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Progress of Theoretical Physics

A Thermodynamic Perturbation Theory of the Anharmonic Oscillator. I

S. Naya

Published Aug 1, 1972• 3 citations
Publisher

Abstract

A new formalism of the thermodynamic perturbation theory for the quantum-anharmonic oscillator is proposed. By this method, for each order of the perturbation series, the partition function and the free energy of the system can be obtained in a compact form which is expressed by the temperature Green's function of the harmonic quanta. Based upon a unified method, the various expressions of the perturbation series for the partition function and the free energy are also given. It is shown that the perturbation series for the free energy can be naturally formulated in terms of a cluster expansion.

DOI
10.1143/ptp.48.407

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A new formalism of the thermodynamic perturbation theory for the quantum-anharmonic oscillator is proposed. By this method, for each order of the perturbation series, the partition function and the free energy of the system can be obtained in a compact form which is expressed by the temperature Green's function of the harmonic quanta. Based upon a unified method, the various expressions of the perturbation series for the partition function and the free energy are also given. It is shown that the perturbation series for the free energy can be naturally formulated in terms of a cluster expansion.
A thermodynamic perturbation method for the quantum-anharmonic oscillator is studied, based upon the Bloch equation. By this method the density matrix (Green's function) for the anharmonic oscillator can be obtained in a compact form for each order of the perturbation series. It is shown that the density matrix of the anharmonic oscillator can be readily expanded in series of the Hermite polynomials. The various expressions of the expansion series for the density matrix are also derived by a unified method.
The path-integral approach is used for determining quantum corrections to the free energy of nonlinear systems. An effective potential, to be inserted in the configurational integral, is constructed by a new variational scheme which gives the quantum modification of the potential due to the anharmonic part together with the frequency renormalization. The single anharmonic oscillator is studied first. Then the extension to fields is presented and a new general expression for the partition function is given. The method is applied to the sine-Gordon chain and explicit calculations of the specific heat are shown.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
In this article the anharmonic oscillator in quantum mechanics is discussed. First propagators of less complex systems are found by using the path integral method. With perturbation theory an expression, up to any order, has been found for the vacuum propagator of the anharmonic oscillator. In this article only the first and second order were calculated and by looking at the n-pointfunctions the Feynman diagrams of these orders were found. It appears that the anharmonic term up to first order causes a change in the frequency of the system. Looking at second order perturbation theory gives five diagrams. The first four of these diagrams are simple diagrams and again these diagrams alter the frequency of the system. The change in the frequency looks a lot like a series and it is possible to ’guess’ a general expression up to any order for all simple diagrams. This expression seems to fit qualitatively for higher orders however the prefactors were not checked nor is there a proof formulated. The fifth diagrams is a complex diagram and this diagram has not been solved in a satisfying way. Finally some recommendations for additional research are given.
Abstract We derive an integral-free thermodynamic perturbation series expansion for quantum partition functions which enables an analytical term-by-term calculation of the series. The expansion is carried out around the partition function of the classical component of the Hamiltonian with the expansion parameter being the strength of the off-diagonal, or quantum, portion. To demonstrate the usefulness of the technique we analytically compute to third order the partition functions of the 1D Ising model with longitudinal and transverse fields, and the quantum 1D Heisenberg model.
The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.