Feynman Path Integral Formulation Research Articles

Quantum propagator for some classes of three-dimensional three-body systems

In this work we solve exactly a class of three-body propagators for the most general quadratic interactions in the coordinates, for arbitrary masses and couplings. This is done both for the constant as the time-dependent couplings and masses, by using the Feynman path integral formalism. Finally, the energy spectrum and the eigenfunctions are recovered from the propagators.

Path-integral formulation of optical beam propagation

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.

The sources of Schwinger's Green's functions

Julian Schwinger's development of his Green's functions methods in quantum field theory is placed in historical context. The relation of Schwinger's quantum action principle to Richard Feynman's path-integral formulation of quantum mechanics is reviewed. The nonperturbative character of Schwinger's approach is stressed as well as the ease with which it can be extended to finite temperature situations.

Determination of the Electron Transfer Mechanism through Decomposition of the Density Matrix

We present a modified Feynman and Vernon's path integral formalism which allows independent consideration of coherent superexchange and incoherent hopping pathways of charge transfer processes in order to determine the type of transport mechanism. By classifying the pathways between donor and acceptor into different mechanisms and by decomposing the density matrix of donor−bridge−acceptor triads into corresponding partial matrices, the contribution of each mechanism is obtained separately. Numerical tests confirm that the scheme is valid and efficient in exploring the transport mechanism and that the incoherent hopping mechanism tends to govern charge transfer processes even in systems with high-energy bridge states.

Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions

Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. We use this fact to find a unique semiclassical representation without the Van Vleck determinant, applicable to a large class of correlation functions expressible as quantum fidelity. As in the Feynman path integral formulation of quantum mechanics, all contributing trajectories have the same amplitude: that is why we denote it the "dephasing representation." By relating our approach to the problem of existence of true trajectories near numerically-computed chaotic trajectories, we make the approximation rigorous for any system in which the shadowing theorem holds. Numerical implementation only requires computing actions along the unperturbed trajectories and not finding the shadowing trajectories. While semiclassical linear-response theory was used before in quasi-integrable and chaotic systems, here its validity is justified in the most generic, mixed systems. Dephasing representation appears to be a rare practical method to calculate quantum correlation functions in nonuniversal regimes in many-dimensional systems where exact quantum computations are impossible.

Path integral approach in the plasma radiation theory

In this paper, we add a supplementary contribution and support to the path integral method by exploring the theory of radiation in plasma physics. This method, based on the Feynman path integral formalism, has the advantage of treating the electrons and ions on the same physical basis and both time-independent and time-dependent problems on the same footing. Since the main quantity in the problem of plasma radiation is the dipolar autocorrelation function of the radiator, we derive it using the path integral technique in its perturbative form. We show that the dipolar autocorrelation function could be written as a generalized expression taking into account the time-dependency effects.

An elementary derivation of the harmonic oscillator propagator

The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

Nuclear degrees of freedom in calculated isotropic hyperfine coupling constants of the ethyl radical: a Feynman path integral – ab initio study

The Feynman path integral quantum Monte Carlo formalism has been combined with an electronic Hamiltonian of the B3LYP-EPRIII type to study the isotropic hyperfine coupling (hfs) constants of the ethyl radical in thermal equilibrium ( T=25 K). The nuclear vibrations and the intramolecular rotation around the C–C bond have a sizeable influence on the hfs parameters. The thermally averaged numbers differ from single-nuclear configuration values calculated at the minimum of the potential energy surface. The analysis of the distribution functions for the isotropic hfs constants of C 2H 5 offers insight into the strong coupling between calculated hfs parameters and the nuclear configuration adopted.

