Abstract Path integral formulae for the (p, q) system and (a†, a) of the Bose and Fermi systems can only be calculated exactly in the harmonic oscillator case. Other physical situations require some approximation schemes. In this chapter we try to make explicit calculations for the harmonic oscillator and discuss a perturbation technique and the WKB approximation that is considered to be most suited to the path integral formalism. As an illustrative example, a detailed computation is performed for the case where non-trivial solutions, called instantons, exist in the classical equation of motion.
Abstract Path integral formulae for the (p, q) system and (a†, a) of the Bose and Fermi systems can only be calculated exactly in the harmonic oscillator case. Other physical situations require some approximation schemes. In this chapter we try to make explicit calculations for the harmonic oscillator and discuss a perturbation technique and the WKB approximation that is considered to be most suited to the path integral formalism. As an illustrative example, a detailed computation is performed for the case where non-trivial solutions, called instantons, exist in the classical equation of motion.
Perturbation theory for non-degenerate and degenerate systems, variational methods, and WKB approximation are covered in this chapter. First, second, and kth order corrections to energy and wave function ket of non-degenerate states, energy eigenvalues, and wave functions kets of anharmonic oscillator, first-order perturbation theory for degenerate states and linear stark effect are written under perturbation theory. Second part of the chapter includes the variational methods for computation of the energy for ground state of helium atom, Rayleigh-Ritz variational method and its application to hydrogen molecule ion, calculation of energy of excited states, and its application to 1D harmonic oscillator. The topics which are covered under WKB approximation are the classical and non-classical regions of particle motion, alternative derivation of WKB formulae, connecting formulae, and quantum condition for bound state. The kth order corrections to energy and wave function kets for non-degenerate perturbation theory, treatment of WKB approximation from two different approaches and the detailed derivation of connecting formulae, application of variational method to compute energy for excited states are topics which are not generally found in existing text books on quantum mechanics. A good number of solved examples illustrating the importance of approximation methods in contemporary fields are added at the end of chapter.
Abstract The path integral discussed in the preceding chapters can be exactly calculated only in a few restricted systems such as harmonic oscillators. The WKB method and perturbational expansions based on the above path integral method are in fact applicable to systems close to classical ones and weakly interacting systems, respectively. However, in order to study interesting phenomena such as high Tc superconductivity, we have to treat strongly correlated systems in which the quantum effect plays an important role. It is effective in these systems to extract some physically crucial contributions numerically. This will be possible if we use the quantum Monte Carlo method. Although there are many different types of quantum Monte Carlo methods, we explain here mainly the quantum statistical Monte Carlo method based on a Trotter-like formula and on the important sampling method applied to transformed classical systems.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. After an introduction including a very brief historical overview of the subject, we derive a path integral expression for the propagator in quantum mechanics, including the free particle and harmonic oscillator as examples. We then discuss a variety of applications, including path integrals in multiply-connected spaces, Euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in quantum field theory, and instantons via path integrals. For the most part, the emphasis is on explicit calculations in the familiar setting of quantum mechanics, with some discussion (often brief and schematic) of how these ideas can be applied to more complicated situations such as field theory.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.