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Nonexponential decay law
ANNALS OF PHYSICS128, 501-502 (1980)
Abstracts
of Papers
to Appear
in Future
Issues
Doorway State Approach to Optical Potential Scattering. F. LENZ, Schweizerisches Institut fur Nuklearforschung, Villigen, Switzerland, and Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; AND E. J. MONIZ, Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; AND K. YAZAKI, Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan. We extend and apply the doorway state approach in the context of optical potential scattering. This entails construction of doorway basis states for resolving the transition operator. We focus on analytic solutions to comparatively simple problems in optical potential scattering. Both low and high energy limits are considered, and absorptive interactions are treated; both on- and off-shell partial wave amplitudes are constructed. Further, the full scattering amplitude in the high energy limit is calculated directly in the doorway expansion. With our analytic results, it is possible to identify the physical parameters controlling convergence of the doorway expansion. The same parameters apply over the entire range of cases studied. These parameters are related simply to the target geometry and to the interaction strength. For interactions appropriate to hadron-nucleus scattering, convergence of both on- and off-shell amplitudes is very rapid. The Random Phase Approximation: Its Role in Restoring Symmetries Lacking in the Hartree-Fock Approximation. A. M. LANE, T. P. 8.9, Atomic Energy Research, Establishment, Harwell, Oxon, United Kingdom; AND J. MARTORELL, Fisica Atomica y Nuclear, Facultad de Ciencias, Universidad de Valladolid, Spain. Hartree-Fock wave-functions often lack symmetries possessed by the Hamiltonian. It is often said that the Random Phase Approximation (RPA) restores the missing symmetries. Since the RPA does not readily lead to explicit wave-functions, it is not a trivial matter to verify this assertion. We analyse the situation, and show that, while RPA restores symmetry in some respects, it does not do so completely. Besides the normal RPA, we discuss the generalisation of RPA that describes modes in isobars of the given nucleus. This is needed to enable us to discuss the case of isospin symmetry, which is analysed in detail. The Eikonal Approximation without Ambigrrity in Direction. ALTON C. WILLIAMS. Department Physics, Alabama A. & M. University, Normal, Alabama 35762,
of
The eikonal approximation is formulated by utilizing Feynman’s path integral methods. By requiring conservation of energy, the direction of travel of a particle along the straight line path that it is assumed to take in the eikonal approximation is uniquely determined to be that of the average momentum. Nonexponential Decay Law. ASHER PERES. Department Technology, Haifa, Israel.
of Physics, Technion-Israel
Institute
of
The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have
501 All
Copyright 0 1980 rights of reproduction
by Academic Press, Inc. in any form reserved.
502
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by (dH)2 = J- W(E)&, and the inverse lifetime by r := 2nW(E,), where E0 is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth (1 dW/dE 1 < 1) except for a sharp threshold at E = Ethr , the transition from quadratic to exponential decay occurs for t N l/(& - Et&, and the transition from exponential to inverse power law when
rt = bWo
- &,Jlrl.
Inelastical Efeets in Classical Field-Theoretical Models with Confinement. Yu. A. .%MONOV, STEP. Moscow, USSR; AND J. A. TJON, Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, P.O. Box 80006, 3508 TA Utrecht, The Netherlands. Solitons in a class of relativistic field-theoretical models with confinement of spreading waves are considered and their stability is proven under certain conditions for any space dimension. Numerical studies are presented about the collision of solitons and antisolitons in one spatial dimension. In these processes different types of localized structures are generated which are pulsating in time and appear to be stable. The interaction of these new objects, called ss- and sa-breathers, with solitons and antisolitons is also studied. The Renormalization of sin2 Bw in Gauge Hierarchies. SALLY DAWSON. Lyman Laboratory Harvard University, Cambridge, Massachusetts 02138.
of Physics,
I explain the renormalization of the coupling constants, especially sin, Bw , in grand unified theories with complicated gauge hierarchies. The necessary techniques are illustrated in an SCr(l6) model.
