maths bonus question: what are the asymptotes of $\exp(\tanh(x))$ for $x\rightarrow\pm\infty$ and what is the slope at $x=0$?

physics bonus question: can you derive the energy-time-uncertainty from the momentum-position-uncertainty in quantum mechanics?

physics bonus question: the energy-time uncertainty is merely a consequence of the momentum-position uncertainty and *not* a genuine relation in itself... the reason being that there is no time-operator in classical quantum mechanics as time is a global parameter. if you use the dispersion relation $E=p^2/(2m)$ and the velocity $\upsilon = p/m$ for a particle of mass m, you can write $\Delta E=\partial E/\partial p\times\Delta p = p/m\times\Delta p$ and $\Delta t = \Delta x / \upsilon = m\Delta x / p$, and collecting the results yields:

ReplyDelete\begin{equation}

\Delta E\Delta t = \frac{p}{m}\Delta p\:\frac{m}{p}\Delta x = \Delta p\Delta x\simeq \hbar

\end{equation}

main question: one could notice energy input into the CMB: if that energy is deposited before decoupling, it gives rise to a nonzero chemical potential leading to a different blackbody shape, the so-called $\mu$-distortion. energy input after decoupling does not conserve the Planck spectrum but gives rise to $y$-distortions like in the thermal Sunyaev-Zel'dovich effect, due to incomplete thermalisation.

ReplyDeletemaths bonus question: the asymptotes are $\mathrm{e}$ for $x\rightarrow+\infty$ and $1/\mathrm{e}$ for $x\rightarrow-\infty$. the slope at the origin is $2/\mathrm{e}$.

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