On some peculiarities of quantum mechanics

Abstract

General regularities related toLagrangian andHamiltonian equations are revealed. Probability distributions for functions ofHamiltonian random variables are considered. It is shown that all probability distributions of this kind are fully determined by the probability distributions for the random variables satisfying the corresponding Lagrangian equations. Some formulas related tocanonically conjugate operators are given. The similarity of these formulas to those related to Hamiltonian random variables is demonstrated. The “quantum approach” to the treatment of Hamiltonian random variables is discussed, and the origin of some peculiarities related to this approach is elucidated; it is explained, in particular, why it is impossible to form the joint probability density for canonically conjugate random variables when using this approach. The peculiarities revealed prove to be common for any objects possessing Hamiltonian random variables, irrespective of the nature of the objects, and coincide, therefore, with those in quantum mechanics. The existence of joint probability distributions for canonically conjugate random variables in the general case is demonstrated through the calculation of the corresponding joint mathematical expectations in an illustrative example. This proves, in particular, that joint probability distributions for canonically conjugate coordinates and momenta exist indeed in the case of mechanical microsystems. The results obtained prove once again that the pecularities of quantum mechanics are not related to the specificity of the measurements of physical quantities for microsystems.