Standard Diagonalization Research Articles

Calculation of reduction factors for the Γ8-(e+t) Jahn-Teller case with applications to acceptor defects in semiconductors

The reduction factors of the general one-mode Gamma 8-(e+t) Jahn-Teller (JT) case are calculated throughout the whole coupling range. For small JT coupling parameters the authors use standard matrix diagonalisation procedures combined with a group theoretical projection technique. The application of a unitary transformation method diagonalises the Hamiltonian exactly for infinite couplings. The resulting vibronic Glauber ground states are used for the reduction factor calculation at high coupling values. A combination with perturbational methods extends the validity of the approach to the whole range of coupling strengths. They apply the method to the case of acceptor defects in semiconductors within a one-mode approximation of the defect Hamiltonian. Results for the reduction of the acceptor deformation potential constants are given for the systems Si(In), Si(B) and GaAs(Mn).

The dynamical basis set

A new Cartesian basis set is defined that is suitable for the representation of molecular vibration-rotation bound states. The Cartesian basis functions are superpositions of semiclassical states generated through the use of classical trajectories that conform to the intrinsic dynamics of the molecule. Although semiclassical input is employed, the method becomes ab initio through the standard matrix diagonalization variational method. Special attention is given to classical-quantum correspondences for angular momentum. In particular, it is shown that the use of semiclassical information preferentially leads to angular momentum eigenstates with magnetic quantum number ‖M‖ equal to the total angular momentum J. The present method offers a reliable technique for representing highly excited vibrational-rotational states where perturbation techniques are no longer applicable.

Two-step approach to one-dimensional anharmonic oscillators

We propose a two-step approach to one-dimensional anharmonic oscillators. A generalized coherent-state ansatz is introduced for the first step. A theorem of Wick's ordering and a Bogoliubov transformation are used to simplify the derivation. This is shown to be equivalent to the Hartree approximation. In the second step, standard diagonalization is used for the transformed Hamiltonian. The method yields a clear physical picture and is capable of producing accurately all the low-lying energy levels in a single diagonalization. Asymptotic expansions for the energy levels are easily obtained. The connection with a pure quartic oscillator is pointed out and as a by-product no calculation is necessary in that model. We also include cubic couplings and apply the method successfully for the two-well oscillator.

Microscopic structure of the pairing vibrational band in the Pb isotopes

The corpus of the (t, p) and (p, t) data around 208Pb is used to check the predictions of the pairing vibrational model. The wave functions associated with the pair addition and pair removal modes were calculated by diagonalizing monopole and multipole pairing forces in the Tamm-Dancoff and in the random phase approximation. The multiphonon wave functions obtained as renormalized product states of these wave functions, eliminating Pauli principle violating components, give a rather accurate description of the data. These “weak coupling” wave functions, which take into account in an approximate way the anharmonicities in the pairing channel, compare well with the wave functions obtained by a standard shell-model diagonalization of monopole and multipole pairing forces, and with wave functions calculated in a two-quasiparticle basis.

Nonperturbative potential model for light and heavy quark-antiquark systems

A potential model based on quantum-chromodynamics (QCD) considerations is developed. The model attempts to overcome the relativistic limitations associated with earlier models by introducing an effective size for quarks. Application of the model Hamiltonian to both light and heavy mesons yields an accurate description of the mass spectrum, radiative transitions, and annihilation widths for a large number of known mesons. Spin-dependent interactions are treated nonperturbatively using standard diagonalization procedures with an oscillator basis set and relativistic kinematics are adopted throughout. The scaling of the potential parameters is found to be similar to simple QCD predictions. Much of the anomalous behavior for pseudoscalar mesons appears to be resolved both for light and heavy mesons.

Direct numerical integration of coupled one-dimensional Schrödinger equations

Physical problems which can usually be described with the help of sets of coupled second-order differential equations are outlined. In the case of scattering problems, the direct numerical integration of these equations is the unique method of obtaining a solution. For bound-state calculations this integration is in direct competition with powerful conventional techniques such as the variational principle and the standard matrix diagonalization. We are developing two methods for solving by direct integration the eigenvalue problem associated with sets of coupled one-dimensional Schrödinger equations. These methods are successfully used to calculate the bound states of nuclear three- and four-body systems with two-body pure central forces.

The April Fool Turing Test

This paper explores certain issues concerning the Turing test; non-termination, asymmetry and the need for a control experiment. A standard diagonalisation argument to show the non-computability of AI is extended to yields a socalled “April fool Turing test”, which bears some relationship to Wizard of Oz experiments and involves placing several experimental participants in a symmetrical paradox – the “April Fool Turing Test”. The fundamental question which is asked is whether escaping from this paradox is a sign of intelligence. An important ethical consideration with such an experiment is that in order to place humans in such a paradox it is necessary to fool them. Results from an actual April Fool Turing Test experiment are reported. It is concluded that the results clearly illustrate some of the difficulties.

The April Fool Turing Test

This paper explores certain issues concerning the Turing test; non-termination, asymmetry and the need for a control experiment. A standard diagonalisation argument to show the non-computability of AI is extended to yields a socalled “April fool Turing test”, which bears some relationship to Wizard of Oz experiments and involves placing several experimental participants in a symmetrical paradox – the “April Fool Turing Test”. The fundamental question which is asked is whether escaping from this paradox is a sign of intelligence. An important ethical consideration with such an experiment is that in order to place humans in such a paradox it is necessary to fool them. Results from an actual April Fool Turing Test experiment are reported. It is concluded that the results clearly illustrate some of the difficulties.