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Journal of Inner Mongolia University

The Calculating of Ground State Energy of the Two Dimensional Acoustic Polaron by Using Bound State Wave Function

Liang Xi-Xia

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The characters of ground state energy of the two dimensional acoustic poalron as a function of the electron-phonon coupling constant have been investigated by using the Gaussian wave function.The results indicate that the Huybrechts-like approach is equivalent to the variational approach by using the bound state wave function.It was verified that the Huybrechts approach is applicable for the whole coupling range by theory deducing and numerical calculating.

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The characters of ground state energy of the two dimensional acoustic poalron as a function of the electron-phonon coupling constant have been investigated by using the Gaussian wave function.The results indicate that the Huybrechts-like approach is equivalent to the variational approach by using the bound state wave function.It was verified that the Huybrechts approach is applicable for the whole coupling range by theory deducing and numerical calculating.
Taking the same trial wave function, the ground-state energy of Fröhlich polaron is investigated by variational method and coherent-state expansion (CSE) one, respectively. Within the accuracy to α2α^{2}α2(the electron-phonon coupling constant), both methods can give the same analytic expressions of polaron ground-state energy as the function of electron-phonon coupling constant. We find that the CSE method can much simplify the calculation and shows more advantages in higher order approximations.
Based on the unitary transformation, a general treatment of the bound polaron is proposed. In distinction from earlier calculations, the present method gives explicit expressions for not only the energy but also the wave functions both in the ground state and in the excited states such as the relaxed excited state and the phonon scattering state. These expressions involve a summation on the complete set of eigenstates of a trial Hamiltonian, and in actual calculations the Coulomb Hamiltonian and the harmonic oscillator Hamiltonian are chosen as trial Hamiltonians. For the ground state, the energy expression is the same as that obtained by Matsuura with the path integral method. The present method is valid in the whole region of the electron-phonon coupling constant α and Coulomb force β except for β→0 and α>6.
The ground-state energy of the optical polaron is calculated by the generalized path-integral formulation. The ground-state approximation is made to simplify the complicated expression, and the Ritz variational and direct integration methods are used to obtain the ground-state energy in the whole range of coupling strength. The results agree with previous work, and it is found that there is a transition for coupling constant ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2$. The ground-state energy obtained by harmonic approximation, which is equivalert to taking Gaussian-like trial wave functions, is compared with those obtained by Pekar's trial wave function.
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.