Feynman Path Integral Formulation Research Articles

Power-duality in path integral formulation of quantum mechanics

Power duality in Feynman's path integral formulation of quantum mechanics is investigated. The power duality transformation consists of a change in coordinate and time variables, an exchange of energy and coupling, and a classical angular momentum replacement. Two physical systems connected by the transformation form a power-dual pair. The propagator (Feynman's kernel) expressed by Feynman's path integral cannot be form-invariant under the transformation, whereas the promotor constructed by modifying Feynman's path integral is found form-invariant insofar as the angular momentum is classical. Upon angular quantization, the power duality breaks down. To save the notion of power duality, the idea of quasi power duality is proposed, which constitutes of an ad hoc angular momentum replacement. The power-dual invariant promotor leads to the quasi-dual invariant Green function. A formula is proposed, which determines the Green function for one of a dual pair by knowing the Green function for the other. As examples, the Coulomb-Hooke dual pair and a family of two-term confinement potentials for a zero-energy state are discussed.

Galilean relativity and the path integral formalism in quantum mechanics

Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.

Untying the Gordian Knot: Process, Reality, and Context

Untying the Gordian Knot: Process, Reality, and Context

A study on the scattering of matter waves through slits

Scattering of matter waves through slits has been explored using the Feynman Path Integral formalism. We explicitly plot the near-zero probability densities to analyse the behaviour near the slit. Upon doing so, intriguing patterns emerge, most notably the braid-like structure in the case of double slits, whose complexity increases as one increases the number of slits. Furthermore, the plot shows the existence of a transition region, where the distribution of near-zero probability points changes from the braided to the fringe-like structure, which has been analysed by explicitly expressing the wavefunction as a hypergeometric function. These patterns are analysed while considering the continuity equation and its consequences for the regions with zero probability density. As a result, we find quasi-traps in the region whose size can be controlled and made much smaller than the wavelength of matter waves.

Feynman kernel analytical solutions for the deformed hyperbolic barrier potential with application to some diatomic molecules

In this paper, we derive the ℓ-states energy spectrum of the q-deformed hyperbolic Barrier Potential. Within the Feynman path integral formalism, we propose an appropriate approximation of the centrifugal term. Then, using Euler angles and the isomorphism between Λ3 and SU(1, 1), we convert the radial path integral into a maniable one. The obtained eigenvalues are in very good agreement with the numerical results. In addition, we applied our results to some diatomic molecules and obtained accurate results compared to the experimental (RKR) values.

Chaotic Einstein–Podolsky–Rosen pairs, measurements and time reversal

We consider a situation when evolution of an entangled Einstein–Podolsky–Rosen (EPR) pair takes place in a regime of quantum chaos being chaotic in the classical limit. This situation is studied on an example of chaotic pair dynamics described by the quantum Chirikov standard map. The time evolution is reversible even if a presence of small errors breaks time reversal of classical dynamics due to exponential growth of errors induced by exponential chaos instability. However, the quantum evolution remains reversible since a quantum dynamics instability exists only on a logarithmically short Ehrenfest time scale. We show that due to EPR pair entanglement a measurement of one particle at the moment of time reversal breaks exact time reversal of another particle which demonstrates only an approximat time reversibility. This result is interpreted in the framework of the Schmidt decomposition and Feynman path integral formulation of quantum mechanics. The time reversal in this system has already been realized with cold atoms in kicked optical lattices in absence of entanglement and measurements. On the basis of the obtained results, we argue that the experimental investigations of time reversal of chaotic EPR pairs are within reach of present cold atom capabilities.

Beyond the Fröhlich Hamiltonian: Path-integral treatment of large polarons in anharmonic solids

The properties of an electron in a typical solid are modified by the interaction with the crystal ions, leading to the formation of a quasiparticle: the polaron. Such polarons are often described using the Fr\"ohlich Hamiltonian, which assumes the underlying lattice phonons to be harmonic. However, this approximation is invalid in several interesting materials, including the recently discovered high-pressure hydrides which superconduct at temperatures above $200$K. In this paper, we show that Fr\"ohlich theory can be extended to eliminate this problem. We derive four additional terms in the Fr\"ohlich Hamiltonian to account for anharmonicity up to third order. We calculate the energy and effective mass of the new polaron, using both perturbation theory and Feynman's path integral formalism. It is shown that the anharmonic terms lead to significant additional trapping of the electron. The derived Hamiltonian is well-suited for analytical calculations, due to its simplicity and since the number of model parameters is low. Since it is a direct extension of the Fr\"ohlich Hamiltonian, it can readily be used to investigate the effect of anharmonicity on other polaron properties, such as the optical conductivity and the formation of bipolarons.

Modeling of a Neural System Based on Statistical Mechanics.

