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Journal of Guangxi University of Technology

Time-dependent variational approach to dispersion relation of the quantum Klein-Gordon model

Hong Zhong

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The phonon dispersion relation of the quantum Klein-Gordon model with symmetric single-well anharmonic substrate potential is studied by means of the time-dependent variational principle combined with Hartree-type trial wavefunction for system and Jackiw-Kerman wavefunction for single particle.Under the condition of minimum uncertainty relation,equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation.It is shown that the coefficient of square term of the substrate potential is strengthened and the band gap is broadened due to the quantum fluctuations in comparison with those of the classical model.In addition,the coefficient of square term and the band gap increase with increasing the quantum fluctuations and the coefficient of quartic term of the substrate potential.

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The phonon dispersion relation of the quantum Klein-Gordon model with symmetric single-well anharmonic substrate potential is studied by means of the time-dependent variational principle combined with Hartree-type trial wavefunction for system and Jackiw-Kerman wavefunction for single particle.Under the condition of minimum uncertainty relation,equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation.It is shown that the coefficient of square term of the substrate potential is strengthened and the band gap is broadened due to the quantum fluctuations in comparison with those of the classical model.In addition,the coefficient of square term and the band gap increase with increasing the quantum fluctuations and the coefficient of quartic term of the substrate potential.
The phonon dispersion relation of the quantum FPU model is studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction.The single-particle state is taken to be a frozen Jackiw-Kerman wavefunction.Under the condition of minimum uncertainty relation,equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation.It is shown that the harmonically coupling strength is strengthened and the width of the phonon band is broadened due to the quantum fluctuations in comparison with those of the classical FPU model.In addition,the harmonically coupling strength and the width of the phonon band increase with increasing the quantum fluctuations and quartic coupling strength.
The phonon dispersion relation of the commensurate quantum Frenkel–Kontorova model is studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction for the particles. The single-particle state is taken to be a frozen Jackiw–Kerman wavefunction. Under the condition of minimum uncertainty, equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation. It is shown that the strength of the substrate potential and the phonon excitation gap are reduced due to the quantum fluctuations in comparison with those of the classical model. We also compare our results with those previously obtained by using the path-integral molecular dynamics.
The ground-state phase transition and the phonon dispersion relation of the quantum double-well model are studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction. The single-particle state is taken to be a frozen Jackiw–Kerman wavefunction. Under the condition of minimum uncertainty relation, we obtain an effective classical Hamiltonian for the system and equations of motion for the particle's expectation values. It is shown that the effective substrate potential transits from a symmetric double-well potential to a symmetric single-well potential, and the ground state exhibits a transition from a broken symmetry phase to a restored symmetry phase as increasing the strength of quantum fluctuations. We also obtain the phonon dispersion relations and the phonon gaps at the two phases.
We present a simple and complete variational approach to the one-dimensional quantum Frenkel-Kontorova model. Dirac's time-dependent variational principle is adopted together with a Hartree-type many-body trial wave function for the atoms. The single-particle state is assumed to have the Jackiw-Kerman form. We obtain an effective classical Hamiltonian for the system, which is simple enough for a complete numerical solution for the static ground state of the model. Numerical results show that our simple approach captures the essence of the quantum effects first observed in quantum Monte Carlo studies.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.