Feynman Operator Research Articles

Exact results for the Schrödinger equation with moving localized potential

We have provided an alternative method to solve the time-dependent Schrödinger equation for a moving delta function potential with a time-varying strength. We have employed the Galilean transformation to get the equation corresponding to a static delta potential with a time-varying strength. We apply the Feynman's operator technique along with the Green's function method to solve the equation. One of the observation is if a particle initially in the non eigenstate, can posses eigen state after encountered by a attractive potential well in the same initial location. By applying our proposed method, we have derived the analytical time domain solution for a constant and time inversely decaying strength of a uniformly moving well. We observed that the particle is unlikely to be found in the moving well for this decaying strength at large times unlike the previously observed in the case of constant strength.

A Family of Banach Spaces Over ℝ∞

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text].

The Feynman-Dyson view

This paper is a survey of our work on the mathematical foundations for the Feynman-Dyson program in quantum electrodynamics (QED). After a brief discussion of the history, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations. As a by-product, we also prove Feynman’s conjecture about the relationship between the operator calculus and has path integral. Thus, providing the first rigorous justification for the Feynman formulation of quantum mechanics.

Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock–Kurzweil integral, construct the operator calculus and extend the Hille–Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter–Kato theory.

Benchmark quantum Monte Carlo calculations of the ground-state kinetic, interaction andtotal energy of the three-dimensional electron gas

We report variational and diffusion quantum Monte Carlo ground-state energies of thethree-dimensional electron gas using a model periodic Coulomb interaction and backflow correctionsfor N = 54, 102, 178, and 226 electrons. We remove finite-size effects by extrapolation and we findlower energies than previously reported. Using the Hellman–Feynman operator samplingmethod introduced in Gaudoin and Pitarke (2007 Phys. Rev. Lett. 99 126406), we computeaccurately, within the fixed-node approximation, the separate kinetic and interactioncontributions to the total ground-state energy. The difference between the interactionenergies obtained from the original Slater-determinant nodes and the backflow-displacednodes is found to be considerably larger than the difference between the correspondingkinetic energies.

Feynman disentangling of noncommuting operators and group representation theory

Feynman's method for disentangling noncommuting operators is discussed and applied to nonstationary problems in quantum mechanics, including the excitation ofa harmonic oscillator by an external force and/or by time-varying frequency; spin rotation in a time-varying magnetic field; the disentangling of an atom (ion) Hamiltonian in a laser field; a model with the hidden symmetry group of the hydrogen atom; and the theory of coherent states. The Feynman operator calculus combined with simple group-theoretical considerations allows disentangling the Hamiltonian and obtaining exact transition probabilities between the initial and final states of aquantum oscillator in analytic form without cumbersome calculations. The case of a D-dimensional oscillator is briefly discussed, in particular, in application to the problem of vacuum pair creation in an intense electric field.

Refined formulation of quantum analysis,q-derivative and exponential splitting

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential df(A) satisfying the Leibniz rule. This formulation with use of the inner derivation ?A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q-derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q-derivative is also given.

Foundations for relativistic quantum theory. I. Feynman’s operator calculus and the Dyson conjectures

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations.

Turbulent diffusion of a passive impurity

A solution of the diffusion equation in the case of a medium which is diffusing in an inhomogeneous and non-stationary manner is constructed using the Feynman operator formalism. The functional transformation proposed by Stratonovich is used for the “disentanglement of the operator exponent”. As a result, the solution is represented in the form of a continual integral which differs from that obtained by Wiener in that, instead of an integral along trajectories, an integral of the velocities of the motion along the trajectories occurs in it. A statistical solution of the diffusion equation is obtained after averaging over random velocities. In the case of Gaussian statistics for the velocity fields or in the case of a spatially homogeneous non-stationary velocity field, continual integration can be carried out in an explicit form in the Markov approximation. In the first case, the result reduces to a renormalization of the coefficient of viscosity (the replacement of the coefficient of molecular viscosity by an effective coefficient of viscosity) and, in the second case, to the replacement of real time by an effective time. A number of papers, a list of which can be found /1/, are concerned with the problem of finding a technique for the “summation” of the coefficients of molecular and turbulent transport (to be specific, we shall speak about diffusion).

