Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations (original) (raw)

On the Path Integral Approach to Quantum Mechanics

On the Path Integral Approach to Quantum Mechanics, 2021

We present novel path modeling techniques suitable for use in the Path-Integral formulation of Quantum Mechanics. Our proposed platform aims to address existing challenges encountered in Monte Carlo and other similar path modeling methods. By introducing 'smooth' path modeling techniques, we demonstrate how they can be seamlessly integrated with current approaches, facilitating more accessible amplitude estimations in this invaluable formulation of Quantum Mechanics.

Numerical path integral approach to quantum dynamics and stationary quantum states

Applicability of Feynman path integral approach to numerical simulations of quantum dynamics in real time domain is examined. Coherent quantum dynamics is demonstrated with one dimensional test cases (quantum dot models) and performance of the Trotter kernel as compared with the exact kernels is tested. A novel approach for finding the ground state and other stationary sates is presented. This is based on the incoherent propagation in real time. For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling. We asses the numerical prerequisites for all of the above.

Path Integrals in Quantum Physics

Lecture Notes in Physics Monographs

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.

Path Integrals in Quantum Physics Lectures given at ETH Zurich

2016

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.

Variational Methods for Path Integral Scattering

2009

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 the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.

Path integral formulation of reggeon quantum mechanics

Nuclear Physics B, 1979

The Hamiltonian path integral for reggeon quantum mechanics with cubic and quartic interactions is unambiguously defined. On the basis of a recursion formula on the time lattice, a general~ed Trotter formula is given and proven to be equivalent to a Lagrangian path integral for a Schroedinger problem in an interval. The potential occurring in this formula is the same found in previous papers, and the corresponding path integral exists also in the limit of vanishing quartic coupling. This provides a regularization procedure for the purely cubic case for both c~(0) less or greater than one. All previous results are confirmed, in particular the tunnel-like energy gap. * Note that the Hamiltonian in the x = l(a + a +) representation contains powers of momentum up to the fourth. To our knowledge, only powers up to the second have been treated so far.

Quantum Fluid Dynamics from Path Integrals

2020

In this work, we develop analytical solutions to the general problem of computing Quantum Trajectories, within the framework of quantum fluid dynamics (QFD). The state-of-the-art technique in the field is to simultaneously solve the non-linear, coupled partial differential equations (PDEs) numerically. We, however, set off from Feynman Path Integrals, and analytically compute the propagator for a general system. This, then, is used to compute the Quantum Potential, which can generate Quantum Trajectories. For cases, where a closed-form solution is not possible, the problem is shown to be reducible to a single real-valued numerical integration (linear time complexity). The work formally bridges the Path Integral approach with Quantum Fluid Dynamics. As a model application to illustrate the method, we solve for the Quantum Potential of Quartic Anharmonic Oscillator and delve into seeking insight into one of the long-standing debates with regard to Quantum Tunneling.

Path Integrals in Quantum Physics (English Version)

arXiv:1209.1315v4, 2017

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.

The Path Integral Formulation Of Quantum Mechanics & Its Topological Applications

In this report, we deliver a detailed introduction to the methods of path integration in the focus of quantum mechanics. After an overview of the techniques of integration and the relationship to the familiar results of quantum mechanics such as the Schroedinger equation, we study some of the applications to mechanical systems with non-trivial degrees of freedom and discuss the advantages that path integration has as a formulation in studying these systems. To this end, in chapter 3, we introduce the topological concept of homotopy classes, and apply this to derive the spin statistics theorem, and discuss the phenomenon of parastatistics for systems constrained to dimension d<3. Following on from this, we calculate a general formula for the quantum mechanical propagator in terms of path integrals over different homotopy classes. The final chapter of this report studies the so called instanton solution as a way of describing quantum mechanical vacuum tunneling effects as a semi-classical problem. We conclude with discussing some applications in gauge field theory and describe the topological quality of the classical vacua of SU(2) Yang-Mills gauge theory; apply the path integral method to arrive at the theta-vacuum, and briefly say a few words about its far reaching consequences and applications.\\The content is aimed predominantly at a mathematical audience with a physical interest, moreover, we assume that the reader has a good grounding in topology and in both the classical and quantum mechanical theories. For completeness, essentials of these topics are reviewed in the appendices. (NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)