Bipolaron stability in an ellipsoidal potential well

AbstractBipolaron states in ellipsoidal quantum dots are examined within the framework of Feynman's path‐integral formalism. The following characteristics of these quasiparticles are calculated: the binding energy, the number of phonons in the bipolaron cloud and the radii. The impact of both shape and size of a mesoscopic structure on the bipolaron characteristics is studied. It is shown, that in the case of strong confinement the influence of the confinement geometry is very pronounced. Two different cases are possible. (i) When the radius Rk corresponding to a given k‐axis of the ellipsoidal potential well is fixed to be less than the radius R0, at which the bipolaron binding energy in a spherical quantum dot achieves its maximum, the bipolaron binding energy passes consecutively through a local maximum and a local minimum with increasing the confinement strength corresponding to the other axes (i ≠ k). (ii) When Rk ≫ R0, the bipolaron binding energy shows only a monotonous growth with decreasing the effective sizes of the structure along the other axes (i ≠ k).

A path integral approach to lattice point problems

The main problem in (planar) lattice point theory consists in counting lattice points under the graph of positive functions supported on [0,M] and with radius of curvature comparable to M. We prove that, in some sense motivated by Feynman path integral formulation of Quantum Mechanics, for “most” functions the lattice error term in the area approximation is O(M1/2+ε). This complements Jarnı́k construction of curves with an optimal O(M2/3) error term.

Exchange symmetry in the Feynman path-integral formalism for many-electron systems with spin

The problem of explicit inclusion of the spin-dependent permutation symmetry in the Feynman path-integral method is solved for many-electron systems. As an example, the Monte Carlo numerical calculations for near-degenerate hydrogen plasma are presented. The new method is promising in systematic computations for systems of quantum particles at finite temperatures, with an accuracy similar to that for systems of classical particles.

Coupling between fragment radial motion and the transversal degrees of freedom in cold fission

The tunneling of a quasibound dinuclear system along the main radial mode is studied for the case of cold fission when the very small excitation energy present in the decaying system is distributed among molecular type transversal collective degrees of freedom. Such a collective mode has typical oscillation periods much smaller than the tunneling time of the main radial mode and, therefore, we discuss various approaches that are suitable in dealing with this problem. The modified penetrabilities are obtained at first by solving the coupled set of Schr\"odinger equations in the adiabatic and the diabatic approximations. The comparison with the one-dimensional penetrability is done also in the frame of the Feynman path integral formalism for a very large inertial mass and small transversal vibrations frequency. In all cases the coupling of the radial(fission) mode to the transversal degrees of freedom leads to a change in the tunneling probability that is, however, not so strong to determine significant variations in the fission lifetime.

Polaron effects in electron channels on a helium film

Using the Feynman path-integral formalism we study the polaron effects in quantum wires above a liquid helium film. The electron interacts with two-dimensional (2D) surface phonons, i.e. ripplons, and is confined in one dimension (1D) by an harmonic potential. The obtained results are valid for arbitrary temperature ($T$), electron-phonon coupling strength ($\alpha $), and lateral confinement ($\omega_{0}$). Analytical and numerical results are obtained for limiting cases of $T$, $\alpha $, and $\omega_{0}$. We found the surprising result that reducing the electron motion from 2D to quasi-1D makes the self-trapping transition more continuous.

Klein–Gordon equation from the path integral formalism

By using Feynman's path integral formalism in the second order for the relativistic Lagrangian for a spinless particle in a gauge field and applying the covariant derivative instead of the commonly used derivative, but without knowing the operator expressions for the momentum and energy, one can obtain the Klein–Gordon equation.

Nuclear quantum effects in calculated NMR shieldings of ethylene; a Feynman path integral – ab initio study

The Feynman path integral Monte-Carlo formalism has been combined with the gauge-including atomic orbital (GIAO) approach to study the absolute magnetic shieldings of C 2H 4 under consideration of the thermal and quantum degrees of freedom of the nuclei. An ab initio Hamiltonian has been employed for the statistical averaging of NMR parameters. The spatial fluctuations of the atoms around their equilibrium positions lead to a deshielding of both types of nuclei relative to their shieldings at the minimum of the potential energy surface. This behavior is caused by quantum mechanical effects. It is supported by bondlength elongations in thermal equilibrium.

