måndag 17 mars 2014

New Uncertainty Principle as Wien's Displacement Law

The recent series of posts based on Computational Blackbody Radiation suggest that Heisenberg's Uncertainty Principle can be understood as a consequence of Wien's Displacement Law expressing high-frequency cut-off in blackbody radiation scaling with temperature according to Planck's radiation law:
• $B_\nu (T)=\gamma\nu^2T\times \theta(\nu ,T)$,
where $B_\nu (T)$ is radiated energy per unit frequency, surface area, viewing angle and second, $\gamma =\frac{2k}{c^2}$ where $k = 1.3806488\times 10^{-23} m^2 kg/s^2 K$ is Boltzmann's constant and $c$ the speed of light in $m/s$, $T$ is temperature in Kelvin $K$,
• $\theta (\nu ,T)=\frac{\alpha}{e^\alpha -1}$,
• $\alpha=\frac{h\nu}{kT}$,
where $\theta (\nu ,T)\approx 1$ for $\alpha < 1$ and $\theta (\nu ,T)\approx 0$ for $\alpha > 10$ as high frequency cut-off with $h=6.626\times 10^{-34}\, Js$ Planck's constant. More precisely, maximal radiance for a given temperature occurs $T$ for $\alpha \approx 2.821$ with corresponding frequency
• $\nu_{max} = 2.821\frac{T}{\hat h}$ where $\hat h=\frac{h}{k}=4.8\times 10^{-11}\, Ks$,
with a rapid drop for $\nu >\nu_{max}$.

The proof of Planck's Law in Computational Blackbody Radiation explains the high frequency cut-off as a consequence of finite precision computation introducing a dissipative effect damping high-frequencies.

A connection to Heisenbergs Uncertainty Principle can be made by noting that a high-frequency cut-off condition of the form
• $\nu < \frac{T}{\hat h}$,
can be rephrased in the following form connecting to Heisenberg's Uncertainty Principle:
• $u_\nu\dot u_\nu > \hat h$                  (New Uncertainty Principle)
where $u_\nu$ is position amplitude, $\dot u_\nu =\nu u_\nu$ is velocity amplitude of a wave of frequency $\nu$ with $\dot u_\nu^2 =T$.

The New Uncertainty Principle expresses that observation/detection of a wave, that is observation/detection of amplitude $u$ and frequency $\nu =\frac{\dot u}{u}$ of a wave, requires
• $u\dot u>\hat h$.
The New Uncertainty Principle concerns observation/detection amplitude and frequency as physical aspects of wave motion, and not as Heisenberg's Uncertainty Principle particle position and wave frequency as unphysical complementary aspects.