Feynman Path Integral Method Research Articles

Spin states of electrons in quantum dots upon heating. Simulation by the Feynman path integral method. Structure

Electrons in ellipsoidal model quantum dots under the conditions of thermal fluctuations are numerically simulated by the Feynman path integral method. This method makes it possible to perform calculations with fundamentally accurate consideration of quantum-mechanical indistinguishability of electrons and spin without applying the single-electron approximation. Electron density distributions depending on the quantum dots shape and temperature variations are obtained. An increase in temperature primarily affects the electron–electron correlations in the quantum dots central region. This influence is the main mechanism of the dependence of the spin state on temperature.

Quantum particle motion in absorbing harmonic trap

The motivation of this work is to get an additional insight into the irreversible energy dissipation on the quantum level. The presented examination procedure is based on the Feynman path integral method that is applied and widened towards the calculation of the kernel of a quantum mechanical damped oscillator. Here, it is shown that the energy loss of the oscillator can be generated by the introduced harmonic complex potential. The related damped wave function, however, does not pertain to the probability meaning as it is usual in the case of complex absorbing potentials. This decrease of the wave function is evaluated, moreover, the energy dissipation and the measure the irreversibility are expressed.

Schwinger Method and Feynman Path Integral for a Harmonic Oscillator with Mass Growing with Time

The exact propagator for a harmonic oscillator with mass growing like m(1+ατ )2 where m and α are constants and τ is time, is derived by the Schwinger method and the Feynman path integral. In the Schwinger formalism, the propagator can be calculated by basic operator algebra and elementary integrations. In the Feynman path integral method, it can be shown that such a propagator can be evaluated from that for a classical action with the application of a Pauli-Van Vleck formula.

Exchange and spin states in quantum dots under strong spatial correlations. Computer simulation by the Feynman path integral method

The fundamental laws in the behavior of electrons in model quantum dots that are caused by exchange and strong Coulomb correlations are studied. The ab initio path integral method is used to numerically simulate systems of two, three, four, and six interacting identical electrons confined in a three-dimensional spherical potential well with a parabolic confining potential against the background of thermal fluctuations. The temperature dependences of spin and collective spin magnetic susceptibility are calculated for model quantum dots of various spatial sizes. A basically exact procedure is proposed for taking into account the permutation symmetry and the spin state of electrons, which makes it possible to perform numerical calculations using modern computer facilities. The conditions of applicability of a virial energy estimator and its optimum form in exchange systems are determined. A correlation estimator of kinetic energy, which is an alternative to a basic estimator, is suggested. A fundamental relation between the kinetic energy of a quantum particle and the character of its virtual diffusion in imaginary time is demonstrated. The process of natural “pairing” of electron spins during the compression of a quantum dot and cooling of a system is numerically reproduced in terms of path integrals. The temperature dependences of the spin magnetic susceptibility of electron pairs with a characteristic maximum caused by spin pairing are obtained.

Wave function for time-dependent harmonically confined electrons in a time-dependent electric field

The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the Harmonic Potential Theorem wave function for the case of the time-independent harmonic confining potential.

Time Fractional Schrodinger Equation Revisited

The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle” obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have “fractional” dimensions, chosen such that the “energy” has the correct dimension . The action is defined as a fractional time integral of the Lagrangian, and a “fractional Planck constant” is introduced. The TFSE corresponds to a “subdiffusion” equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber’s work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a “free particle” is obtained. The total probability for a “free” particle is shown to go to zero in the limit of infinite time, in contrast with Naber’s result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.

SPATIAL PROPAGATION OF QUANTUM LIGHT IN NONLINEAR WAVEGUIDING DEVICES: THEORY AND APPLICATIONS

We present the main theoretical results on spatial propagation of quantum light in nonlinear waveguiding devices together with a few applications. We show that the quantization of the classical momentum provides a consistent quantum mechanical formulation of both the linear and the nonlinear quantum light propagation in waveguiding devices. The momentum operator allows us to derive the correct spatial Heisenberg's equations for the forward and backward absorption and emission operators and consequently to analyze the spatial propagation in the multimode optical-field strength space. For that purpose we use the Feynman's path integral method in order to derive the quantum spatial optical propagators of different nonlinear waveguiding devices and accordingly to calculate the spatial propagation of the optical-field strength probability amplitude of quantum states in these nonlinear devices. Likewise, we present a preliminary and heuristic formulation of quantum dissipation in order to take into account the losses under spatial quantum propagation in lossy nonlinear waveguiding devices. It must be stressed that these optical-field strength probabilities have the advantage of being measured by homodyne techniques and therefore their theoretical analysis is of a remarkable interest in the optical characterization of quantum states obtained under linear and nonlinear propagation in waveguiding devices.

