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Symmetry properties, tests, and reduction of the crossing matrix
ANNALS
OF PHYSICS:
61, 530-531 (1970)
Abstracts
of Papers
to Appear
in Future
Issues
Application of the Coherent State Representation in the Theory of Magnetism: I. The Heisenberg Model. KENNETH H. DOUGLASS. Pennsylvania State University, Schuylkill Campus, Schuylkill Haven, Pennsylvania 17972. By transforming the Heisenberg model Hamiltonian to Bose operators it becomes possible to define a coherent state representation for the spin system. This representation proves to be a useful basis for evaluating the partition function in a manner similar to that used previously for a superfluid. A hard core repulsive interaction is used to include the effects of the spin commutators exactly. The scattering of two spin waves is solved to all orders of perturbation theory in this formalism and used to rederive the low temperature series expansions due to Dyson. The results using the hard core interaction are compared with those using Dyson’s dynamical interaction alone. The method is extended to give some results for the case of an anisotropic exchange interaction. Antisymmetry Ejgects in Deuteron Stripping Reactions. LUTZ D~HNERT. Escuela de Fisica, Matematicas y Computation, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela. In this present work we study deuteron stripping reactions from a microscopic point of view. We define an open channel wavefunction which has the same asymptotic behaviour as the total wavefunction describing the total system (target plus incident deuteron) by simply extending the asymptotic form of the respective open channel wavefunctions to all space. We work with explicitly antisymmetrized wavefunctions so that the Pauli principle is always taken into account. By means of the Unified Theory of Reactions we obtain a system of coupled Schrodinger equations for the elastic deuteron scattering and the inelastic proton stripping amplitudes, after having analyzed carefully the respective projection operators. We obtain effective stripping potentials. We estimate the relative importance of corrections to the usual coupling potentials, which arise due to antisymmetry and nonorthogonality of the channel wavefunctions, by means of a DWBA. The effect of nonorthogonality becomes important when the channel wavefunctions are antisymmetrized. Symmetry Properties, Tests, and Reduction of the Crossing Matrix. JAMIL DABOLJL. Department of Physics, Temple University, Philadelphia, Pennsylvania 19122. We derive symmetry properties of the crossing matrix from general symmetry arguments. These properties can be used: (a) To test the known crossing matrices. We find that the crossing matrix of C-TMN [l] satisfies these conditions only if its overall phase factor is modified. We also show that the crossing matrix of Trueman and Wick [2], as we interpret it, is equivalent to that of C-TMN if one takes into account the different continuation paths. (b) To reduce the crossing matrix into two or more submatrices, such that one submatrix connects “symmetry-conserving” amplitudes of the s and t channels with each other, whereas the other submatrix connects only the “symmetry-breaking” amplitudes of the two channels with each other. This fact may be useful for bootstrap calculation.
530
ABSTRACTS
OF
PAPERS
TO APPEAR
IN
FUTURE
ISSUES
531
In addition, we derive in a simple way the crossing matrices for the crossing of any two particles in terms of that for particles 1 and 4. Furthermore, we derive in the appendices the exact symmetry relations of the c.m. helicity amplitudes under T, CPT, El2 , Ea4, and E for general reactions using consistent intrinsic phases, as these relations are not available in the literature. Our method is simpler than that of Jacob and Wick [3], since it doesn’t involve any partial wave amplitudes. Impurities in an Imperfect Bose Gas. I. The Condensate. TIMOTHY C. PADMORE AND ALEXANDER L. FETTER. Institute of Theoretical Physics, Department of Physics Stanford, University, Stanford, California 94305. A variational principle is used to evaluate the change in the condensate energy of an imperfect Bose gas arising from the introduction of stationary impurities. Moving impurities are incorporated by performing a Galilean transformation from a frame with bulk flow at infinity to one with asymptotically stationary fluid. The corresponding effective mass is calculated numerically and compared with that of He3 impurities in He II. A generalization to charged impurities exhibits the anomalous flow pattern suggested by Gross and allows a model calculation of the eflective mass of positive ions in He II. The Damped Degenerate-Level Atom. AMNON AHARONY. Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Israel. Equations are derived for the time evolution of the reduced density operator for an (N + l)state atom which is damped by its coupling to a finite-temperature boson bath. General formal solutions are discussed in the limit of low temperature. Exact solutions are given for the decay of a p level in an electric field. The effects of a driving field and of transitions between the excited states on the equations are discussed. The application for the decay of elementary particles is mentioned.
