=Academic Press}}
}}

== 外部链接 ==
* {{lang|en|[http://topex.ucsd.edu/rs/radiation.pdf Summary of Radiation]}}
* {{lang|en|[http://www.vias.org/simulations/simusoft_blackbody.html Radiation of a Blackbody] - interactive simulation to play with Planck's law}}
* {{lang|en|[http://scienceworld.wolfram.com/physics/PlanckLaw.html Scienceworld entry on Planck's Law]}}

[[Category:统计力学|P]]
[[Category:量子力学|P]]
[[Category:物理定律|P]]