By Mohsen Razavy
This e-book discusses concerns linked to the quantum mechanical formula of dissipative platforms. It starts off with an introductory evaluate of phenomenological damping forces, and the development of the Lagrangian and Hamiltonian for the damped movement. it really is proven, as well as those tools, that classical dissipative forces is additionally derived from solvable many-body difficulties. an in depth dialogue of those derived forces and their dependence on dynamical variables can also be offered. the second one a part of this booklet investigates using classical formula within the quantization of dynamical platforms less than the effect of dissipative forces. the implications convey that, whereas a passable technique to the matter can't be stumbled on, diversified formulations characterize diverse approximations to the entire resolution of 2 interacting platforms. The 3rd and ultimate a part of the publication makes a speciality of the matter of dissipation in interacting quantum mechanical platforms, in addition to the relationship of a few of those versions to their classical opposite numbers. a couple of vital purposes, akin to the idea of heavy-ion scattering and the movement of a radiating electron, also are mentioned.
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Additional resources for Classical and Quantum Dissipative Systems
112) Bibliography     P. Cardirola, Nuovo Cimento 18, 393 (1941). E. Kanai, Prog. Theor. Phys. 3, 440 (1948). P. Havas, Nuovo Cimento Supp. 5, 363 (1957). H. Goldstein, Classical Mechanics, Second Edition (Addison-Wesley Reading, 1980) Chapter 8. M. Santilli, Foundations of Theoretical Mechanics, vol. 1 (SpringerVerlag, New York, 1978).  M. Razavy, Z. Phys. B26, 201 (1977). A. Kobussen, Acta Phys. Austriaca, 59, 293 (1979).  P. Caldirola, Rend. 1st. Lomb. Sc, A 93, 439 (1959).
85) where Ei(y) -dz. 86) is the exponential integral function . The position of the rocket as a function of time can be found by integrating v(t) with respect to t. 68) with fi(t) = ext. e. j ^ , Eq. 51), must vanish. Now if we multiply L by a constant number, a, and add the total time derivative of a function of Xi and t to it we find a new Lagrangian X, dg(xj,t) dt ' aL (3. e. j ^ = 0 gives us the same equation of motion as Eq. 51). However the Lagrangian L is not the most general Lagrangian for the motion.
23) 18 Classical Dissipative Motion T as the sum of the quadratic terms in velocities, 1 3 2 ! C i<-M*J'*J' ( 3 - 24 ) and in addition we can have a "gyroscopic" term, 3 1 — y ^ XiGijXj. 25) Here the matrix elements M^ and Vij are real and symmetrical, while Gy s are asymmetrical, but all are constants. 2,3, (3-26) where B^ = Rij + Gij. 27) The theory of damped heavy ion scattering offers an interesting example of the application of the above-mentioned Lagrangian formulation. While we expect that in the domain of nuclear scattering the problem has to be treated quantum mechanically, but under the conditions where semi-classical approximations such as WKB are valid, we can justify a classical description for the motion of the nuclei.
Classical and Quantum Dissipative Systems by Mohsen Razavy