söndag 16 mars 2014

Uncertainty Principle, Whispering and Looking at a Faint Star

The recent series of posts on Heisenberg's Uncertainty Principle based on Computational Blackbody Radiation suggests the following alternative equivalent formulations of the principle:
  1. $\nu < \frac{T}{\hat h}$,
  2. $u_\nu\dot u_\nu > \hat h$,
where $u_\nu$ is position amplitude, $\dot u_\nu =\nu u_\nu$ is velocity amplitude of a wave of frequency $\nu$ with $\dot u_\nu^2 =T$, and $\hat h =4.8\times 10^{-11}Ks$ is Planck's constant scaled with Boltzmann's constant.

Here, 1 represents Wien's displacement law stating that the radiation from a body is subject to a frequency limit scaling with temperature $T$ with the factor $\frac{1}{\hat h}$.

2 is superficially similar to Heisenberg's Uncertainty Principle as an expression of the following physics: In order to detect a wave of amplitude $u$, it is necessary that the frequency $\nu$ of the wave satisfies $\nu u^2>h$. In particular, if the amplitude $u$ is small, then the frequency $\nu$ must be large.

This connects to (i) communication by whispering and (ii) viewing a distant star, both being based on the possibility of detecting small amplitude high-frequency waves.

The standard presentation of Heisenberg's Uncertainty Principle is loaded with contradictions:
  • The uncertainty principle is certainly one of the most famous and important aspects of quantum mechanics. 
  • But what is the exact meaning of this principle, and indeed, is it really a principle of quantum mechanics? And, in particular, what does it mean to say that a quantity is determined only up to some uncertainty? 
  • So the question may be asked what alternative views of the uncertainty relations are still viable.
  • Of course, this problem is intimately connected with that of the interpretation of the wave function, and hence of quantum mechanics as a whole. 
  • Since there is no consensus about the latter, one cannot expect consensus about the interpretation of the uncertainty relations either. 
In other words, today there is no consensus on the meaning of Heisenberg's Uncertainty principle. The reason may be that it has no meaning, but that there is an alternative which is meaningful.

Notice in particular that the product of two complementary or conjugate variables such as position and momentum is questionable if viewed as representing a physical quantity, while as threshold it can make sense.

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