Commutation relations characterized

The interaction Hamiltonian is decomposed into a sum of terms characterized by their commutation relations with Sz ... 

Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations... 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

This observation has importance when we take into account the irreversibility. Due to irreversibility, the damped oscillator proceeds to thermal equilibrium with the thermal bath. This thermal equilibrium can be characterized in terms of classical statistic theory. However, in classical statistics, random variables have a joint distribution function, which could exist in the case of quantum theory if the operators are compatible. The commutator relation (Equation (100)) is compatible this physical picture, but from Equations (100) and (101), we obtain... 

In summary, on the one hand, classical mechanics was able to presume that the constructive properties were attributes of matter even if the experiments that were necessary for their determination were not accepted. On the other hand, in quantum physics, this was no longer possible due to the limitation of Heisenberg s indeterminacy relation, for any couple and conjugated variables. Weyl accepted it as a fundamental insight, different from Heisenberg s mathematical characterization of the commutation relation. In the case of electrochemistry and electrocatalysis, the fundamentals of... 

Tensors (15.39)—(15.41) meet commutation relations (14.2) for irreducible components of the momentum operator, and, in addition, they commute with the operators of orbital (14.15) and spin (14.16) angular momenta for the lN configuration, since they are scalars in their respective spaces. Accordingly, the states of the lN configuration can be characterized by the eigenvalues of operators L2, Lz, S2, Sz, Q2, Qz. 

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