Wednesday, April 16, 2014

Wien displacements in many dimensions

can you show that the Wien displacement law for frequencies for a Planck-spectrum in $n$ dimensions is identical to the displacement law for wave lengths in $n+2$ dimensions?

1 comment:

  1. the Planck-law in $n$ dimension for the frequency $\nu$ is
    \begin{equation}
    S(\nu) = \frac{\nu^n}{\exp(\nu)-1}
    \end{equation}
    which yields for the Wien-displacement law $\mathrm{d}S/\mathrm{d}\nu=0$ the relation
    \begin{equation}
    n = \frac{\nu\exp(\nu)}{\exp(\nu)-1}
    \end{equation}
    whereas the representation in terms of wave length $\lambda$ requires the Jacobian for keeping the normalisation conserved: $\nu=1/\lambda\rightarrow\mathrm{d}\nu = -\mathrm{d}\lambda/\lambda^2$:
    \begin{equation}
    S(\lambda) = \frac{1}{\lambda^{2+n}}\frac{1}{\exp(1/\lambda)-1}.
    \end{equation}
    looking for the maximum $\mathrm{d}S/\mathrm{d}\lambda=0$ yields the relation
    \begin{equation}
    2+n = \frac{1}{\lambda}\frac{\exp(1/\lambda)}{\exp(1/\lambda)-1}
    \end{equation}
    which corresponds to the previous relation by substitution $x=1/\lambda$, albeit with the dimension being larger by two.

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