Showing Wien's Displacement Law from Wien's LawDoes anyone know how I would show that λ ∗ T is constant, using only Wien's Law? That ρ ( λ , T ) = 1 / λ 5 ∗ f ( λ T ) I differentiated, but all I could get was λ T = 5 f ( λ T ) / f ′ ( λ T ), which I don't think means it's necessarily a constant.

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Showing Wien&#039;s Displacement Law from Wien&#039;s LawDoes anyone know how I would show that   λ  ∗  T is constant, using only Wien&#039;s Law? That   ρ  (  λ  ,  T  )  =  1      /        λ    5    ∗  f  (  λ  T  ) I differentiated, but all I could get was   λ  T  =  5  f  (  λ  T  )      /        f    ′    (  λ  T  ), which I don&#039;t think means it&#039;s necessarily a constant.

Kosyging1j7u · Accepted Answer

You are essentially there. Wien&#039;s law constricts the form of the blackbody spectrum to  ρ  (  λ  ,  T  )  =            f      (      λ      T      )              λ      5        ,while Wien&#039;s displacement law talks about the peak of the spectrum at a given temperature. Thus you are correct that you need to set                                                   ∂          ρ                                                                  ∂          λ                      =  0, and that this leads to the equationThis equation will look a lot more friendly if you introduce some more notation. If you set   μ  =  λ  T, then you can reduce (1) to an equation that&#039;s exclusively in   μ:                    (2)                    5        f        (        μ        )        =        μ                  f          ′                (        μ        )        .            Depending on   f, this may have one, many, or no solutions, and each solution will mark a peak in the blackbody spectrum. Since Wien&#039;s law does not specify   f, we can&#039;t tell yet, but we expect that there will be a unique solution       μ    0   to (2). This implies that the peak wavelength       λ    peak   must obey      λ    peak    =            μ      0        T    .Note that here       μ    0   is a (dimensionful) constant that is determined by the final form of   f, but we know it can&#039;t depend on anything other than fundamental constants.It&#039;s also important to note that this fact is part of a stronger result which is what Wien&#039;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&#039;s Law to show spectral distruibution function of one temperature represents all temperatures.

sg101cp6vv · Accepted Answer

I think you need more info, specifically the form of   f:  f  =  a      e          −      b              /            (      λ      T      )      (Although not defined by Wien, this form of   f is now conventionally understood to be part of Wien&#039;s law.)Update: With thanks to Emilio Pisanty and Pulsar, I see that a more general formulation is possible: if   f is such that   ρ has a maximum at       λ    1   at temperature       T    1  , the scaling argument shows that the displacement law is valid. The form I initially used, from Wien&#039;s approximation, is only a specific case.

Showing Wien's Displacement Law from Wien's Law Does anyone know how I would show that &#x03BB

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