Feynman Path Integral Representation Research Articles

Spin and exchange in mixed quantum states from first principles in Feynman path integral quantum statistical mechanics representation

The exchange and space correlations in the system of indistinguishable electrons against the background of thermal fluctuations are studied from first principles. A complete set of mutually orthogonal spin-coordinate basis functions symmetric with respect to permutations in coordinate and spin variables is constructed by the action of Young symmetry operators. As an illustration of a general approach, Path Integral Monte Carlo (PIMC) computer simulation of mixed quantum states of four electrons in a model ellipsoidal nanoscopic trap with soft walls is performed. The equilibrium value of the spin is calculated as well as the mechanism of its formation depending on the temperature and shape of the trap in connection with the evolution of the spatial organization of electrons is analyzed. The Quantum-statistical calculation, executed with a fundamentally accurate description of the permutation symmetry of electrons and the spin variable, includes in explicit form the Coulomb and exchange correlations of all orders and the energy exchange with a thermal bath. The equilibrium values ​​of the spin in dependence of the shape of the potential well and temperature as well as statistical distributions over the eigenfunctions of the spin squared operator are obtained. A general, fundamentally accurate and numerically implementable on computers available today computers approach for describing the spin states of systems of different numbers of indistinguishable fermions with spin 1/2 in the representation of path integrals is formulated.The regularities in dependence of spin on the shape of the system and on the temperature are studied. The variations in the geometry of the trap in direction of elongated shapes are accompanied by the transfer of a part of the electron cloud first from the equator to the poles of the ellipsoidal potential well, and then in the opposite direction with the subsequent collapse of the ring-shaped form into a compact clot in the center and its decomposition into two fragments localized on the axis of rotation. The spin of the mixed quantum state of four electrons at room temperature varies depending on the geometry of the system in the opposite direction, compared with the system of two and three electrons, reaching its maximum at a spherical shape. Spin states of electron ensembles several nanometers in size are unstable to thermal perturbations already in the region of room temperatures. Compression leads to a stabilization of spin states. The calculation results indicate a strong dependence of the energy of collective mixed spin states of electrons on a detailed account of exchange and Coulomb high-orders spatial correlations.

Ray-wave duality in classical optics: crossing the Feynman bridge.

Classical ray and wave optics share a common structure with quantum mechanics. Much of this results from the representations given to abstract operators. It is possible to build a mathematical bridge that allows one to transform back and forth between the ray and wave picture. This is given by the Feynman path integral representation. The map between the ray and wave description is given in terms of a sum over histories integration. In particular, it allows for the computation of wave phenomena by the summation of multiple ray paths of a phase function connected to the classical optical path length.

A white noise approach to the Feynman integrand for electrons in random media

Using the Feynman path integral representation of quantum mechanics it is possible to derive a model of an electron in a random system containing dense and weakly coupled scatterers [see F. Edwards and Y. B. Gulyaev, “The density of states of a highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964)]. The main goal of this paper is to give a mathematically rigorous realization of the corresponding Feynman integrand in dimension one based on the theory of white noise analysis. We refine and apply a Wick formula for the product of a square-integrable function with Donsker's delta functions and use a method of complex scaling. As an essential part of the proof we also establish the existence of the exponential of the self-intersection local times of a one-dimensional Brownian bridge. As a result we obtain a neat formula for the propagator with identical start and end point. Thus, we obtain a well-defined mathematical object which is used to calculate the density of states [see, e.g., F. Edwards and Y. B. Gulyaev, “The density of states of a highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964)].

The theory of quantum dots in Feynman path integral representation: Spin states and magnetic susceptibility

The theory of quantum dots in Feynman path integral representation: Spin states and magnetic susceptibility

Angular smoothing and spatial diffusion from the Feynman path integral representation of radiative transfer

The propagation kernel for time dependent radiative transfer is represented by a Feynman path integral (FPI). The FPI is approximately evaluated in the spatial-Fourier domain. Spatial diffusion is exhibited in the kernel when the approximations lead to a Gaussian dependence on the Fourier domain wave vector. The approximations provide an explicit expression for the diffusion matrix. They also provide an asymptotic criterion for the self-consistency of the diffusion approximation. The criterion is weakly violated in the limit of large numbers of scattering lengths. Additional expansion of higher-order terms may resolve whether this weak violation is significant.

A Feynman Path Integral Representation for Elastic Wave Scattering by Anisotropic Weakly Perturbations

We write a space-time Feynman Path Integral representation for scattered elastic wave fields from a weakly anisotropic non-homogeneity.

Path integral evaluation of equilibrium isotope effects

A general and rigorous methodology to compute the quantum equilibrium isotope effect is described. Unlike standard approaches, ours does not assume separability of rotational and vibrational motions and does not make the harmonic approximation for vibrations or rigid rotor approximation for the rotations. In particular, zero point energy and anharmonicity effects are described correctly quantum mechanically. The approach is based on the thermodynamic integration with respect to the mass of isotopes and on the Feynman path integral representation of the partition function. An efficient estimator for the derivative of free energy is used whose statistical error is independent of the number of imaginary time slices in the path integral, speeding up calculations by a factor of approximately 60 at 500 K and more at room temperature. We describe the implementation of the methodology in the molecular dynamics package AMBER 10. The method is tested on three [1,5] sigmatropic hydrogen shift reactions. Because of the computational expense, we use ab initio potentials to evaluate the equilibrium isotope effects within the harmonic approximation and then the path integral method together with semiempirical potentials to evaluate the anharmonicity corrections. Our calculations show that the anharmonicity effects amount up to 30% of the symmetry reduced reaction free energy. The numerical results are compared with recent experiments of Doering et al., [J. Am. Chem. Soc. 128, 9080 (2006); J. Am. Chem. Soc.129, 2488 (2007)] confirming the accuracy of the most recent measurement on 2,4,6,7,9-pentamethyl-5-(5,5-(2)H(2))methylene-11,11a-dihydro-12H-naphthacene as well as concerns about compromised accuracy, due to side reactions, of another measurement on 2-methyl-10-(10,10-(2)H(2))methylenebicyclo[4.4.0]dec-1-ene.

