Transition probabilities for the forced, undamped quantum harmonic oscillator are obtained by expressing S-matrix elements entirely in terms of Heisenberg states and operators. The application of this method to this elementary-field theoretic model provides a simple example of the contraction technique of modern field theory whereby particles are converted into their corresponding currents. This problem was recently analyzed in the interaction picture from a pedagogical point of view and the treatment presented here is given in the same spirit.
Static dielectric screening in undoped semiconductors at zero temperature is formulated within the framework of the Thomas-Fermi-Dirac (TFD) model of a homogeneous and isotropic solid. At each point in the solid the valence electrons are treated as a degenerate gas in statistical equilibrium in the space-varying self-consistent potential of a point-charge impurity. The theory involves the electrostatic, kinetic, and exchange energies of the electrons in the development of a nonlinear TFD equation for the screened potential. The Thomas-Fermi (TF) theory of dielectric screening is recovered when exchange effects are neglected. Closed analytical expressions for the wave-vectordependent dielectric function and the spatial dielectric function are obtained by linearization of the TFD equation and the range of validity of approximation investigated. Numerical solutions of the nonlinear TFD equation for point-charge screening show an increasing departure from linear behavior with impurity charge. These properties of the nonlinear TFD theory are already manifest in the TF scheme. A comparison between TFDand TF-model dielectric functions shows important differences due to exchange. In the linear screening regime, it is found that impurity potentials are more effectively reduced when exchange effects are included. As a result, the TF theory compares more favorably with accurate band-structure calculations of the dielectric functions for silicon and germanium.It is expected that improvement in the TFD dielectric functions depends on extending the treatment to include correlation and/or the quantum correction. In the nonlinear regime, attractive potentials are more effectively screened in the TFD theory, while the opposite is not generally true for repulsive potentials. Finally, it is seen that donor-acceptor asymmetry is stronger in the presence of exchange effects.
The Thomas-Fermi statistical theory, including the Dirac-Slater local-density treatment of exchange correlation, has been applied to the problem of nonlinear screening of a donor point charge embedded in an electron-gas-model semiconductor. The nonlinear screening equation is solved numerically, giving spatial dielectric functions and screening radii with exchange-correlation strength and ion-charge state as parameters. Illustrations and tabulations of these results are given for five semiconductors, four ion charges, and two nonzero values of the exchange-correlation strength corresponding to the Kohn-Sham and Slater exchange potentials. A variational principle equivalent of the nonlinear equation leads to approximate analytical expressions for the spatial dielectric functions which are in close agreement with the exact results. Variational parameters are given for a subset of the semiconductors and charge states. Dielectric functions of silicon are used to illustrate typical comparisons between the two methods of solution.
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