PATH INTEGRAL SOLUTIONS OF RLC MESOSCOPIC CIRCUIT WITH SOURCE

In this paper, mesoscopic RLC circuit with source is studied with Feynman's path integral method. Resistance and source in the circuits make the quantization process rather complicated. To solve the problem, fluctuation analysis method is proposed to calculate the path integral propagator. Furthermore, the wave function and quantum fluctuation of the system are obtained, and time evolution characters of the system are discussed. The methods used in the paper would be helpful to the application of mesoscopic quantum theory.

The loss of quantum coherence induced by a Gaussian random potential

We investigate the source of decoherence from a random external potential by the Feynman path integral method. We find that the off-diagonal elements of the density matrix decay with two characteristic time scales, which are separated by the correlation length. The heavy and light mass cases are demonstrated.

Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman Path Integrals Method

This paper discusses the teaching of basic quantum mechanics in high school. Rather than following the usual formalism, our approach is based on Feynman's path integral method. Our presentation makes use of simulation software and avoids sophisticated mathematical formalism.

Feynman path-integral treatment of the BEC-impurity polaron

The description of an impurity atom in a Bose-Einstein condensate can be cast in the form of Frohlich's polaron Hamiltonian, where the Bogoliubov excitations play the role of the phonons. An expression for the corresponding polaronic coupling strength is derived, relating the coupling strength to the scattering lengths, the trap size and the number of Bose condensed atoms. This allows to identify several approaches to reach the strong-coupling limit for the quantum gas polarons, whereas this limit was hitherto experimentally inaccessible in solids. We apply Feynman's path-integral method to calculate for all coupling strengths the polaronic shift in the free energy and the increase in the effective mass. The effect of temperature on these quantities is included in the description. We find similarities to the acoustic polaron results and indications of a transition between free polarons and self-trapped polarons. The prospects, based on the current theory, of investigating the polaron physics with ultracold gases are discussed for lithium atoms in a sodium condensate.

JOINT ENTROPY OF THE HARMONIC OSCILLATOR WITH TIME-DEPENDENT MASS AND/OR FREQUENCY

Time-dependent joint entropy is obtained for harmonic oscillator with the time-dependent mass and frequency case. It is calculated by using time-dependent wave function obtained via Feynman path integral method. Variation of time dependence is investigated for various cases.

THE LINEARIZED POLARON MODEL SYSTEM IN A MAGNETIC FIELD

This article is devoted to the calculation of the free energy on electron-phonon interaction in the external magnetic field according to Green's function method. It should be noted that creation of structures with predicted features occurred (with the use of experimental techniques). To sum up, it should be noted that the given result coincides with the result of Devreese and Brosens, which was obtained by Feynman's path integral method.

Time Dependence of Joint Entropy of Oscillating Quantum Systems

The time dependent entropy (or Leipnik’s entropy) of harmonic and damped harmonic oscillator systems is studied by using time dependent wave function obtained by the Feynman path integral method. The Leipnik entropy and its envelope change as a function of time, angular frequency and damping factor. Our results for simple harmonic oscillator are in agreement with the literature. However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time for certain values of damping factor. The envelope of the joint entropy curve increases with time monotonically. These results show the general properties of the envelope of the joint entropy curve for quantum systems.

Observing the attosecond dynamics of nuclear wavepackets in molecules by using high harmonic generation in mixed gases

We probe the attosecond dynamics of nuclear wavepackets in H2 and D2 molecules by measuring the relative phase of high harmonics generated in each molecule by using a novel method with a mixed gas of H2 and D2. We find that not only the single molecule responses but also the propagation effects of harmonics differ between the two isotopes and we conclude that in order to discuss the dynamics of molecules in the single molecule responses, the propagation effects need to be excluded from the raw harmonic signals. The measured relative phase as well as the intensity ratio are found to be monotonic functions of the harmonic order and are successfully reproduced by applying Feynman's path integral method fully to the dynamics of the nuclei and electrons in the molecules.

