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About 2,013 papers for “Path integral formulation”

Path integral formulation of quantum mechanics

Michael Strickland · Morgan & Claypool Publishers eBooks · 2019

In this chapter, I introduce the concept of the path integral applied to non-relativistic quantum mechanics. I derive a compact expression for the path integral formulated in terms of both the Hamiltonian and Lagrangian, show how to add sources, and show how to compute expectation values of quantum operators. Finally, I discuss the Euclidean formulation and the connection to statistical mechanics.

Cited by 1Open access

Bosonization in the path integral formulation

Fujikawa, Kazuo, Suzuki, Hiroshi · arXiv (Cornell University) · 2015

We establish the direct $d=2$ on-shell bosonization $ψ_{L}(x_{+})=e^{iξ(x_{+})}$ and $ψ_{R}^{\dagger}(x_{-})=e^{iξ(x_{-})}$ in path integral formulation by deriving the off-shell relations $ψ_{L}(x)ψ_{R}^{\dagger}(x)=\exp[iξ(x)]$ and $ψ_{R}(x)ψ_{L}^{\dagger}(x)=\exp[-iξ(x)]$. Similarly, the on-shell bosonization of the bosonic commuting spinor, $ϕ_{L}(x_{+})=ie^{-iξ(x_{+})}\partial^{+}e^{-iχ(x_{+})}$, $ϕ^{\dagger}_{R}(x_{-})=e^{-iξ(x_{-})-iχ(x_{-})}$ and $ϕ_{R}(x_{-})=ie^{iξ(x_{-})}\partial^{-}e^{+iχ(x_{-})}$, $ϕ^{\dagger}_{L}(x_{+})=e^{iξ(x_{+})+iχ(x_{+})}$, is established in path integral formulation by deriving the off-shell relations $ϕ_{L}(x)ϕ^{\dagger}_{R}(x)=ie^{-iξ(x)}\partial^{+}e^{-iχ(x)}$ and $ϕ_{R}(x)ϕ^{\dagger}_{L}(x)=ie^{iξ(x)}\partial^{-}e^{iχ(x)}$.

Cited by 0Open access

Path Integral Formulation of Quantum Mechanics

P. C. Deshmukh · 2023

We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.—Richard P. Feynman

Cited by 0

Electronegativity Scale from Path Integral Formulation

Mihai V. Putz, Nino Russo · ArXiv.org · 2003

In the framework of the density functional theory a new electronegativity formulation via the Feynman-Kleinert path integral formalism in the markovian limit is proposed. The computation of the electronic density follows, in terms of partition function, the same procedure of the Levy constrained-search for the wave function. The obtained electronegativity scale seems to respect the main criteria largely used for its acceptability.

Cited by 1Open access

Path integral formulation of quantum mechanics

Dey, Arpan · Zenodo (CERN European Organization for Nuclear Research) · 2025

In this article, we present a rigorous introduction to Feynman’s path integral formulation of quantum mechanics. We start by outlining the motivation behind the path integral formulation of quantum mechanics. Then we build the mathematics required for defining the sum-over-paths and derive the free particle propagator, as well as the propagator for a particle with a non-zero potential energy, with particular focus on linear and quadratic potential energies. We use these results to arrive at some standard results in quantum physics using path integrals: namely Planck-Einstein equation, de-Broglie equation and Schrödinger equation. This helps us appreciate the equivalence between the path integral and the canonical formulation, as well as understand the difference in the approaches employed by the two formulations to arrive at the results. We also use Python to generate plots of the free particle wave function and propagator as functions of position and time, and explore the conceptual difference between the wave function in canonical quantum mechanics and the propagator used in the path integral formulation. We also introduce the basic idea behind perturbation theory, by using the free particle propagator to study a particle that moves between two potential-free points in spacetime, but via intermediate points with non-zero potentials, in one of the appendices. In the end, we discuss path integrals in a broad perspective, and make some general comments on the future of theoretical physics.

