The path integral formalism gives a very illustrative and intuitive understanding of quantum mechanics but due to its difficult sum over phases one usually prefers Schrödinger's approach. We will show that it is possible to calculate simple quantum phenomena by performing Feynman's sum over all paths staying entirely in real time. Once the propagator is obtained it is particularly easy to get the energy spectrum or the evolution of any wavefunction.
The path integral formalism gives a very illustrative and intuitive understanding of quantum mechanics but due to its difficult sum over phases one usually prefers Schrödinger's approach. We will show that it is possible to calculate simple quantum phenomena by performing Feynman's sum over all paths staying entirely in real time. Once the propagator is obtained it is particularly easy to get the energy spectrum or the evolution of any wavefunction.
The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.
We will derive a rigorous real-time propagator for the non-relativistic quantum mechanical L 2 transition probability amplitude and for the non-relativistic wavefunction. The propagator will be given explicitly in terms of the time evolution operator. The derivation will be for all self-adjoint non-vector potential Hamiltonians. For systems with potentials that carry at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real-time, time-sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in a non-standard analysis formulation. Finally, we will compute the propagator for the harmonic oscillator using the non-standard analysis Feynman path-integral formulation; we will compute the propagator without using any knowledge of the classical properties of the harmonic oscillator.
This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.
Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem.