Analysis of a nondegenerate decay system

Neumann, H; Handelsman, Richard A.; Horwitz, L. P.

doi:10.1007/bf02730187

Cited by 16 publications

(13 citation statements)

References 15 publications

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“…The well-known demonstration of the Zeno effect for very short times t relies on an expansion of the survival probability P (t) of a unstable state |ψ in a series [21] …”

Section: Equivalence Of Exact Markovian Coupling and Globally Flat Anmentioning

confidence: 99%

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

2011

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We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation the evolution of a total system subspace is not an exact semigroup for multichannel decay, unless the projectors into eigenstates of the reduced evolution generator W (z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations zα = z β , and since the orthogonality relation does not generally hold at different values of z, the semigroup evolution is a poor approximation for the multi-channel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W (z) projectors regardless the number of the poles. Such a possibility occurs when W (z), and hence its eigenvectors, are independent of z, which corresponds to the Markovian limit of the coupling to the continuum spectrum.

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“…The well-known demonstration of the Zeno effect for very short times t relies on an expansion of the survival probability P (t) of a unstable state |ψ in a series [21] …”

Section: Equivalence Of Exact Markovian Coupling and Globally Flat Anmentioning

confidence: 99%

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

2011

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“…The saddlepoint approximation may be considered a version of Laplace's method applied in the complex domain (see, e.g., [9]), which makes it suitable for application to the characteristic function * of a sample mean, or some other statistic * , say T. By inversion, the saddlepoint approximation to the characteristic function of T produces an approximation to the corresponding limiting distribution of T which, following Daniels [12], is also called a saddlepoint approximation. The resulting distributional expansion is closely related to the Edgeworth expansion (see CORNISH-FISHER AND EDGEWORTH EXPANSIONS) but is more accurate.…”

Section: The Saddlepoint Approximationmentioning

confidence: 99%

“…The most elegant applies the fully exponential approximation (9) to the moment generating function M(s) = E[exp{sg(θ )}] (using the positivity of exp{sg(θ )}) to get, say,M(s), and then computesM (0) −M (0) 2 . Each of these may be shown to have multiplicative error of O(n −2 ).…”

Section: Second-order Approximations For Expectations and Variancesmentioning

confidence: 99%

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Laplace's Method

Kass¹

2005

Encyclopedia of Statistical Sciences

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“…Let us assume that the eigenvalues are ordered so that one pole appears at z a such that z a − z 1 (z a ) = 0 and the second at z b such that z b − z 1 (z b ) = 0. In distorting the contour below the real axis across the cut into the second sheet, these pole contributions may dominate the intermediate time behavior of the amplitude (the very short time behavior is controlled by the Taylor expansion at small t) 3 . For semibounded spectrum, in (0, ∞), the long time behavior is controlled by the contribution at the branch point, and goes as t −n , where n is the dimensionality of space.…”

Section: Introductionmentioning

confidence: 99%

Stark effect in Lax–Phillips scattering theory

Ari

Horwitz

2004

Physics Letters A

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Abstract:The scattering theory of Lax and Phillips, originally developed to describe resonances associated with classical wave equations, has been recently extended to apply as well to the case of the Schrödinger equation in the case that the wave operators for the corresponding Lax-Phillips theory exist. It is known that the bound state levels of an atom become resonances (spectral enhancements) in the continuum in the presence of an electric field (on all space) in the quantum mechanical Hilbert space. Such resonances appear as states in the extended Lax-Phillips Hilbert space. We show that for a simple version of the Stark effect, these states can be explicitly computed, and exhibit the (necessarily) semigroup property of decay in time. The widths and location of the resonances are those given by the poles of the resolvent of the standard quantum mechanical form. 1 IntroductionThe description commonly used for an unstable system is that of Wigner and Weisskopf 1 . They assumed that the unstable system is represented by a vector, say φ, in the Hilbert space of states, and that the Hamiltonian evolution of this state permits the development of other components which represent the "decayed" system. One may assume a Hamiltonian H 0 for which φ is an eigenstate, and the full Hamiltonian is constructed by adding a pertubation V which induces transitions from this state. The basic assumption of their physical model is that these transitions carry the state of the system from that of some specific system to a state in the continuous spectrum which contains the decay products. For example, one may think of an atom in an excited state as the unstable system; in the absence of electromagnetic interaction, this state would be a stable bound state. A term added to the Hamiltonian corresponding to electromagnetic interaction induces a transition from this state, and the square of the corresponding transition amplitudes, which may be generally computed perturbatively, give the probability for decay. This very standard technique is rigorously correct for reversible quantum transitions, according to the laws of quantum theory. When applied to the decay of an unstable system, however, for which the evolution is irreversible, it consitutes an approximation which may be inadequate. In the following we discuss some of the difficulties of the application of the Wigner-Weisskopf method to the treatment of irreversible phenomena, such as particle decay, and in Section 3 we describe the general structure of the Lax-Phillips theory, which yields an exact semigroup evolution and appears to be a more appropriate description of such phenomena. In Section 2, we apply the Wigner-Weisskopf theory to a simple model for the Stark effect, for comparison, and work out the Lax-Phillips theory for this model in Section 4. Some of the calculations in Section 2 coincide with those needed in Section 4; the approximate pole approximation decay law of the Wigner-Weisskopf theory contains exactly the same pole as the singularity of the Lax-Phillips S m...

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Analysis of a nondegenerate decay system

Cited by 16 publications

References 15 publications

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

Laplace's Method

Stark effect in Lax–Phillips scattering theory

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