Quantum Gravity 396.3: How Heisenberg's Uncertainty Principle (HUP) and Born's Probabilistic Interpretation Directly Involve Probability
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From: Osher Doctorow on 21 May 2010 01:43 From Osher Doctorow Most readers are aware that Born's Probabilistic Interpretation: 1) ww* = P(finding particle in a volume of space) where w = Schrodinger's wave function is a Probability, but the same is true of Heisenberg's Uncertainty Principle (HUP), for the simple reason that: 2) "Uncertainty" is measured as a variance or standard deviation, which does not exist without a Probability Distribution - in simple language, without a Probability. Regarding (2), an introductory college course in Probability makes explicit the fact that Random Variables have Probabilities associated with them, and that the Probabilities in turn when graphed have Measures of Central Tendency (Means, Medians, and/or Modes) and Measures of Variation (Standard Deviations, Variances, or similar measures). It might be argued that for continuous Random Variables, Probability Densities are not the same as Probabilities because they sometimes exceed 1, but continuous Random Variables are equally characterized by Cumulative Distribution Functions (cdfs) which ARE Probabilities because the cdf is FX(x) = P(X < = x) for random variable X and value x. Thus, the problem of Conditional Probability and Probable Causation/ Influence (PI) yielding often "opposite" results is immediately a difficulty in anything that involves either Born's Probabilistic Interpretation or Heisenberg's HUP. This provides further indications for the claims in the last few posts that Quantum Mechanics is plagued by defects and involves hidden assumptions (and ignoring alternatives!) not specified overtly in its Axioms and definitions. Osher Doctorow |