Operator method for the perturbative solution of ordinary differential equations

We adapt the Feynman operator calculus to the Lie operators connected with the solution of ordinary, coupled, first order differential equations, particularly those arising from non-linear oscillations or from interactions between modes. Combining this calculus with the method of averaging we obtain approximate solutions which are algebraic and exponential functions of the appropriate expansion parameter rather than power series of this parameter. In addition to illustrating the method for the damped harmonic oscillator and for the van der Pol oscillator we present an approximate solution of the Lotka-Volterra problem of two competing species. The solutions are compared numerically with those of a variant of the Bogoliubov-Krylov-Mitropolsky (BKM) method of averaging. The bulk of the analytic calculations both for the operator algorithm and for the BKM method have been automated.

Identities involving angular momentum operators operating in the space of many-nucleon intrinsic wavefunctions

An angular momentum operator identity which was previously established using matrix representations is re-derived using Feynman's operator differential equation technique.

Intramolecular Nonradiative Transitions in the ``Non-Condon'' Scheme

In this paper we consider the nonradiative intramolecular decay of a large molecule utilizing Born-Oppenheimer wavefunctions as a zero order basis, and bypassing the conventional Condon approximation for the calculation of the electronic coupling matrix matrix elements. The electronic adiabatic wavefunctions are expanded in terms of the Wigner-Brillouin perturbation series in the weak electronic-vibrational coupling limit. We have applied a generalized version of Feynman's operator calculus to derive general expressions for the nonradiative decay probability of a statistical harmonic molecule characterized by displaced potential surfaces. Numerical calculations were performed for the decay of the vibrationless excited electronic state in the ``non-Condon'' scheme. The numerical data for the decay rate in a two electronic level system in the weak electronic vibrational coupling limit exceed the results obtained invoking the Condon approximation by 2–3 orders of magnitude; the exact correction factor depends on the molecular parameters, and, in particular, is roughly proportional to the square of the (frequency normalized) electronic energy gap. Finally, the relevant off resonance coupling terms in the adiabatic representation are shown to be appreciably smaller than the near resonance coupling terms, demonstrating the superiority of the adiabatic basis over the crude adiabatic basis in describing electronic relaxation processes.

Optical Selection Studies of Radiationless Decay in an Isolated Large Molecule. II. Role of Frequency Changes

In this paper we provide an extension of the theoretical study of nonradiative decay of a single vibronic level of a large molecule. We have derived theoretical expressions for the dependence of the electronic relaxation rate on the excess vibrational energy in the excited electronic state of a ``harmonic molecule'' which is characterized by displaced and frequency modified potential surfaces. The simple case of displaced potential surfaces was handled by relating the potential surfaces via a simple displacement operator. To handle frequency changes in optical selection we have applied Feynman's operator techniques to disentangle exponential operators which involve nonlinear terms. The role of frequency changes in optical selection experiments was elucidated.

Optical Selection Studies of Radiationless Decay in an Isolated Large Molecule

In this paper we present the results of a theoretical study of the nonradiative decay probability of a single vibronic level of a large isolated molecule. Utilizing Feynman's operator techniques, we were able to derive a theoretical expression of the dependence of the electronic relaxation rate on the excess vibrational energy in the excited electronic state for a “harmonic molecule” which is characterized by displaced potential surfaces. For a large effective electronic energy gap the nonradiative decay probability increases with increasing excess vibrational energy, while for a small energy gap the nonradiative decay in higher vibronic levels may be retarded. Our rough numerical calculations are found to be consistent with recent experimental data on optical selection studies in the isolated benzene molecule.

Symmetrical Formulation of the Quantum Theory of Three-Wave Optical Parametric Interactions

The quantum theory of three-wave optical parametric interactions in a crystal such as LiNb${\mathrm{O}}_{3}$ is formulated in a symmetrical manner with respect to interaction time and distance. The formalism leads to a unique treatment of parametric interactions of both cavity type and propagation type, in that the resemblance between time in the cavity type and distance in the propagation type of interaction in the classical description is preserved. Three-wave effects of harmonic generation, parametric amplification (including spontaneous emission), and frequency conversion are treated in a unique and symmetric manner. Using Feynman's operator techniques, quantum states resulting from parametric interactions, previously unavailable, are derived, from which characteristics of the processes are deduced.