Cited by 0Open access

Bosonization in the path integral formulation

Kazuo Fujikawa, Hiroshi Suzuki · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D, Particles, fields, gravitation, and cosmology · 2015

We establish the direct $d=2$ on-shell bosonization ${\ensuremath{\psi}}_{L}({x}_{+})={e}^{i\ensuremath{\xi}({x}_{+})}$ and ${\ensuremath{\psi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{\ensuremath{-}})={e}^{i\ensuremath{\xi}({x}_{\ensuremath{-}})}$ in path integral formulation by deriving the off-shell relations ${\ensuremath{\psi}}_{L}(x){\ensuremath{\psi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=\mathrm{exp}[i\ensuremath{\xi}(x)]$ and ${\ensuremath{\psi}}_{R}(x){\ensuremath{\psi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=\phantom{\rule{0ex}{0ex}}\mathrm{exp}[\ensuremath{-}i\ensuremath{\xi}(x)]$. Similarly, the on-shell bosonization of the bosonic commuting spinor, ${\ensuremath{\phi}}_{L}({x}_{+})=\phantom{\rule{0ex}{0ex}}i{e}^{\ensuremath{-}i\ensuremath{\xi}({x}_{+})}{\ensuremath{\partial}}^{+}{e}^{\ensuremath{-}i\ensuremath{\chi}({x}_{+})}$, ${\ensuremath{\phi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{\ensuremath{-}})={e}^{\ensuremath{-}i\ensuremath{\xi}({x}_{\ensuremath{-}})\ensuremath{-}i\ensuremath{\chi}({x}_{\ensuremath{-}})}$ and ${\ensuremath{\phi}}_{R}({x}_{\ensuremath{-}})=i{e}^{i\ensuremath{\xi}({x}_{\ensuremath{-}})}{\ensuremath{\partial}}^{\ensuremath{-}}{e}^{+i\ensuremath{\chi}({x}_{\ensuremath{-}})}$, ${\ensuremath{\phi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{+})={e}^{i\ensuremath{\xi}({x}_{+})+i\ensuremath{\chi}({x}_{+})}$, is established in path integral formulation by deriving the off-shell relations ${\ensuremath{\phi}}_{L}(x){\ensuremath{\phi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=i{e}^{\ensuremath{-}i\ensuremath{\xi}(x)}{\ensuremath{\partial}}^{+}{e}^{\ensuremath{-}i\ensuremath{\chi}(x)}$ and ${\ensuremath{\phi}}_{R}(x){\ensuremath{\phi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=i{e}^{i\ensuremath{\xi}(x)}{\ensuremath{\partial}}^{\ensuremath{-}}{e}^{i\ensuremath{\chi}(x)}$.

Cited by 1Open access

Validity of the path-integral formulation

Frank E. Richards · Physical review. B, Solid state · 1974

The widely used path-integral formulation of transport problems, which has previously been found valid only for the case of isotropic elastic scattering, is rigorously proved to be valid for general scattering in any material for which the Boltzmann equation is valid, for a wide variety of transport problems. In particular, it is valid for arbitrary space and time dependence of the driving forces and of the scattering (including surface scattering) even in the presence of a uniform static magnetic field of any value.

Cited by 3

Path integral formulation of evolving ecosystems

BAGNOLI, FRANCO, Michele Bezzi · Florence Research (University of Florence) · 1999

The master equation of an ecosystem in evolution on a static fitness landscape can be written as a sum over all possible phylogenetic generations of individuals. Such a time trajectory represents alternating phases of selection (the fitness landscape) and mutation (during reproduction). This approach permits to restrict the sum to the most probable paths. For medium and small time scales, a smooth landscape and long-range mutations it is shown that the equilibrium distribution is of Boltzmann type, with the fitness as an energy and mutations as temperature.

Cited by 0

Berry’s phase through the path integral formulation

M.A. Alves, M. T. Thomaz · American Journal of Physics · 2007

We study the evolution of a quantum system under adiabatic conditions using the path integral formulation. We obtain the Berry phase for systems with nondegenerate and degenerate energy spectra and recover well-known results. We also present a simple derivation of the adiabatic theorem.

Cited by 1

Path-integral formulation of optical beam propagation

S. R. Vatsya · Journal of the Optical Society of America B · 2005

Propagation of optical beams described by the Schrödinger-type equations can be expressed in terms of the propagator obtained in the framework of the Feynman path-integral formulation of quantum mechanics, which can also be obtained from a standard representation of the propagator by an exact integration. A numerical scheme is developed to evaluate the path-integral propagator by the fast Fourier transform method. The resulting procedure is convenient, efficient, versatile, and accurate, comparing favorably with other methods. Numerical operations required are about half of those in a commonly used method employing fast Fourier transforms. Further, a transformation is shown to eliminate inconvenience associated with a beam of rapidly varying radius. The scheme is illustrated with several examples, including the soliton propagation and the converging beams in a self-focusing medium together with the impact of plasma it generates.

Cited by 6Open access