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Journal of Physics C Solid State Physics

A non-perturbative approach to the bound polaron. III

S. Mukhopadhyay, Tapas Mitra

Published Feb 20, 1981• 8 citations
Publisher

Abstract

For pt.II see ibid., vol.13, p.3193 (1980). The formal difficulties of the non-perturbative iteration variational method proposed recently for the ground state of the bound polaron problem are removed by a rigorous method of canonical transformation which not only yields essentially equivalent results but also extends its validity to excited states.

DOI
10.1088/0022-3719/14/5/010

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For pt.I see Phys. Lett. A, vol.71, p.385 (1979). A recently developed non-perturbative iterative variational method has been applied to the bound polaron problem. The electron-phonon Hamiltonian is reduced in successive orders resulting in a transformed electronic Hamiltonian which gets an attractive phonon-induced potential from each order of reduction. The transformed electronic Hamiltonian has been solved numerically for a series of values of the parameters alpha , beta . Relation to perturbative solutions is discussed. Interesting localisation criteria derived from the calculated wavefunctions are also discussed.

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The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.

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Representation of quantum mechanical wavefunctions by transformation generators. III. Canonical transformations in an extended phase space

C Jung, H. Krüger

For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.

Journal of Physics A Mathematical and General•Apr 1, 1984•0 citations
For pt.II see ibid., vol.13, p.3193 (1980). The formal difficulties of the non-perturbative iteration variational method proposed recently for the ground state of the bound polaron problem are removed by a rigorous method of canonical transformation which not only yields essentially equivalent results but also extends its validity to excited states.
For pt.I see Phys. Lett. A, vol.71, p.385 (1979). A recently developed non-perturbative iterative variational method has been applied to the bound polaron problem. The electron-phonon Hamiltonian is reduced in successive orders resulting in a transformed electronic Hamiltonian which gets an attractive phonon-induced potential from each order of reduction. The transformed electronic Hamiltonian has been solved numerically for a series of values of the parameters alpha , beta . Relation to perturbative solutions is discussed. Interesting localisation criteria derived from the calculated wavefunctions are also discussed.
The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.
For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.