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The Certainty Principle (review) Historical Comments

The certainty principle (2005) allowed to conceptualize from the more fundamental grounds both the Heisen-berg uncertainty principle (1927) and the Mandelshtam-Tamm relation (1945). In this review I give detailed explanation and discussion of the certainty principle, oriented to all physicists, both theorists and experimenters. Uncertainty principle. A reader interested in the history of the uncertainty principle should read, for example, the review [1]. Here I give only some notes, that are important for the following recital. The uncertainty principle was suggested by Heisenberg in 1927 [2]. Heisenberg formulated it as follows: • The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. For uncertainties of coordinate and momentum the following relation was suggested: ∆ X ∆ P ∼. (1) Considering concrete examples, Heisenberg gave just qualitative formulation. But for the notion of " uncertainty " he did not give exact definition. Soon Kennard [3] gave exact mathematical formulation for the case of coordinate and momentum. Assuming " commutation relation " [X, P ] = i , he has shown that for an arbitrary quantum state the following relation takes place: ∆ X ∆ P 2. (2) And here It means that, in accordance with probabilistic interpretation of quantum mechanics, it was suggested to understand the uncertainties of coordinate and momentum as standard deviations of these observables. Because of simplicity of the proof, the relation (2) became conventional mathematical expression of the uncertainty principle and appeared in all textbooks on quantum mechanics. All criticism (including that about correspondence to practical experiment) [1] was mainly ignored. But soon we will see that the certainty principle allows to get alternate inequalities, describing the uncertainty principle, which turn out to be more sapid from practical point of view. Later the mathematical proof was generalized in many ways. And it was extended to other pairs of non-commuting observables. So, in order to avoid ambiguity in the term uncertainty principle, let us give here some other, more up-to-date and general, formulation of this principle 1 Such a formulation not only allows to better understand, what exactly is " uncertain " , but also allows to clearly see the contrast with the certainty principle. 1 • If one tries to describe the dynamical state of a quantum particle by methods of classical mechanics, then precision of such description is limited in principle. The classical state of the …