Mathtest

How about $

This is test for typing (and correction) of formulas, while they do not appear as follows.

Q2. The evolution of state with time is determined by the Hamiltonian \\[\hat H\\]of the system; the Hamiltonian is Hermitian operator acting on the elements of the linear space mentioned in the Axiom Q1; in one of representations, the evolution of the state \\[\psi\\]obeys the Schroedinger equation
 * $$\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \mathrm i ~\hbar~ \partial \psi /\partial t = \hat H ~\psi\$$

where time \\[t\\]is real-valued parameter, and \\[\hbar\\]is universal constant, called the Planck constant.

Q3. The measurable quantities of classical mechanics correspond to the linear Hemitian operators acting on the space of the states; the Hamiltonian corresponds to the classical energy.

Q4. The results of measurements of observable quantity \\[\hat F\\]can be interpreted in terms of classical mechanics in the following way: the mean value of the quantity \\[\hat F\\]measured at the state \\[\psi\\]is expressed with
 * $$\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~ \langle~ \hat F~ \rangle = \psi ^{\dagger} ~\hat F~ \psi\$$

Q5. If \\[\hat x\\]corresponds to the classical coordinate, and \\[\hat p\\]corresponds to the classical momentum, then
 * $$\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ \hat p ~\hat x ~–~ \hat x~ \hat p = - \mathrm i ~ \hbar\$$

As it is common in linear algebra, the sign of operation of association is omitted, and such association is called "product of operators". In particular, \\[\hat p^2(\psi)\\]means $$\hat p ( \hat p (\psi))\$$, but not $$\hat p(\psi)^2\$$. In general, in TORI, the superscript after the name of operation means the iteration, not a power function of the result of operation .

History and interpretation of Quantum mechanics
The need of Quantum Mechanics appeared after the hypothesis of the Nobel laureate A.Einstein about the quantum character of emission and absorption of light. The parameters that determine these processes are callet the Einstein coefficients.

The Einstein's hypothesis gave good explanation of the Aleksandr Stoletov's observation of the photo effect. However, such a hypothesis is not compatible with basic interpretation of a coordinate, momentum and trajectory, used in the classical Newtonian theory.

Problems of Quantum Mechanics in the USSR
The vector of state is not observable, this contradicts the basic materialistic postulate of marxism about absence of non–measurable quantities. For this reason, in the USSR, since the beginning, the Quantum Mechanics was declared as preudo–science; this had catastrophic sequences for the researchers who worked in this area. The "rehapilitation" of quantum mechanics begun only in late 1930s, when it became clear, that bolshevicks cannot make a nuclear bomb without formalism of quantum mechanics; many physicists were withdrawn from the regular concentration to the special "closed cities", where the prisoners were forced to work on the scientific and military projects. As the Soviet archives are closed and falsified, various scientific hypothesis about that epoch should be considered.

Interpretation of quantum mechanics
In Centiry 20, it was popular to "interpret" the quantum mechanics as some probabilistic approach.

Later, the appearance of the quantum standard of frequency, showed that namely probabilistic theory is completely deterministic, and the probability appears only at the moment the results of the quantum consideration are interpreted in terms of classical mechanics. There are no divergent trajectories in quantum mechanics; in principle, the evolution of an isolated quantum system can be predicted for arbitrary period of time.

In classical mechanics, contrary, the trajectories with positive Lyapunov's increment exist, and the prediction of the evolution requres exponential decrease of errors of determination of the initial state. Soon such improvement of precision meets the limits of applicability of the Classical theory. In this sense, the Classical mechanics is intrinsically probabilistic, while the quantum mechanics is intrinsically deterministic.

The misunderstanding of the deterministic character of the quantum mechanics and the probapilistic character of the classical mechanics caused a lot of paradoxes. Practically, every quantum phenomenon, that has no classical analogy, had been qualified as a "paradox", and these paradoxes were considered as a proof of inconsistency of quantum mechanics. However, that time, the axioms of TORI were not yet elaborated, and the crytics od Quantum Mechanics was considered as "sceincific". Practically, and especially in the USSR, such a critics was only a pretext for the physical elimination of opponents and political competitors.

In century 21, it is recognized, that the interpretation of quantum mechanics has no scientific meaning and can be qualified as pseudo-science. According to the Axioms of TORI the quantum mechanics is considered as self–consistent theory. The internal contradictions or the contradictions with experiments could be used to limits its range of applicability, but the contradictions with the classical concepts (so called "paradoxes") cannot be used as an argument against the Quantum mechanics.