Approximation Of Green's Function Research Articles

Generalized perturbation series

Abstract An approximate solution of the Green's function equation may be used to generate an exact solution of the Schrodinger equation. This is accomplished through an iterative procedure. The procedure is equivalent to a perturbation expansion if the approximate Green's function is exact with respect to some reference potential.

Spectral variational principle for Green's functions

For a suitable approximation $$\tilde K$$ (q, q′, τ) to the Dirac-Feynman Green's function of a quantummechanical system, the parameter $${\mathcal{L}\tilde K}$$ is defined, where ℒ≡i∂/∂τ−ℋ. It is shown thatΔ≧0 andΔ→0 asK→K, the exact Green's function, thus providing a criterion on approximate Green's functions analogous in its role to the variational principle for wavefunctions. A second somewhat weaker criterion is also proposed, based on $$\Sigma \equiv \left[ {tr\tilde K*tr\mathcal{L}^2 \tilde K - |tr\mathcal{L}\tilde K|^2 } \right]_{\tau avg} \geqq 0$$ . Recipes are given for projecting out continuum contributions toΔ or∑ and for analyzing for the discrete eigen-value spectrum.

A Quantum Theoy of Double Exchange. II. The Green Function of a Hole.

The one-particle states of a hole in a double exchange ferromagnet are discussed by making use of the Green function. An approximate Green function of a hole in a system with randomly oriented localized spins is obtained by applying the CPA extended as to take into account the effects of exchange scatterings. The density of states of a hole is numerically calculated for a simple case. The narrowing of the width of the density of states is examined for the s.c. lattice. In the case S =1/2, which is equivalent to the Hubbard model with infinitely strong correlations, the present result agrees with that of Kawabata.

Moment-conserving decoupling of green functions via pade approximants

A generalization of the Padé approximant technique obtained by inner projections leads to approximate Green functions whose spectral densities satisfy a certain number of frequency moments. This approach contains decoupling schemes previously given and their explicit representation in terms of the moments.

Green's function approximation for an equation of state describing liquid-gas phase transitions

The formalism of thermodynamic Green's functions is used to derive a theoretical equation of state for real fluids which shows both the volume exclusion effect and the appearance of a liquid-gas phase transition and of a critical point.

On the short-wave asymptotic motion due to a cylinder heavinǵ on water of finite depth. I

AbstractThis paper is a first investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we present a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The reduction of the potential problem by the use of two auxifiary potentials, introduced by the subtraction of formal solutions constructed from the limit potential, enables us to find (i) an integral expression, by using a subsidiary approximate Green's function, for the potential in the far field which is evaluated after making plausible assumptions about the asymptotic value of the first auxiliary potential on the cylinder and below the free surface; and (ii) a form for the virtual mass, using the second auxiliary potential whose value on the cylinder is deduced from that of the first to which it is related. Thus we obtain the asymptotic evaluation of the coefficients describing the wave-making and virtual mass of the heaving cylinder, which depend on the limit potential and have no formal dependence on the depth of the bottom.

Approximate Green's functions and the solution of related integral equations

Approximate Green's functions and the solution of related integral equations

Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. II

The phase-integral approximation of the Green's function in momentum space is investigated for a particle of negative energy (bound state) which moves in a spherically symmetric potential. If this potential has a Coulomb-like singularity at the origin, it is shown that any two momenta can be connected by an infinity of classical trajectories with a fixed energy. The summation of the usual phase and amplitude factors over these trajectories is the approximate Green's function. If there is orbital precession, there are not only poles along the negative energy axis, but also weaker singularities which are not examined in detail. The poles are found at the energies which are given by the semiclassical quantum conditions: angular momentum = (l + ½)ℏ and action integral for the radial motion = (n + ½)2πℏ, where l and n are integers ≥ 0. The residues at these poles give the approximate bound-state wavefunctions as a product of the asymptotic formula for Legendre polynomials with the asymptotic solution of the radial Schrödinger equation. It is conjectured that the occurrence of poles in the approximate Green's function is a direct consequence of the periodic character of the classical motion.

