By Mohsen Razavy

This e-book discusses matters linked to the quantum mechanical formula of dissipative platforms. It starts with an introductory evaluation of phenomenological damping forces, and the development of the Lagrangian and Hamiltonian for the damped movement. it really is proven, as well as those equipment, that classical dissipative forces is also derived from solvable many-body difficulties. a close dialogue of those derived forces and their dependence on dynamical variables is additionally provided. the second one a part of this ebook investigates using classical formula within the quantization of dynamical structures less than the impression of dissipative forces. the consequences exhibit that, whereas a passable strategy to the matter can't be discovered, various formulations signify various approximations to the entire resolution of 2 interacting structures. The 3rd and ultimate a part of the publication specializes in the matter of dissipation in interacting quantum mechanical structures, in addition to the relationship of a few of those versions to their classical opposite numbers. a couple of very important purposes, equivalent to the idea of heavy-ion scattering and the movement of a radiating electron, also are mentioned.

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**Extra resources for Classical and Quantum Dissipative Systems **

**Example text**

For this group of Hamiltonian 37 Formulation problems it is more difficult to construct gp-equivalent Hamiltonians (see below). e. a force proportional to x(t), Eq. 15), we can find explicitly time-dependent Lagrangian and Hamiltonian or time-independent Hamiltonian. Let us consider the Lagrangian function given by Cardirola [8] and by Havas [9] (see Eq. 21) where x(t) is the velocity of the particle and f(x) is the conservative force acting on it. e. 15). \ =mxext. 23); H(x,p, t) = (px - L)i=i{pt) = | ^ e - A t - ext f f (*') dx'.

23); H(x,p, t) = (px - L)i=i{pt) = | ^ e - A t - ext f f (*') dx'. 25) For these systems the Hamiltonian function is not unique. 25) is: H2(x,p) = — I — I xp — \ncos(upx) + lnx, (4-26) where cu2 = WQ — ^-. This Hamiltonian corresponds to the Lagrangian given by Eq. 111). Let us discuss some of the similarities and differences of the Hamiltonians Hi(x,p,t) and H2(x,p). 27) 38 Classical Dissipative Motion but if we eliminate x and x between the Hamilton canonical equations p=- — , a n d * = ^ > (4-28) then we find different equations for p.

A. Bromley, (Plenum Press, New York, 1984) p. 138. R. R. N. De and D. Sperber, Phys. Rev. Lett. 40, 1123 (1978). V. I. D. A. Komar (Nova Science, Commack, 1988) p. 57. A. Pars, A Treatise on Analytical Dynamics, (John Wiley & Sons, New York, 1965). [16] H. Helmholtz, J. reine angew. Math. 100, 137 (1887). [17] J. Douglas, Trans. Am. Math. Soc. 50, 71 (1940). [18] M. Henneaux, Ann. Phys. (NY) 140, 45 (1982). [19] V. I. D. Skarzhinsky, Lebedev Physical Institute Preprint No (216), Moscow (1978). D.