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?

3 comments:

  1. bonus question: you can write down the variation of the action $S=\int\mathrm{d}t\:\mathcal{L}$ to second order,
    \begin{equation}
    \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)
    \end{equation}
    which can be rewritten as a bilinear form,
    \begin{equation}
    \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)
    \end{equation}
    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 $\Phi\propto x^2$.

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  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.

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  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... :)

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