All Effective Mass Approximation for Several Bands

AbstractThe effective mass formalism is generalized to include the effect of several bands by constructing an approximate Green's function for the periodic lattice. Energy levels are obtained for a hydrogenic impurity in a periodic lattice described by two and three parabolic bands. In addition to the energy levels predicted by effective mass theory a deep level is found. An explicit expression for the deep level is given in terms of the band parameters and a comparison with the experimentally observed deep levels in gallium arsenide is made.

Derivation and Analysis of the Pariser–Parr–Pople Model

Second quantization formalism is used to derive the Pariser–Parr–Pople model for planar unsaturated molecules. The model is analyzed in detail in the atomic limit and the equivalence with an Heisenberg antiferromagnet is demonstrated by perturbation theory. An approximate Green's-function method is developed and its properties in the same limit are derived and shown to be correct. The case of a linear chain is calculated explicitly and compared with other results. Limitations and difficulties in the establishment of the results are discussed.

A Green function approach to effective mass theory

Effective mass theory is reformulated in terms of one-electron Green's functions. This enables a generalization of effective mass theory to be made and leads to an approximate Green function for the periodic lattice. The Green function is evaluated for a two-band model and used to calculate the energy levels of a hydrogenic impurity.

Phase-Integral Approximation in Momentum Space and the Bound States of an Atom

The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrödinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman-type functional integral which was recently proposed by Garrod for the energy-momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound-state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.

Phonon-Broadened Optical Spectra: Urbach's Rule

Starting from Fr\olich's Hamiltonian for conduction-electron-optical-phonon interactions, an approximate Green's function $G(p, t)$ is derived. The spectral density derived from this Green's function is shown to possess an exponential tail at low energies, $\frac{E}{{\ensuremath{\omega}}_{0}}\ensuremath{\ll}0$. This demonstrates that Urbach's rule can be derived from Fr\olich's Hamiltonian. The approximate Green's function $G(p, t)$ is shown to be related to the intermediate-coupling models for polarons.

Antiferromagnetism in the Face-Centered Cubic Lattice. I. The Random-Phase Green's Function Approximation

The random-phase Green's function approximation is used to derive equations for zero-field magnetic susceptibility and two-spin correlation functions which are valid, at temperatures above the Curie or N\'eel point, for all ferromagnets and antiferromagnets which can be described by any isotropic Heisenberg Hamiltonian. At high temperatures, these expressions are also expanded as series in inverse powers of temperature. Detailed numerical calculations are carried out for the face-centered-cubic lattice with antiferromagnetic nearest- and next-nearest-neighbor exchange parameters ${J}_{1}$ and ${J}_{2}$. Susceptibility results are compared with molecular-field estimates at temperatures near the N\'eel point, and with the known terms of the exact power-series expansion at high temperatures. The spin correlations are computed for the first four shells of nearest neighbors. Finally, the sublattice-magnetization curves at temperatures below the N\'eel point are computed, in the random-phase Green's function approximation, for the type-2 antiferromagnetic order of the face-centered-cubic lattice. The curve shapes are found to be very insensitive to $\frac{{J}_{2}}{{J}_{1}}$ and approximate closely the shape of the molecular-field Brillouin-function curves. The significance of this result in connection with the biquadratic-exchange question in MnO is discussed in detail in the following paper by Lines and Jones.

Konsistente Temperaturentwicklungen f�r den Ferromagneten

Ferromagnetism is studied inDyson's model of ideal spinwaves and in the Heisenberg model by means of temperature dependentGreen's functions. As is well known the low temperature magnetization of the free spinwave gas has a series expansion inT (3+2n)/2. The “Hartree-Fock” approximation forGreen's function gives an additional term inT 4 forDyson's model, further contributions toT 4 are furnished by higher approximations for the two spinwave correlation and the summation of all ladder graphs leads back toDyson's result, though the contribution of three and higher spin wave correlations remains obscure. For the Heisenberg model a discussion of the temperature dependence of 〈S i z S j z 〉-correlations, total energy and specific heat shows that the well-known decoupling process (-“Hartree”-approximation for the spin operator two point function-) is not consistent unlessT≪T c .

