Eigenvalues are determined for the plane pendulum problem by the WKBJ method in one- and four-term approximations, and the results are compared. It is found that at high quantum numbers, the four-term WKBJ approximation can yield eigenvalues of eight-significant-figure accuracy, but for low quantum numbers the results continue to be poor.
Eigenvalues are determined for the plane pendulum problem by the WKBJ method in one- and four-term approximations, and the results are compared. It is found that at high quantum numbers, the four-term WKBJ approximation can yield eigenvalues of eight-significant-figure accuracy, but for low quantum numbers the results continue to be poor.
The path integral formulation of quantum mechanics in the \VKB approximation (quasi-classical) gives to us a useful tool to relate the classical systems (with or without chaos) with its quantum analogs. \Ve use this point of view to compute the eigenfunctions and energy eigenvalues for the quantum pendulum comparing with those obtained with the standard \VKB approximation. The method that we employed in this work was developed mainly by Gutzwiller, and one of its main features is that this method can be used in integrable and no integrable problems.
The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity (tau) is also dramatic, having the special limiting case of tautoinfty at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.
A detailed quantum mechanical treatment of the motion of a plane pendulum is given. Exact solution of the Schrödinger equation shows that the energy eigenfunctions and eigenvalues go over to those of a plane rotator for vanishing acceleration due to gravity g and to those of a harmonic oscillator in the limit of infinitely massive bob. In view of this, perturbation calculations have been done using rotator and oscillator basis, and conditions for convergence of the perturbation series in both cases have been discussed.
The JWKB approximation is widely used in many problems in quantum mechanics for its simplicity and also because a closed-form solution often provides more insight than a numerical one. Unfortunately, like all approximations, the JWKB method has its limitations: Firstly, it may fail near or at a classical turning point, and secondly, it employs solutions for a potential which deviates from the actual potential W(x). In this work, a new method is proposed to avoid the limitations of the JWKB method while keeping its advantages. First, the singularity around the classical turning point is resolved not only for a linear potential as proposed by Langer, but also for any potential with a linear leading term through the turning point, and second, a solution is constructed in closed form which is an exact solution for the actual potential W(x). The new method consists in finding a substitute potential U(x) in such a way that, together with its (DELTA){U,x} it is equal to the actual W(x). The solution is then based on U(x) but actually is the solution for W(x) because W(x) = U(x) + (DELTA){U,x}. The new method is first applied to potentials with only two turning points. The eigenvalue problem is reduced to the solution of a linear equation and the energy eigenvalues are exactly given. Then, potentials with more than two turning points are studied and the eigenvalue problem is again reduced to the solution of a polynomial equation whose order depends upon the number of turning points. Then the method is applied to the hindered or restricted rotation. The energy levels are found to be associated with the N-th roots of 1 in the complex plane, i.e. on the unit circle. The method described here is of rather general nature and can be used to solve a number of problems in quantum mechanics, where a solution for the Schrodinger equation in closed form is needed. The applications provided in this work are only given as examples of the applicability and the relative simplicity of the method.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1980 .S223. Source: Dissertation Abstracts International, Volume: 41-03, Section: B, page: 1002. Thesis (Ph.D.)--University of Windsor (Canada), 1980.
In quantum mechanics, there exist a lot of problems from the domain of atomic, molecular, solid-state physics, physics of magnetic phenomena, nuclear physics, etc. The number of problems which may be solved exactly are few such as some one-dimensional problems, problems in the rectangular well (barrier), the oscillator, problems of motion in a central field such as motion in a Coulomb field. These methods are well formulated and are used in evaluations to a greater approximation. However, in some cases, these may create problems that are very difficult, if not impossible, to be solved exactly. In such problems, the Hamiltonian cannot be exactly diagonalized in a simple manner. For such problems, we resort to approximate methods for finding the eigenvalues and eigenfunctions of the Hamiltonian. Besides, these approximate methods yield results closer to experimental ones. Amongst the approximate methods, we examine the variational method.
An improved algorithm to implement our previously proposed static quantum embedding method MPCC (J. Chem. Phys. 161, 164107 (2024)) is presented. It uses the density-fitting (DF) approximation for the electron-repulsion integrals (ERIs) across all layers of the embedding method: the low-level method, the screened interaction, and the high-level method. Additionally, an approximate factorized solution of the Sylvester equation is employed based on the Laplace transformation for the low-level method. To build the screened interaction, we have explored an approximate construction of it based on the distinguishable cluster approximation (DCA) by Kats and Manby (J. Chem. Phys. 141, 061101 (2014)), which amounts to neglecting certain higher-order exchange and ladder diagrams. Combining all these techniques, we have obtained an O(N4) algorithm for the low-level method; a non-iterative NfragO(N4) algorithm for the screened interaction; and an iterative O(Nfrag6) algorithm for the fragment solver, where Nfrag is the number of fragment orbitals and N is the total number of orbitals. The resulting low-scaling implementation of the MPCC method is applied to trans polyacetylenes, that is, C2nH2n+2 molecules for chain lengths n=1−8, and the timing and accuracy of the method are compared to the original unfactorized implementation. Moreover, the approximations are further validated by applying the method to the potential energy surface of the \ce{O3} molecule. Finally, we apply the method to the selected molecules in the S22 dataset for which the performance of the MP2 method is known to be poor. The results demonstrate that the DF-based implementation of the MPCC method provides a significant reduction in computational cost while maintaining accuracy comparable to that of the original implementation.