Euclidean Quantum Mechanics and Universal Nonlinear Filtering

An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr¨odinger equation.

Feynman path integrals for the inverse quartic oscillator

The Feynman path integral representation for the weak solution of the Schrödinger equation with an inverse quartic oscillator potential is given in terms of a well defined infinite dimensional oscillatory integral. An analytically continued Wiener integral representation for the solution is provided and an explicit description of the quantum dynamics associated with a nonessentially self-adjoint Hamiltonian is given.

The time-dependent quartic oscillator—a Feynman path integral approach

The Schrödinger equation with a time-dependent quadratic plus quartic Hamiltonian is considered. A rigorous Feynman path integral representation for its solution is given in terms of a well-defined infinite-dimensional oscillatory integral.

Feynman path integrals for the time dependent quartic oscillator

The Schrödinger equation with a time dependence in both a quadratic and a quartic potential is considered. Existence of solutions is shown and a rigorous Feynman path integral representation for the solution is given in terms of well-defined infinite-dimensional oscillatory integrals. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Bayesian approach to inverse quantum statistics: Reconstruction of potentials in the Feynman path integral representation of quantum theory

The Feynman path integral representation of quantum theory is used in a non–parametric Bayesian approach to determine quantum potentials from measurements on a canonical ensemble. This representation allows to study explicitly the classical and semiclassical limits and provides a unified description in terms of functional integrals: the Feynman path integral for the statistical operator and its derivative with respect to the potential, and the integration over the space of potentials for calculating the predictive density. The latter is treated in maximum a posteriori approximation, and various approximation schemes for the former are developed and discussed. A simple numerical example shows the applicability of the method.

Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.

Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E/ c 2 moving in the two-dimensional space orthogonal to the classical path for a time τ= L/ c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the “creeping” case to the classically allowed motion in the “whispering gallery” case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.

Generalized infinite-dimensional Fresnel integrals

A generalized infinite dimensional oscillatory integral with a polynomially growing phase function is defined and explicitly computed in terms of an absolutely convergent Gaussian integral. The results are applied to the Feynman path integral representation for the solution of the Schrödinger equation with an anharmonic oscillator potential. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Schrödinger operators with singular potentials and magnetic fields

A formal Schrödinger operator of the form in is considered, where is a bounded measurable vector-valued function and both and are measures satisfying certain additional conditions. It is shown that one can give meaning to such an operator as a lower bounded self-adjoint operator in . The corresponding heat kernel is constructed and its small-time asymptotics are obtained. A rigorous Feynman path integral representation for the solutions of the heat and Schrödinger's equations with generator is given.

Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

AbstractAn interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Variational Method for the Calculation of Plasma Phase Diagrams in the Path Integral Representation

The use of variational method in functional integral approach is discussed for fermion and boson systems with Coulomb interaction. The formal general expression of thermodynamic potential is obtained by Feynman path integral technique and representation of Coulomb interaction with functional integrals. Introduced additional complex field show to transform the problem to calculation of functional integrals containing third order vertices. The thermodynamic potential can be found from variational principle with respect to field cumulants. The calculation of the equation of state and critical properties is demonstrated for symmetrical plasma by variation of finite number of parameters in the propagator.

Stochastic construction of a Feynman path integral representation for Green’s functions in radiative transfer

A construction of a stochastic, Lagrangian path integral representation is presented for classical Green’s functions or propagators for radiative transfer in random scattering medias. As Fermat rays comprising the initial collimated pulse (e.g., from a laser) enter the medium, they evolve into a bundle of random paths or trajectories due to scattering. Stochastically, this can be interpreted as a bundle of randomly evolving Markov trajectories traced out by a gas or ensemble of “Brownian-type” classical point photons undergoing multiple scatterings. The path integral is structurally of the same form as a Wick rotated Euclidean quantum Feynman integral with direct optical analogs of the Hamiltonian and Lagrangian emerging. However, the optical stochastic integral is real and is defined in real time becoming exponentially damped rather than oscillatory. The calculation also constitutes an alternative mathematical derivation of the time dependent diffusion equation from the radiative transfer theory. The limits of the path integral representation give a maximally diffused Gaussian distribution of the heat kernal form (the large scatter limit) and a Beer’s law exponential decay corresponding to the extremal Fermat rays obeying Euler–Lagrange equations (the zero scatter limit). The approach also highlights the direct structural analogs between classical radiative transfer and optical diffusion in real time (t) and ordinary quantum mechanics in Euclidean time (√(−1)t=it).

Feynman path-integral for the damped Caldirola-Kanai action

We propose a local Feynman path-integral representation for the damped Caldirola-Kanai action.