The minimization of a free energy is often regarded as the key principle in understanding how the brain works and how the brain structure forms. In particular, a statistical-mechanics-based neural network model is expected to allow one to interpret many aspects of the neural firing and learning processes in terms of general concepts and mechanisms in statistical physics. Nevertheless, the definition of the free energy in a neural system is usually an intricate problem without an evident solution. After the pioneering work by Hopfield, several statistical-mechanics-based models have suggested a variety of definition of the free energy or the entropy in a neural system. Among those, the Feynman machine, proposed recently, presents the free energy of a neural system defined via the Feynman path integral formulation with the explicit time variable. In this study, we first give a brief review of the previous relevant models, paying attention to the troublesome problems in them, and examine how the Feynman machine overcomes several vulnerable points in previous models and derives the outcome of the firing or the learning rule in a (biological) neural system as the extremum state in the free energy. Specifically, the model reveals that the biological learning mechanism, called spike-timing-dependent plasticity, relates to the free-energy minimization principle. Basically, computing and learning mechanisms in the Feynman machine base on the exact spike timings of neurons, such as those in a biological neural system. We discuss the consequence of the adoption of an explicit time variable in modeling a neural system and the application of the free-energy minimization principle to understanding the phenomena in the brain.

Measuring the deviation from the superposition principle in interference experiments

The Feynman path integral formalism has long been used for calculations of probability amplitudes. Over the last few years, it has been extensively used to theoretically demonstrate that the usual application of the superposition principle in slit based interference experiments is often incorrect. This has caveat in both optics and quantum mechanics where it is often naively assumed that the boundary condition represented by slits opened individually is same as them being opened together. The correction term comes from exotic sub-leading terms in the path integral which can be described by what are popularly called non-classical paths. In this work, we report an experiment where we have a controllable parameter that can be varied in its contribution such that the effect due to these non-classical paths, which we will refer to as sub-leading paths, can be increased or diminished at will. Thus, the reality of these sub-leading paths is brought forth in a classical experiment using microwaves, thereby proving that the boundary condition effect being investigated transcends the classical-quantum divide and that the Feynman path integral formalism is an overarching framework. We report the first measurement of a deviation (as big as 6%) from the superposition principle in the microwave domain using antennas as sources and detectors of the electromagnetic waves. We also show that our results can have potential applications in astronomy.

Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions

We present results for the solution of the large polaron Fr\"ohlich Hamiltonian in 3-dimensions (3D) and 2-dimensions (2D) obtained via the Diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A.S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from 3D$\to$2D, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small coupling regime and verify the first order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.

An open-chain imaginary-time path-integral sampling approach to the calculation of approximate symmetrized quantum time correlation functions.

We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.

Ro-vibrational energies of the shifted Deng-Fan oscillator potential with Feynman path integral formalism

In this paper, we derive approximate analytical solutions of the D-dimension Feynman propagator in presence of the shifted Deng-Fan oscillator by using an appropriate approximation of the centrifugal term. Energy eigenvalues and the corresponding eigenfunctions are obtained for different quantum numbers and some diatomic molecules. In the three-dimensional case, we calculate the energy level values and compare our results with those given in the literature. Our calculated energy levels are in much better agreement with the ones obtained numerically.

Perturbation expansions of stochastic wavefunctions for open quantum systems.

Based on the stochastic unravelling of the reduced density operator in the Feynman path integral formalism for an open quantum system in touch with harmonic environments, a new non-Markovian stochastic Schrödinger equation (NMSSE) has been established that allows for the systematic perturbation expansion in the system-bath coupling to arbitrary order. This NMSSE can be transformed in a facile manner into the other two NMSSEs, i.e., non-Markovian quantum state diffusion and time-dependent wavepacket diffusion method. Benchmarked by numerically exact results, we have conducted a comparative study of the proposed method in its lowest order approximation, with perturbative quantum master equations in the symmetric spin-boson model and the realistic Fenna-Matthews-Olson complex. It is found that our method outperforms the second-order time-convolutionless quantum master equation in the whole parameter regime and even far better than the fourth-order in the slow bath and high temperature cases. Besides, the method is applicable on an equal footing for any kind of spectral density function and is expected to be a powerful tool to explore the quantum dynamics of large-scale systems, benefiting from the wavefunction framework and the time-local appearance within a single stochastic trajectory.

The uniform quantized electron gas revisited

In this article we continue and extend our recent work on the correlation energy of the quantized electron gas of uniform density at temperature . As before, we utilize the methods, properties, and results obtained by means of classical statistical mechanics. These were extended to quantized systems via the Feynman path integral formalism. The latter translates the quantum problem into a classical polymer problem in four dimensions. Again, the well known RPA (random phase approximation) is recovered as a basic result which we then modify and improve upon. Here we analyze the condition of thermodynamic self-consistency. Our numerical calculations exhibit a remarkable agreement with well known results of a standard parameterization of Monte Carlo correlation energies.

