Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

physica status solidi (b)

The path‐integral method and the motion of a phonon–impurity system in a magnetic field

Sh. N. Gifeǐsman, Yu. G. Malkoch

Published Jun 1, 1975• 2 citations
Publisher

Abstract

Abstract It is shown that the variational principle connected with a quadratic trial Lagrangian in Feynman's path‐integral representation can be applied for calculating the energy eigenvalues and thermodynamic functions of systems in a uniform arbitrary external magnetic field. As an example the ground‐state energy of a hydrogen‐like atom in a magnetic field is calculated. A theory of shallow F‐centres in the presence of an arbitrary external magnetic field is developed. The results are valid for arbitrary magnitudes of the polaron coupling constant.

DOI
10.1002/pssb.2220690246

Related papers

Free Energy of a Polaron in a Strong Magnetic Field

Motohiko Saitoh

New results for the free energy of a polaron in a strong magnetic field are presented. The method used here is the extended version of Feynman's pathintegral formalism in which the most general quadratic form containing an infinite number of variational parameters is employed to simulate the variational trial action. A set of nonlinear integral equations is established to determine the upper bound for the free energy in the entire range of electron-phonon coupling constants, temperatures and magnetic fields. Analytical solutions are provided for some limiting cases of interest, and are compared critically with existing calculations of the ground-state energy at zero temperature. The free energy thus obtained is exact to the lowest order in the coupling constant and provides the good interpolation up to the strong coupling regime.

Journal of the Physical Society of Japan•Jul 15, 1981•27 citations

Statistical properties of polarons in a magnetic field. I. Analytic results

François Maria Peeters, J. T. Devreese

The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature ($T$), and magnetic field strength ($\mathcal{H}$). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, $T$, and $\mathcal{H}$.

Physical review. B, Condensed matter•Jun 15, 1982•94 citations

Variational calculation of two-dimensional-polaron energy levels in a magnetic field

D. Ninno, Giuseppe Iadonisi

Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.

Physical review. B, Condensed matter•May 15, 1989•11 citations

THE LINEARIZED POLARON MODEL SYSTEM IN A MAGNETIC FIELD

A. A. Ghazaryan

This article is devoted to the calculation of the free energy on electron-phonon interaction in the external magnetic field according to Green's function method. It should be noted that creation of structures with predicted features occurred (with the use of experimental techniques). To sum up, it should be noted that the given result coincides with the result of Devreese and Brosens, which was obtained by Feynman's path integral method.

International Journal of Modern Physics B•Nov 10, 2008•0 citations

Path-Integral Solutions of the Fröhlich Hamiltonian

Ashok Chatterjee, Soma Mukhopadhyay

This chapter presents a brief introduction to the path-integral method before applying it to the polaron problem. The coupling theory is based on the path-integral formulation of quantum mechanics developed by Feynman in 1955. The path-integral formalism proves to be a useful tool for many field-theoretic problems. Feynman's path-integral method as applied by Feynman to determine the polaron effective mass has an intuitive rather than formal basis. The comparison revealed unequivocally the superiority of the Feynman model over the entire range of the coupling constant. Schultz also simulated the Feynman approximation by replacing the whole lattice with a second, fictitious particle harmonically bound to the electron by a spring of constant k. Thus, the Feynman's strong-coupling energy is better than the Landau-Pekar energy with the Gaussian trial function. The numerical results reported by Marshall and Mills show that the second-order correction to the Feynman result for the ground state energy is less than 2% for the entire coupling rang.

CRC Press eBooks•Oct 9, 2018•0 citations

Rigorous Ground State Energy of a Linearized Model of a Polaron in a Magnetic Field

F. Brosens, J. T. Devreese

Abstract An explicit analytical expression is derived for the ground state energy of a linearized polaron model in a constant and uniform magnetic field. In this model, the validity of the Feynman inequality in a magnetic field is investigated. In a fixed magnetic field, it is shown that even for arbitrarily small electron—phonon coupling, the presence of a magnetic field is prohibitive for the application of the Feynman inequality.

physica status solidi (b)•Feb 1, 1988•11 citations
Abstract It is shown that the variational principle connected with a quadratic trial Lagrangian in Feynman's path‐integral representation can be applied for calculating the energy eigenvalues and thermodynamic functions of systems in a uniform arbitrary external magnetic field. As an example the ground‐state energy of a hydrogen‐like atom in a magnetic field is calculated. A theory of shallow F‐centres in the presence of an arbitrary external magnetic field is developed. The results are valid for arbitrary magnitudes of the polaron coupling constant.
New results for the free energy of a polaron in a strong magnetic field are presented. The method used here is the extended version of Feynman's pathintegral formalism in which the most general quadratic form containing an infinite number of variational parameters is employed to simulate the variational trial action. A set of nonlinear integral equations is established to determine the upper bound for the free energy in the entire range of electron-phonon coupling constants, temperatures and magnetic fields. Analytical solutions are provided for some limiting cases of interest, and are compared critically with existing calculations of the ground-state energy at zero temperature. The free energy thus obtained is exact to the lowest order in the coupling constant and provides the good interpolation up to the strong coupling regime.
The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, TTT, and H\mathcal{H}H.
Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.
This chapter presents a brief introduction to the path-integral method before applying it to the polaron problem. The coupling theory is based on the path-integral formulation of quantum mechanics developed by Feynman in 1955. The path-integral formalism proves to be a useful tool for many field-theoretic problems. Feynman's path-integral method as applied by Feynman to determine the polaron effective mass has an intuitive rather than formal basis. The comparison revealed unequivocally the superiority of the Feynman model over the entire range of the coupling constant. Schultz also simulated the Feynman approximation by replacing the whole lattice with a second, fictitious particle harmonically bound to the electron by a spring of constant k. Thus, the Feynman's strong-coupling energy is better than the Landau-Pekar energy with the Gaussian trial function. The numerical results reported by Marshall and Mills show that the second-order correction to the Feynman result for the ground state energy is less than 2% for the entire coupling rang.