Planck's Derivation of Quantization: Summation vs Integrand

[ENDLESSYOU]

When Planck first derived the concept of quantization, he treated the integrand for average energy =$\int_{0}^{\infty} \epsilon*P(\epsilon) d\mu$ , where $P(\epsilon)$ is the Boltzmann distribution as a summation nh\mu, and derived the Planck law. While when we use it to derived the Stefan-Boltzmann law, we integrate the variable \mu. I'm puzzled about why we use integrand here. It just like we treat frequency to be continuous in the Stefan-Boltzmann law.( But I do a summation here and find that the summation for the Stefan-Boltzmann law is almost the same as what we obtained by integrating. )

 

[Fredrik]

Try ## instead of $. More information (about LaTeX at PF) here.

 

[ENDLESSYOU]

When Planck first derived the concept of quantization, he treated the integrand for average energy \bar{\varepsilon}=\int_{0}^{\infty} \varepsilon P(\varepsilon) d\varepsilon , where P(\varepsilon) is the Boltzmann distribution as a summation nh \nu, and derived the Planck law. While when we use it to derived the Stefan-Boltzmann law, we integrate the variable \nu. I'm puzzled about why we use integrand here. It just like we treat frequency to be continuous in the Stefan-Boltzmann law.( But I do a summation here and find that the summation for the Stefan-Boltzmann law is almost the same as what we obtained by integrating. )