The Feynman effective classical potential in the Schr\"odinger formulation

We show that the computational effort for the numerical solution of fermionic quantum systems, occurring e.g., in quantum chemistry, solid state physics, field theory in principle grows with less than the square of the particle number for problems stated in one space dimension and with less than the cube of the particle number for problems stated in three space dimensions. This is proven by representation of effective algorithms for fermion systems in the framework of the Fey...

In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, ...

In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schr\"odinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical H...

We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path \gamma connecting points q_0 and q_1 in time t, we define a formal power series V_\gamma(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_\gamma) satisfies Schr\"odinger's equation, and explain in what sense the t\to 0 lim...

Utilizing operational dynamic modeling [Phys. Rev. Lett. 109, 190403 (2012); arXiv:1105.4014], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechan...

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last ten years or so, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given. Explicit formul\ae\ for the so-called basic path integrals are presented on which our general scheme to classify and calculate pa...

We briefly review a hamiltonian path integral formalism developed earlier by one of us. An important feature of this formalism is that the path integral quantization in arbitrary co-ordinates is set up making use of only classical hamiltonian without addition of adhoc $\hbar^2$ terms. In this paper we use this hamiltonian formalism and show how exact path integration may be done for several potentials.

Functional Schr\"{o}dinger equations for interacting fields are solved via rigorous non-perturbative Feynman type integrals.

The effects of dissipation on the thermodynamic properties of nonlinear quantum systems are approached by the path-integral method in order to construct approximate classical-like formulas for evaluating thermal averages of thermodynamic quantities. Explicit calculations are presented for one-particle and many-body systems. The effects of the dissipation mechanism on the phase diagram of two-dimensional Josephson arrays is discussed.

By an extension of the Feynman-Kleinert variational approach, we calculate the temperature-dependent effective classical potential governing the quantum statistical properties of a hydrogen atom in a uniform magnetic field. In the zero-temperature limit, we obtain ground state energies which are accurate for all magnetic field strengths from weak to strong fields.