We evaluate perturbatively the density matrix in the low-temperature limit and thus the ground-state wave function of the anharmonic oscillator up to second order in the coupling constant. We then employ Kleinert's variational perturbation theory to determine the ground-state wave function for all coupling strengths.
The variational approximation to the thermal properties of a quantum-mechanical system proposed by Giachetti and Tognetti [Phys. Rev. Lett. 55, 912 (1985); Phys. Rev. B 33, 7647 (1986)], Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)], and others is taken as the starting point, and a systematic perturbation series is developed for both the free energy and the particle density. Generalizations of the usual Feynman propagator appear naturally in the formalism. The anharmonic oscillator is treated as an example; the second-order correction to the free energy and the first-order correction to the particle density are evaluated explicitly and shown to lead to significant improvements over the variational solution.
It is observed that variationally calculated wavefunctions do not satisfy orthogonality relation when tested for the potentials like anharmonic oscillator and deviated hydrogen atom type. A new approach is suggested using quasi-states only to correlate between variationally calculated wavefunctions and orthogonality relation using perturbation theory considering the anharmonic oscillator example. Analytically expression for energy levels of the anharmonic oscillator are calculated up to the second order. A test for convergency of the perturbation theory is also discussed.
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.
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.
Recent applications of the coupled-cluster method to the (${\ensuremath{\varphi}}^{4}$)2 quantum field theory showed the necessity of a detailed analysis of its convergence behavior. The anharmonic oscillator has always served as a good test case for different approximation schemes. We treat the model both in the maximum-overlap condition and in the Hartree approximation. The ground-state energy is reproduced very well for all values of the coupling strength and already for a low-order truncation scheme. The expansion in correlation amplitudes can be carried out in extremely high order and shows a divergent tendency of the amplitudes. Introducing temperature dependence allows us to select the stable ground state out of a variety of solutions of the hierarchy of equations.