## Wednesday, December 4, 2013

### quantum microwave background

CQW celebrates its second anniversary!

would a classical description of the microwave background today be applicable? please justify your answer using the thermal wavelength.

bonus question: motion extremises the action in classical Lagrangian mechanics. is that extremum a maximum or a minimum?

1. bonus question: you can write down the variation of the action $S=\int\mathrm{d}t\:\mathcal{L}$ to second order,

\delta^2S=
\frac{1}{2}\int\mathrm{d}t\:
\left(
\frac{\partial^2\mathcal{L}}{\partial x^2}(\delta x)^2 +
2\frac{\partial^2\mathcal{L}}{\partial x\partial\dot{x}}\delta x\delta\dot{x} +
\frac{\partial^2\mathcal{L}}{\partial\dot{x}^2}(\delta\dot{x})^2
\right)

which can be rewritten as a bilinear form,

\delta^2S=
\frac{1}{2}\int\mathrm{d}t\:
\left(
\begin{array}{c}
\delta x\\
\delta\dot{x}
\end{array}
\right)^t
\left(
\begin{array}{cc}
m & 0 \\
0 & -\mathrm{d}^2\Phi/\mathrm{d}x^2
\end{array}
\right)
\left(
\begin{array}{c}
\delta x \\
\delta\dot{x}
\end{array}
\right)

for a standard Lagrange function of the form $\mathcal{L}=m\dot{x}^2/2-\Phi$. this bilinear form is positive if all sub-determinants is positive, yielding the condition $\mathrm{d}^2\Phi/\mathrm{d}x^2<0$ for positive definiteness and for $\delta^2S$ to be a minimum, i.e. the action is maximised for bound systems, for example the harmonic potential $\Phi\propto x^2$.

2. let's have a look at the thermal wavelength: setting $\epsilon=cp=c\hbar k=\hbar\omega$ for photons as ultrarelativistic particles equal to $k_BT/2$ assuming equilibration yields $\omega=k_BT/2\hbar$. this typical thermal angular frequency can be compared to the position of the maximum of the Planck-spectrum: $\mathrm{d}S(\omega)/\mathrm{d}\omega=0$ gives the equation $(3-x)\exp(x)=3$, which can be solved numericall for $x\simeq2.82$, i.e. $\omega_m=2.82k_BT/\hbar$, roughly a factor $5$ larger than the thermal angular frequency. from that one can conclude that right of the maximum the Planck-spectrum is dominated by thermal occupation statistics and that left of the maximum the system is a quantum system of low energy photons.

3. CQW likes to add that somehow it seems very weird that objects as large, hot and energetic as the Sun are dominated by quantum effects, at least in the low-energy part of their spectrum... :)