How the New Scale Relativity Theory resolves some quantum paradoxes

It is explicitly shown that within the framework of the New Relativity Theory, some Quantum Mechanical Paradoxes like the Einstein–Rosen Podolsky and the Black Hole Information Loss, are easily resolved. Such New Relativity Theory requires the introduction of an infinite-dimensional quantum space-time as has been shown recently by one of us. This can be viewed as just another way of looking at Feynman's path integral formulation of Quantum Mechanics. Instead of having an infinite-dimensional functional integral over all paths, smooth, forwards and backwards in time, random and fractal, in a finite-dimensional space-time, one has a finite number of paths in an infinite-dimensional quantum space-time. We present a few-lines proof why there is no such thing as an EPR Paradox in this New Relativity theory. The reason is not due to a superluminal information speed but to a divergent information charge density. In the infinite-dimensional limit, due to the properties of gamma functions, the hypervolume enclosed by a D-dim hypersphere, of finite non-zero radius, shrinks to zero: to a hyperpoint, the infinite-dimensional analog to a point. For this reason, Information flows through the infinite-dimensional hypersurface of non-zero radius, but zero size, the hyperpoint, in an instant. In this fashion we imbue an abstract mathematical “point” with a true physical meaning: it is an entity in infinite dimensions that has zero hypervolume at non-zero radius. A plausible resolution of the Information Loss Paradox in Black Holes is proposed.

Isotope Effect in the Mott Transition; a Prediction on the Basis of Molecular All-Quantum Simulations

We have linked an ab initio Hartree-Fock Hamiltonian to the Feynman path integral quantum Monte Carlo formalism in order to derive electronic expectation values under consideration of nuclear degrees of freedom. This approach yields electronic expectation values which depend on the atomic masses. On the basis of combined path integral – ab initio calculations we predict an isotope effect in the correlation driven Mott transition. The nuclear degrees of freedom lead to an enhancement in the electronic correlation strength, an effect which supports the transition conditions. This enhancement is negatively correlated with the atomic masses. Implications of an isotope effect in the Mott transition for the explanation of the superconducting pairing are mentioned concisely.

Isotope Effect in the Mott Transition; a Prediction on the Basis of Molecular All-Quantum Simulations

We have linked an ab initio Hartree-Fock Hamiltonian to the Feynman path integral quantum Monte Carlo formalism in order to derive electronic expectation values under consideration of nuclear degrees of freedom. This approach yields electronic expectation values which depend on the atomic masses. On the basis of combined path integral – ab initio calculations we predict an isotope effect in the correlation driven Mott transition. The nuclear degrees of freedom lead to an enhancement in the electronic correlation strength, an effect which supports the transition conditions. This enhancement is negatively correlated with the atomic masses. Implications of an isotope effect in the Mott transition for the explanation of the superconducting pairing are mentioned concisely.

NELSON'S STOCHASTIC MECHANICS BY MEANS OF FEYNMAN PATH INTEGRALS

We show that Nelson's stochastic mechanics suitably formulated as a Hamilton–Jacobi first-order equation leads straightforwardly to the Feynman path integral formulation of quantum mechanics.

Foundations of quantum mechanics (II): equilibrium, Bohr–Sommerfeld rules and duality

The present paper is divided into two parts. In the first part we present a new axiomatic derivation of the Schrödinger equation from three basic postulates. This new derivation sheds some light on the statistical character of the quantum formalism. We also show the formal connection between this derivation and the one done by other means in paper I of this series. The role of the entropy in all these developments is discussed at length. A discussion about the wave-particle duality problem is also presented. It is shown that the present approach avoids this problem. In the second part we show how to derive the Bohr–Sommerfeld quantization rules from the infinitesimal Wigner–Moyal transformation of paper I. We will thus show to what extent these rules are constituent of the modern quantum formalism and interpretation. The Feynman path integral formalism will also be derived from some slight modification of this approach, thus providing a high level of formal unification for our developments. These results will then be applied to provide fully corpuscular explanations of diffraction and interference experiments. When connecting these applications to the developments of the first part we will be able to exemplify quite clearly in what sense the duality problem is avoided by the present approach.