The diagonal variation-perturbation technique to fourth order is applied to the partition function of asymmetric anharmonic oscillators. Their constants are taken from spectroscopic constants of diatomic molecules. In contrast to thermodynamic perturbation theory the series does not diverge for very low and high temperatures over a wide range of anharmonicities, but it oscillates slightly around the second order result. The approximation is excellent, differing only 10−3 to 10−4 from the exact numerical results. This technique can be extended to systems of coupled anharmonic oscillators with polynomial anharmonicity and interactions.

Smearing Formula for Higher-Order Effective Classical Potentials In the variational approach to quantum statistics, a smearing formula describes efficiently the consequences of quantum fluctuations upon an interaction potential. The result is an effective classical potential from which the partition function can be obtained by a simple integral. In this work, the smearing formula is extended to higher orders in the variational perturbation theory. An application to the singular Coulomb potential exhibits the same fast convergence with increasing orders that has been observed in previous variational perturbation expansions of the anharmonic oscillator with quartic potential. Published as: J.Phys.A31:8307-8321,1998 DOI: 10.1088/0305-4470/31/41/005 arXiv categories: quant-ph

We derive the semiclassical series for the partition function of a one-dimensional quantum-mechanical system consisting of a particle in a single-well potential. We do this by applying the method of steepest descent to the path-integral representation of the partition function, and we present a systematic procedure to generate the terms of the series using the minima of the Euclidean action as the only input. For the particular case of a quartic anharmonic oscillator, we compute the first two terms of the series, and investigate their high and low temperature limits. We also exhibit the nonperturbative character of the terms, as each corresponds to sums over infinite subsets of perturbative graphs. We illustrate the power of such resummations by extracting from the first term an accurate nonperturbative estimate of the ground-state energy of the system and a curve for the specific heat. We conclude by pointing out possible extensions of our results which include field theories with spherically symmetric classical solutions.

Phase Space Representations and Perturbation Theory for Continuous-time Histories We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical histories phase space, comparing and contrasting the Q, P and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the histories formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time path (CTP) generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories. Published as: J.Math.Phys.48:072106,2007 DOI: 10.1063/1.2752009 arXiv categories: quant-ph gr-qc

Abstract We use four different methods to calculate an anharmonic correction factor fvib to the conventional RRHO partition functions for H2O, HO2, 3CH2, H2O2, and CH4 over a temperature range up to 3000 K. The exact quantum mechanical method benchmarks the other three approximate methods that are based on classical Monte Carlo phase space integrals, on vibrational perturbation theory, and on conventional harmonic partition functions evaluated with fundamental, rather than harmonic, frequencies. The last two of these methods converge on the exact partition function below temperatures that vary from 1500 K for the least anharmonic system (H2O) to 250 K for the most anharmonic system (H2O2). For 3CH2 and H2O2, both these methods are qualitatively incorrect because they are insensitive to a low energy barrier for internal motion. The classical method qualitatively overestimates quantum mechanical results at low temperatures because of the exclusion of zero point energy. However, here anharmonic corrections are small. At high temperatures, our anharmonic corrections can be large (up to 40% for CH4 at 3000 K) and at high enough temperatures the classical and exact quantum results will converge. Comparing perturbation theory and the classical method, the classical method becomes the approximate method of choice above ∼750 K for H2O2 and CH4, ∼2100 K for 3CH2, ∼2700 K for HO2, and > 3000 K for H2O.

Hilbert Spaces of bounded one dimensional non-linear oscillators are studied. It is shown that the eigenvalue structure of all such oscillators have the same general form. They are dependent only on the ground state energy of the system and a single functional $\lambda(H)$ of the Hamiltonian $H$ whose form depends explicitly on $H$. It is also found that the Hilbert Space of the non-linear oscillator is unitarily inequivalent to the Hilbert Space of the simple harmonic oscillator, providing an explicit example of Haag's Theorem. A number operator for the nonlinear oscillator is constructed and the general form of the partition function and average energy of an non-linear oscillator in contact with a heat bath is determined. Connection with the WKB result in the semi-classical limit is made. This analysis is then applied to the specific case of the $x^4$ anharmonic oscillator.