Blackbody Radiation

The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency. But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Nature knew better. 但预测的随着频率增加而持续增加的辐射能量（被称为‘紫外灾难’）并未发生。自然更了解真相。 在给定的频率范围内，发射的辐射量应与该范围内模式数成正比。经典物理的最佳解释认为，所有模式都有相等的可能性被产生，且模式数与频率的平方成正比。

Cavity Modes

Planck Radiation Formula

From the assumption that the electromagnetic modes in a cavity were quantized in energy with the quantum energy equal to Planck's constant times the frequency, Planck derived a radiation formula. The average energy per "mode" or "quantum" is the energy of the quantum times the probability that it will be occupied (the Einstein-Bose distribution function):

This average energy times the density of such states, expressed in terms of either frequency or wavelength

gives the energy density, the Planck radiation formula.

	Example 示例
	Example 示例

The Planck radiation formula is an example of the distribution of energy according to Bose-Einstein statistics. The above expressions are obtained by multiplying the density of states in terms of frequency or wavelength times the photon energy times the Bose-Einstein distribution function with normalization constant A=1.

To find the radiated power per unit area from a surface at this temperature, multiply the energy density by c/4. The density above is for thermal equilibrium, so setting inward=outward gives a factor of 1/2 for the radiated power outward. Then one must average over all angles, which gives another factor of 1/2 for the angular dependence which is the square of the cosine.

Rayleigh-Jeans vs Planck

Blackbody Intensity as a Function of Frequency

The Rayleigh-Jeans curve agrees with the Planck radiation formula for long wavelengths, low frequencies. 瑞利-金斯曲线在长波长、低频段与普朗克辐射公式一致。 Show 显示

Comments on the Development of the Rayleigh-Jeans Law

Equilibrium standing wave electromagnetic radiation in a cubical cavity of dimension L must meet the condition: 立方腔体边长为L的平衡驻波电磁辐射必须满足条件：		Show 显示
The number of modes in the cavity is: 腔中的模式数为：		Show 显示
The number of modes per unit wavelength is: 单位波长内的模式数是：		Show 显示
The energy per unit volume per unit wavelength is: 单位体积单位波长的能量是:		Show 显示
The average radiated energy per unit wavelength is: 单位波长的平均辐射能量是：		Show 显示
Which when expressed in terms of frequency is: 用频率表示时为：		Show 显示