Relationship between Wien's law and Stefan-Boltzmann's law

Question

I'm studying Quantum mechanics by Bransden and Joachain and in the introduction chapter it says:

Wien showed that the spectral distribution function had to be on the form

$$\rho(\lambda,T)=\lambda^{-5}f(\lambda T)$$

where $f(\lambda T)$ is a function of the single variable $\lambda T$. It is a simple matter to show that Wien's law includes Stefan-Boltzmann law $R(T)=\sigma T^4$.

One of the exercises is to show this and I cannot understand how to.

---

This is what I've tried:

The relationship between spectral emittance and spectral distribution is

$$\rho(\lambda,T)=\frac{4}{c}R(\lambda,T),$$ where c is the speed of light, which inserted in the above equation gives

$$R(\lambda,T)=\frac{c}{4}\lambda^{-5}f(\lambda T).$$

Now, the total spectral emittance is the integral of $R$ over all wavelengths so

$$ R(T)=\frac{c}{4}\int\limits_0^\infty \lambda^{-5}f(\lambda T)d\lambda $$

This is where I'm stuck. Can anyone help me figure this out?

Answer 1

The solution to your problem⁽¹⁾ is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's Law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's Displacement Law), see Emilio Pisanty answer therein : Showing Wien's Displacement Law from Wien's Law.

---

^{(1) "Quantum Mechanics" B. H. Bransden-C. J. Joachain, 2nd Edition 2000, Pearson Education Limited (Problem 1.3, page 45)}