The quasi-quantum treatment of rotationally inelastic scattering from a hard shell potential: its derivation and practical use

The QQT is a quasi-quantum mechanical treatment of the collision between molecules. Instead of a partial wave expansion approach, it uses a kind of Feynman path integral method that exploits the path length differences originating from the different orientations of an anisotropic molecule. As a result, the QQT provides valuable physical insight while requiring very little computational effort. The current paper gives a systematic derivation of the QQT and explains its underlying principles. The expression for the scattering amplitude is shown to be self-consistent, without any normalisation factors, when the rotational energy level spacing is negligible. The constant curvature approximation that is presented makes the QQT conceptually even more simple, and its effect on the calculated differential cross-sections (DCSs) turns out to be small. As examples we present QQT calculations of the DCSs for Ne–CO(1Σ) and He–NO(2Π), at collision energies of, respectively, 511 cm−1 and 514 cm−1. The anisotropy of the hard shell potential energy surface for Ne–CO in terms of the incoming de Broglie wavelength is about twice as large as for He–NO. This leads to state-to-state DCSs that have up to three maxima of comparable amplitude, instead of only one large maximum as is found for He–NO. The QQT results for these two applications are compared with results from close coupling calculations.

Effect of interface roughness on the density of states of finite barrier height quantum wells

We calculate the density of states of a 2D electron gas in finite barrier height quantum wells with the explicit inclusion of the interface roughness effect. By using Feynman path-integral method, the analytic expression is derived. The results show that the 2D density of states is dependent on the RMS of the fluctuation potential. The interface roughness causes localized states below the subband edge. We also apply the theory to model the finite barrier height quantum wells in Al x Ga 1− x As/GaAs.

Numerical simulation of the double slit interference with ultracold atoms

We present a numerical simulation of the double slit interference experiment realized by F. Shimizu, K. Shimizu and H. Takuma with ultracold atoms. We show how the Feynman path integral method enables the calculation of the time-dependent wave function. Because the evolution of the probability density of the wave packet just after it exits the slits raises the issue of interpreting the wave/particle dualism, we also simulate trajectories in the de Broglie–Bohm interpretation.

Calculation of the Equation of State of a Dense Hydrogen Plasma by the Feynman Path Integral Method

A method is developed for calculating the equation of state of a system of quantum particles at a finite temperature, based on the Feynman formulation of quantum statistics. A general analytical expression is found for the virial estimator for the kinetic energy of a system with rigid boundaries at a finite pressure. An effective method is developed for eliminating the unphysical singularity in the electrostatic potential between a discretized Feynman path of an electron and a proton. It is shown that the “refinement” of an expansion of a quantum-mechanical propagator by addition of high powers of time exacerbates, rather than eliminates, the divergence of a Feynman path integral. A brief summary of the current status of the problem is presented. The proposed new approaches are presented in relation to progress made in this field. Path integral Monte Carlo simulations are performed for nonideal hydrogen plasmas in which both indistinguishability and spin of electrons are taken into account under conditions preceding the formation of the electron shells of atoms. The electron permutation symmetry is represented in terms of Young operators. It is shown that, owing to the singularity of the Coulomb potential, quantum effects on the behavior of the electron component cannot be reduced to small corrections even if the system must be treated as a classical system according to the formal de Broglie criterion. Quantum-mechanical delocalization of electrons substantially weakens the repulsion between electrons as compared to protons. In relatively cold plasmas, many-body correlations lead to complex behavior of the potential of the average force between particles and give rise to repulsive forces acting between protons and electrons at distances of about 5 angstroms. Plasma pressure drops with decreasing plasma temperature as the electron shells of atoms begin to form, and the electron kinetic energy reaches a minimum at a temperature of about 31000 K. The minimum point weakly depends on plasma density. Owing to quantum effects, the electron component is “heated” well before electrons are completely bound in the field of protons.

Radiative Corrections to Casimir Force of Maxwell-Chern-Simons Gauge Field**The project supported by National Natural Science Foundation of China under Grant No. 10347104, Natural Science Foundation of Guandong Province of China under Grant No. 033446

We discuss the two-loop radiative correction to the Casimir force of Maxwell–Chern–Simons Abelian gauge field between two parallel ideal conducting wires in terms of Feynman path integral method.