Abstracts
of Papers
to Appear
in Future
Issues
Doorway State Approach to Optical Potential Scattering. F. LENZ, Schweizerisches Institut fur Nuklearforschung, Villigen, Switzerland, and Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; AND E. J. MONIZ, Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; AND K. YAZAKI, Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan. We extend and apply the doorway state approach in the context of optical potential scattering. This entails construction of doorway basis states for resolving the transition operator. We focus on analytic solutions to comparatively simple problems in optical potential scattering. Both low and high energy limits are considered, and absorptive interactions are treated; both on- and off-shell partial wave amplitudes are constructed. Further, the full scattering amplitude in the high energy limit is calculated directly in the doorway expansion. With our analytic results, it is possible to identify the physical parameters controlling convergence of the doorway expansion. The same parameters apply over the entire range of cases studied. These parameters are related simply to the target geometry and to the interaction strength. For interactions appropriate to hadron-nucleus scattering, convergence of both on- and off-shell amplitudes is very rapid. The Random Phase Approximation: Its Role in Restoring Symmetries Lacking in the Hartree-Fock Approximation. A. M. LANE, T. P. 8.9, Atomic Energy Research, Establishment, Harwell, Oxon, United Kingdom; AND J. MARTORELL, Fisica Atomica y Nuclear, Facultad de Ciencias, Universidad de Valladolid, Spain. Hartree-Fock wave-functions often lack symmetries possessed by the Hamiltonian. It is often said that the Random Phase Approximation (RPA) restores the missing symmetries. Since the RPA does not readily lead to explicit wave-functions, it is not a trivial matter to verify this assertion. We analyse the situation, and show that, while RPA restores symmetry in some respects, it does not do so completely. Besides the normal RPA, we discuss the generalisation of RPA that describes modes in isobars of the given nucleus. This is needed to enable us to discuss the case of isospin symmetry, which is analysed in detail. The Eikonal Approximation without Ambigrrity in Direction. ALTON C. WILLIAMS. Department Physics, Alabama A. & M. University, Normal, Alabama 35762,
of
The eikonal approximation is formulated by utilizing Feynman’s path integral methods. By requiring conservation of energy, the direction of travel of a particle along the straight line path that it is assumed to take in the eikonal approximation is uniquely determined to be that of the average momentum. Nonexponential Decay Law. ASHER PERES. Department Technology, Haifa, Israel.
of Physics, Technion-Israel
Institute
of
The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have
501 All
Copyright 0 1980 rights of reproduction
by Academic Press, Inc. in any form reserved.
502
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by (dH)2 = J- W(E)&, and the inverse lifetime by r := 2nW(E,), where E0 is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth (1 dW/dE 1 < 1) except for a sharp threshold at E = Ethr , the transition from quadratic to exponential decay occurs for t N l/(& - Et&, and the transition from exponential to inverse power law when
rt = bWo
- &,Jlrl.
Inelastical Efeets in Classical Field-Theoretical Models with Confinement. Yu. A. .%MONOV, STEP. Moscow, USSR; AND J. A. TJON, Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, P.O. Box 80006, 3508 TA Utrecht, The Netherlands. Solitons in a class of relativistic field-theoretical models with confinement of spreading waves are considered and their stability is proven under certain conditions for any space dimension. Numerical studies are presented about the collision of solitons and antisolitons in one spatial dimension. In these processes different types of localized structures are generated which are pulsating in time and appear to be stable. The interaction of these new objects, called ss- and sa-breathers, with solitons and antisolitons is also studied. The Renormalization of sin2 Bw in Gauge Hierarchies. SALLY DAWSON. Lyman Laboratory Harvard University, Cambridge, Massachusetts 02138.
of Physics,
I explain the renormalization of the coupling constants, especially sin, Bw , in grand unified theories with complicated gauge hierarchies. The necessary techniques are illustrated in an SCr(l6) model.