OF PHYSICS:
61, 530-531 (1970)
Abstracts
of Papers
to Appear
in Future
Issues
Application of the Coherent State Representation in the Theory of Magnetism: I. The Heisenberg Model. KENNETH H. DOUGLASS. Pennsylvania State University, Schuylkill Campus, Schuylkill Haven, Pennsylvania 17972. By transforming the Heisenberg model Hamiltonian to Bose operators it becomes possible to define a coherent state representation for the spin system. This representation proves to be a useful basis for evaluating the partition function in a manner similar to that used previously for a superfluid. A hard core repulsive interaction is used to include the effects of the spin commutators exactly. The scattering of two spin waves is solved to all orders of perturbation theory in this formalism and used to rederive the low temperature series expansions due to Dyson. The results using the hard core interaction are compared with those using Dyson’s dynamical interaction alone. The method is extended to give some results for the case of an anisotropic exchange interaction. Antisymmetry Ejgects in Deuteron Stripping Reactions. LUTZ D~HNERT. Escuela de Fisica, Matematicas y Computation, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela. In this present work we study deuteron stripping reactions from a microscopic point of view. We define an open channel wavefunction which has the same asymptotic behaviour as the total wavefunction describing the total system (target plus incident deuteron) by simply extending the asymptotic form of the respective open channel wavefunctions to all space. We work with explicitly antisymmetrized wavefunctions so that the Pauli principle is always taken into account. By means of the Unified Theory of Reactions we obtain a system of coupled Schrodinger equations for the elastic deuteron scattering and the inelastic proton stripping amplitudes, after having analyzed carefully the respective projection operators. We obtain effective stripping potentials. We estimate the relative importance of corrections to the usual coupling potentials, which arise due to antisymmetry and nonorthogonality of the channel wavefunctions, by means of a DWBA. The effect of nonorthogonality becomes important when the channel wavefunctions are antisymmetrized. Symmetry Properties, Tests, and Reduction of the Crossing Matrix. JAMIL DABOLJL. Department of Physics, Temple University, Philadelphia, Pennsylvania 19122. We derive symmetry properties of the crossing matrix from general symmetry arguments. These properties can be used: (a) To test the known crossing matrices. We find that the crossing matrix of C-TMN [l] satisfies these conditions only if its overall phase factor is modified. We also show that the crossing matrix of Trueman and Wick [2], as we interpret it, is equivalent to that of C-TMN if one takes into account the different continuation paths. (b) To reduce the crossing matrix into two or more submatrices, such that one submatrix connects “symmetry-conserving” amplitudes of the s and t channels with each other, whereas the other submatrix connects only the “symmetry-breaking” amplitudes of the two channels with each other. This fact may be useful for bootstrap calculation.
530
ABSTRACTS
OF
PAPERS
TO APPEAR
IN
FUTURE
ISSUES
531
In addition, we derive in a simple way the crossing matrices for the crossing of any two particles in terms of that for particles 1 and 4. Furthermore, we derive in the appendices the exact symmetry relations of the c.m. helicity amplitudes under T, CPT, El2 , Ea4, and E for general reactions using consistent intrinsic phases, as these relations are not available in the literature. Our method is simpler than that of Jacob and Wick [3], since it doesn’t involve any partial wave amplitudes. Impurities in an Imperfect Bose Gas. I. The Condensate. TIMOTHY C. PADMORE AND ALEXANDER L. FETTER. Institute of Theoretical Physics, Department of Physics Stanford, University, Stanford, California 94305. A variational principle is used to evaluate the change in the condensate energy of an imperfect Bose gas arising from the introduction of stationary impurities. Moving impurities are incorporated by performing a Galilean transformation from a frame with bulk flow at infinity to one with asymptotically stationary fluid. The corresponding effective mass is calculated numerically and compared with that of He3 impurities in He II. A generalization to charged impurities exhibits the anomalous flow pattern suggested by Gross and allows a model calculation of the eflective mass of positive ions in He II. The Damped Degenerate-Level Atom. AMNON AHARONY. Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Israel. Equations are derived for the time evolution of the reduced density operator for an (N + l)state atom which is damped by its coupling to a finite-temperature boson bath. General formal solutions are discussed in the limit of low temperature. Exact solutions are given for the decay of a p level in an electric field. The effects of a driving field and of transitions between the excited states on the equations are discussed. The application for the decay of elementary particles is mentioned.