Showing Wien's Displacement Law from Wien's Law

Question

Does anyone know how I would show that $\lambda * T$ is constant, using only Wien's Law? That $\rho(\lambda,T) = 1/\lambda^5*f(\lambda T)$ I differentiated, but all I could get was $\lambda T = 5f(\lambda T)/f'(\lambda T)$, which I don't think means it's necessarily a constant.

Answer 1

You are essentially there. Wien's law constricts the form of the blackbody spectrum to $$\rho(\lambda,T)=\frac{f(\lambda T)}{\lambda^5},$$ while Wien's displacement law talks about the peak of the spectrum at a given temperature. Thus you are correct that you need to set $\cfrac{\partial \rho}{\partial\lambda}=0$, and that this leads to the equation $$5 f(\lambda T)=\lambda Tf'(\lambda T).\tag 1$$ This equation will look a lot more friendly if you introduce some more notation. If you set $\mu=\lambda T$, then you can reduce (1) to an equation that's exclusively in $\mu$: $$5 f(\mu)= \mu f'(\mu).\tag 2$$ Depending on $f$, this may have one, many, or no solutions, and each solution will mark a peak in the blackbody spectrum. Since Wien's law does not specify $f$, we can't tell yet, but we expect that there will be a unique solution $\mu_0$ to (2). This implies that the peak wavelength $\lambda_\text{peak}$ must obey $$\lambda_\text{peak}=\frac{\mu_0}T.$$ Note that here $\mu_0$ is a (dimensionful) constant that is determined by the final form of $f$, but we know it can't depend on anything other than fundamental constants.

It's also important to note that this fact is part of a stronger result which is what Wien's law really embodies: a change in temperature can only make a scaling transformation on the corresponding blackbody spectrum. This is beautifully explored in Using Wien's Law to show spectral distruibution function of one temperature represents all temperatures.

Answer 2

I think you need more info, specifically the form of $f$:

$$ f = a e^{-b/(\lambda T)} $$

(Although not defined by Wien, this form of $f$ is now conventionally understood to be part of Wien's law.)

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Update: With thanks to Emilio Pisanty and Pulsar, I see that a more general formulation is possible: if $f$ is such that $\rho$ has a maximum at $\lambda_1$ at temperature $T_1$, the scaling argument here shows that the displacement law is valid. The form I initially used, from Wien's approximation, is only a specific case.