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 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.
We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.
Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.
It is interesting to know when to expect the wave nature of a particle propagation to be well approximated by the properties of the trajectory of a classical particle. Usually, whenever there is an integral expression for a given quantity, it is often easier to employ approximate methods compared to a differential equation. It is simply a matter of convenience representing a path by a classical path and fluctuations around it. For the quasi-classical approximation, however, it is critical to expand around the path leading to the dominant contribution, i.e. the classical path. The knowledge of the stationary phase approximation is very appropriate to apply to path integration. This heading examines some specific systems of particular interest in people investigation. The harmonic oscillator has practical applications in a variety of domains of modern physics, such as molecular spectroscopy, solid state physics, nuclear structure, quantum field theory, quantum statistical mechanics and so on.
Using the discretized path integral formalism we develop a numerically convenient and accurate method for evaluating the finite time propagator (and density matrix) of a given system. The eigenfunctions and energies of large number of states of the system are also found. We demonstrate the accuracy of the method by computing eigenvalues, eigenfunctions, position and flux autocorrelation functions and complex time propagators for a large number of systems and comparing these results with those available from other sources. In all the cases we find very good agreement. The method thus provides a very simple and accurate procedure for the study of static and dynamic properties of a quantum system.