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The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-WignerMoyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow's transitions with energy conservation, additional transitions with energy loss, are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.

By using the analytical superasymmetric fission model it is shown that all "stable" nuclei lighter than lead with Z > 40 are metastable relative to the spontaneous emission of nuclear clusters. An even-odd effect is included in the Zero point vibration energy. Half-lives in the range 1040-1050 s are obtained for Z > 62.The region of metastability against these new decay modes is extended beyond that for a decay and in some cases, in the competing region, the emission rates for nuclear clusters are larger than for a decay.During the last few years advances in studies of many nuclear decay modes have gained considerable interest. Recently, these have been reviewed by Hamilton et al. ' We have used (see Refs. 2-4, and references therein) several methods to show that nuclei heavier than a particles (A2 > 4) and lighter than fission fragments (A2 < 70) are spontaneously emitted from various parent nuclides ( A, Z ) leading to the daughters (A1,Z1). A review paper presenting our early work will be published el~ewhere.~ There is, already, experimental evidence concerning two of more than 140 new decay m o d e~:~.~ (1) 14C spontaneous emissiona-l2 from 223Ra andll from 222,224Ra and (2) 24Ne radioactivity13 of 232U and14 of 231Pa. The experimental data are in agreement with the halflives and the branching ratios relative to a decay calculatedS-',Is (see also Refs. 16 and 17) in the framework of the analytical superasymmetric fission model (ASAFM)3.'8 and with the branching ratios computed by Shi and Swiatecki19 using a proximity-plus-Coulomb potential.Up to now only the region of parent nuclides with Z > 82 have been investigated. The purpose of this paper is to extend the domain for nuclides lighter than lead, pointing out that all the so-called "stable" nuclides with atomic numbers Z > 40, are, in fact, metastable with respect to several new cluster decay modes.In order to estimate the half-lives, T' and T, relative to nuclear cluster emission we shall use ASAFM7 with two values of the Zero point vibration energy E,. This energy enters crucially the formula for the lifetime against cluster emission where the standard notations7 are used for the reduced mass, p , the potential interaction energy E ( r ) and E (R,) = E ( Rb) = Q'. We choose on the one hand, which leads the half-life T, regardless of the odd ( 0 ) or even ( e ) character of the neutron ( N I and proton ( Z ) numbers of the parent nuclide, and on the other hand, with (1.105, e-e I 0.947, e-o E; = E u x parent , 1.000, o-e leading to the half-life T', one can obtain better agreement for a decay of 380 emitters. Hence, T' and T are the halflives with or without the even-odd effect taken into acCount, respectively. A similar even-odd effect was observed20 for 14C radioactivity of Ra isotopesll and of 2 2 5 A~: an enhanced cluster emission rate from e-e nuclei, or equivalently a hindrance from o -e, e -0, and o -0 parents. The released energy, Q, is computed with the new version of the mass table.21 We do not consider the relatively small angular momentum c...

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