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Exact Uncertainty

Heisenberg’s Uncertainty Principle is the basis for the Copenhagen interpretation of quantum mechanics (Neils Bohr), and is often one of the best distinguishing characteristics between classical and quantum physics.  The implied degree of fuzziness of the uncertainty provides a degree of freedom from the limitations of classical physics, where probabilities and inequalities are generally frowned upon.  The difficulty however is that the uncertainty also forms a gap between the classical and quantum worlds, where one cannot be derived from the other.  

This gap has been bridged to some extent by Hall and Reginatto [1], who have shown that “an exact form of the uncertainty principle may in fact be formulated, which provides the single key element in moving from the equations of motion of a classical ensemble to those of a quantum ensemble.”  By assuming that a classical ensemble is subject to random momentum fluctuations and the intensity of fluctuation is precisely determined inversely by the uncertainty in position [a very significant assumption!], “then the resulting modified equations of motion are equivalent to the Schroedinger equation.”  [The latter being one of the fundamentals of Quantum Physics.]  

[Of particular note is the assumption of “ensembles”, as opposed to individual elementary particles.  This provides for a statistical approach -- which may or may not be an accurate description of the universe when mere probabilities are not appropriate.]  

Nevertheless, Samuel [2] has described the work as “an improved version of the famous Heisenberg uncertainty principle.”  The latter, of course, is where both the position and momentum of any particle could not be measured simultaneously with perfect accuracy.  But how big the uncertainty might actually be was never clear.  

Hall and Reginatto [1] assumed that the physical momentum was composed of two parts: one an average momentum of the ensemble, and one a fluctuation about this average.  The latter was regarded as “fundamentally nonanalysable, being introduced as a simple device to remove the notion of individual particle trajectories.” [emphasis added]   

This led to two additional assumptions: 1) that the fluctuations were unbiased, and 2) the fluctuations were linearly uncorrelated with the average momentum.”  The rationale goes on to state that an exact uncertainty principle is equivalent to stating that the momentum fluctuation DN is determined by the uncertainty in position, where the latter is characterized by a probability density, P.  The one dimensional example of the latter is:  

dx  =  9 m P [ (1/P) (dP/dx) ]2 dx @ -1/2

which, in turn, results in:

dx DN  =  h / 4p  

Inasmuch as the Cramer-Rao inequality of statistical estimation theory implies Dx ³ dx, and the randomness assumptions in the development implies that (Dp)  ³  (DN)2 then clearly,

Dx Dp  ³ h / 4p  

Hall notes that the equation is "a far stronger relation" than the inequality.  In fact, it is the equation which allows for the derivation of the basics of quantum mechanics; including the Schrödinger equation that describes the behavior of quantum-mechanical wave functions.  

According to Samuel [2], "‘I find it remarkable that the Schrödinger equation no longer has to be God-given,’ says Wolfgang Schleich, who studies the foundations of quantum mechanics at the University of Ulm.  You still have to make the assumption that there is some quantum uncertainty, but this is much simpler than [merely] assuming Schrödinger's equation.”  Fewer Assumptions tend to be far more preferable in any theory.  

“Hall says it implies a tight relationship between uncertainty and energy that makes it easier to understand why, in quantum mechanics, systems have a minimum kinetic energy even if there aren't any forces acting. ‘There's a kind of quantum kinetic energy that comes from the uncertainty,’ he says.” [2]  [emphasis added]  


Connective Physics         Heisenberg’s Uncertainty Principle

Forward to:

Wave-Particle Duality         Quantum Knowing



[1]  Hall, M.J.W. and Reginatto, M., “Schroedinger equation from an exact uncertainty principle,” Journal of Physics A, Vol 35, April 12, 2002, pages 3289-3303.

[2]  Samuel, E., “Exact Uncertainty brought to quantum world,” New Scientist -- available at < http://www.newscientist.com/news/news.jsp?id=ns99992209>.



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