Calculation of precise firing statistics in a neural network model

A precise prediction of neural firing dynamics is requisite to understand the function of and the learning process in a biological neural network which works depending on exact spike timings. Basically, the prediction of firing statistics is a delicate manybody problem because the firing probability of a neuron at a time is determined by the summation over all effects from past firing states. A neural network model with the Feynman path integral formulation is recently introduced. In this paper, we present several methods to calculate firing statistics in the model. We apply the methods to some cases and compare the theoretical predictions with simulation results.

Importance of a Fully Anharmonic Treatment of Equilibrium Isotope Fractionation Properties of Dissolved Ionic Species As Evidenced by Li+(aq)

Equilibrium fractionation of stable isotopes is critically important in fields ranging from chemistry, including medicinal chemistry, electrochemistry, geochemistry, and nuclear chemistry, to environmental science. The dearth of reliable estimates of equilibrium fractionation factors, from experiment or from natural observations, has created a need for accurate computational approaches. Because isotope fractionation is a purely quantum mechanical phenomenon, exact calculation of fractionation factors is nontrivial. Consequently, a severe approximation is often made, in which it is assumed that the system can be decomposed into a set of independent harmonic oscillators. Reliance on this often crude approximation is one of the primary reasons that theoretical prediction of isotope fractionation has lagged behind experiment. A class of problems for which one might expect the harmonic approximation to perform most poorly is the isotopic fractionation between solid and solution phases. In order to illustrate the errors associated with the harmonic approximation, we have considered the fractionation of Li isotopes between aqueous solution and phyllosilicate minerals, where we find that the harmonic approximation overestimates isotope fractionation factors by as much as 30% at 25 °C. Lithium is a particularly interesting species to examine, as natural lithium isotope signatures provide information about hydrothermal processes, carbon cycle, and regulation of the Earth's climate by continental alteration. Further, separation of lithium isotopes is of growing interest in the nuclear industry due to a need for pure 6Li and 7Li isotopes. Moving beyond the harmonic approximation entails performing exact quantum calculations, which can be achieved using the Feynman path integral formulation of quantum statistical mechanics. In the path integral approach, a system of quantum particles is represented as a set of classical-like ring-polymer chains, whose interparticle interactions are determined by the rules of quantum mechanics. Because a classical isomorphism exists between the true quantum system and the system of ring-polymers, classical-like methods can be applied. Recent developments of efficient path integral approaches for the exact calculation of isotope fractionation now allow the case of the aforementioned dissolved Li fractionation properties to be studied in detail. Applying this technique, we find that the calculations yield results that are in good agreement with both experimental data and natural observations. Importantly, path integral methods, being fully atomistic, allow us to identify the origins of anharmonic effects and to make reliable predictions at temperatures that are experimentally inaccessible yet are, nevertheless, relevant for natural phenomena.

Sign Learning Kink-based (SiLK) Quantum Monte Carlo for molecular systems.

The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is used to calculate the ab initio ground state energies for multiple geometries of the H2O, N2, and F2 molecules. The method is based on Feynman's path integral formulation of quantum mechanics and has two stages. The first stage is called the learning stage and reduces the well-known QMC minus sign problem by optimizing the linear combinations of Slater determinants which are used in the second stage, a conventional QMC simulation. The method is tested using different vector spaces and compared to the results of other quantum chemical methods and to exact diagonalization. Our findings demonstrate that the SiLK method is accurate and reduces or eliminates the minus sign problem.

Quantum and isotope effects on hydrogen diffusion, trapping and escape in iron

We calculate the rate coefficient as a function of temperature for lattice diffusion of hydrogen and its isotopes in α-iron; and also for trapping and escape from a vacancy. We employ Monte-Carlo and molecular dynamics methods based around the Feynman path integral formulation of the quantum partition function. We find large quantum effects including tunnelling at low temperature and recrossing at high temperature due to the finite extent of the particle probability density. In particular these serve to increase the rate of trapping and to decrease the rate of escape at low temperature. Our results also show very clear non classical isotope effects.

Relative criterion for validity of a semiclassical approach to the dynamics near quantum critical points.

Based on an analysis of Feynman's path integral formulation of the propagator, a relative criterion is proposed for validity of a semiclassical approach to the dynamics near critical points in a class of systems undergoing quantum phase transitions. It is given by an effective Planck constant, in the relative sense that a smaller effective Planck constant implies better performance of the semiclassical approach. Numerical tests of this relative criterion are given in the XY model and in the Dicke model.

On the superposition principle in interference experiments.

The superposition principle is usually incorrectly applied in interference experiments. This has recently been investigated through numerics based on Finite Difference Time Domain (FDTD) methods as well as the Feynman path integral formalism. In the current work, we have derived an analytic formula for the Sorkin parameter which can be used to determine the deviation from the application of the principle. We have found excellent agreement between the analytic distribution and those that have been earlier estimated by numerical integration as well as resource intensive FDTD simulations. The analytic handle would be useful for comparing theory with future experiments. It is applicable both to physics based on classical wave equations as well as the non-relativistic Schrödinger equation.