Calculation of nucleon form factors through the Green function approximation scheme

The photon-pion-nucleon system is studied by a non-perturbative approximation method based on the truncation of covariant Green functions in field theory. The prescription of spectral representations is used to incorporate correct analyticity requirements. The anomalous magnetic moments and mean square charge radii of the nucleon calculated by this approximation method are an improvement over perturbation theory. The results are compared with those of dispersion theory.

Note on the Generating Function of Two-Time Green Functions

A new approach to the two-time Green function method is proposed. In order to give a systematic way of decoupling the two-time Gr~en functions and to overcome the difficulties often encountered in terminating the hierarchy of Green functions, a generating function is introduced and its properties are examined in detail for a simple model. It is shown that the generating function provides powerful tool even in the case where conventional approximation for Green functions does not work. An example of the generating function is constructed for Fermion system with two-body interaction.

Coulomb Green's Functions and the Furry Approximation

The Coulomb Green's function for the nonrelativistic Schrödinger equation is obtained in closed form starting from the partial-wave expansion and using an integral representation for a product of two Whittaker functions with different arguments. The Neumann's series for Jv(kz) is required in evaluating the sum on states. Using the same methods, the Coulomb Green's functions for the Klein-Gordon and iterated Dirac equations are obtained in closed form in the ``Furry approximation,'' a2/(J + ½)2 ≪ 1, a = Ze2/4πh/c. The Klein-Gordon Green's function in this approximation is shown to be at the same time the exact Green's function for the Klein-Gordon equation without the potential squared term. An alternate and very simple derivation of the approximate Green's function for the iterated Dirac equation is given using perturbation theory. From this Green's function, an approximate Coulomb Green's function in closed form for the Dirac equation itself is constructed. Certain known results for Coulomb wavefunctions with modified plane-wave behavior at large distances are rederived using the foregoing methods and results.

A variational approach to the anharmonic lattice problem (with an application to the theory of ferroelectric transitions)

Abstract An extended variational principle for the free energy is set up which may be applied to the anharmonic lattice problem by the use of a harmonic trial Hamiltonian. This approach which is essentially that of Born and Hooton may be used to derive a generalized anharmonic Einstein model. However, the trial system does not contain phonon damping. This is remedied by following Feynman's approach to the polaron problem and extending the variational principle to include harmonic coupling to an auxiliary set of trial oscillators. For extended trial Hamiltonians H T which are purely harmonic we show that the variational principle may be re-written explicitly in terms of the one particle Green's function defined for the model system: G ij ( β 1 , β 2 ) = [ T { u i ( β 1 ), u j ( β 2 )}] where u i ( β 1 ) = exp ( β 1 H T ) u i exp(– β 1 H T ) and u i , are atomic displacements. The resulting approximate Green's function variational principle is the phonon analog of an approximation to a variational principle due to Klein and others. Variation with respect to G ij leads to a self consistent equation for the one-phonon Green's function for which an approximate solution may be represented in terms of the iteration of a series of first and second order diagrams. Application to a lattice vibration model of ferroelectric transitions of the barium titanate type is discussed. By the introduction of an additional displacement variable, the variational principle can operate through the phase transition and leads to a microscopic analog of the Devonshire equations in which the coefficients are explicit functions of the G ij .

Green's Function Approximation Method. I. The Nucleon

A method for the approximate construction of the nucleon Green's function is presented. The development is such that the approximate Green's function automatically has the same analytical properties as the exact one. This method involves the symmetrical treatment of the Green's function and, ultimately, the assumption that certain particles behave in an uncorrelated manner. The approximation results in a linear integral equation for the Green's function which is completely renormalized. This equation is solved exactly through the use of the spectral representation which, by construction, is consistent with the approximation.