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A White Noise Approach to Phase Space Feynman Path Integrals

Wolfgang Böck, Martin Grothaus

Published Aug 17, 2016• 18 citations
OpenAlex

Abstract

The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrödinger equation are obtained and the canonical commutation relations are satisfied. The latter can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional.

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Derivation of quantum propagator for coupled harmonic oscillator with uniform electric field in a single harmonic oscillator environment using white noise functional approach

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The use of white noise analysis is utilized in this study. We consider an open quantum system of coupled harmonic oscillator with uniform two dimensional electric field in a single harmonic oscillator environment. We obtain the quantum propagator for the said system by considering the Feynman path integral as a white noise functional. In addition, the resulting quantum propagator was consistent with the previous literature and also, this study gives more general form due to the inclusion of the electric field.

Journal of King Saud University - Science•Apr 14, 2018•6 citations

Quantum Propagator Dynamics of a Harmonic Oscillator in a Multimode Harmonic Oscillators Environment using White Noise Functional Analysis

Bienvenido Masirin Butanas Jr., Roland Cristopher F. Caballar

White noise analysis is used to derive the propagator of an open quantum system consisting of a harmonic oscillator which is coupled to an environment consisting of N multimode harmonic oscillators. The quantum propagators are obtained after solving for the normal modes of the system-environment interaction in order to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian is then used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator is then found to consist of a product of N simple harmonic oscillator propagators. N-2 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other 2 correspond to the non-degenerate normal mode frequencies. This method of deriving the propagators greatly simplifies the task of mathematically describing the dynamics of the open quantum system. The ease of use of this method suggests that, together with its inherent mathematical rigor, white noise analysis can be a powerful tool in analyzing the dynamics of an open quantum system.

arXiv:1703.04909•Mar 15, 2017•2 citations

Mathematical aspects of the Feynman path integral

Daniel W. Boutros

Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.

•Jan 1, 2020•0 citations

Dynamics of coupled harmonic oscillators in an environment using white noise analysis

Bienvenido Masirin Butanas Jr.

White noise analysis has gained considerable attention in the past years in the analysis of quantum systems, polymer entanglement, transport, and diffusion phenomena. However, there is dearth on the application of the analysis into open quantum systems specifically coupled harmonic oscillators in an environment. In this work, white noise analysis was used to study the dy- namics of the propagator of an open quantum system consisting of coupled harmonic oscillators that are coupled to an environment consisting of N-2 multimode harmonic oscillators. The quantum propagators were obtained after solving for the normal modes of the system-environment interaction to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian was used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator was found to consist of the product of N simple harmonic oscillator propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other three correspond to the non-degenerate normal mode frequen- cies, two of these nondegeneracies correspond to complex normal modes. This approach of describing the dynamics of the open quantum system greatly simplifies the task mathematically.

AIP conference proceedings•Jan 1, 2020•2 citations

Path Integral Formulation

S. Rajasekar, R. Velusamy

The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.

CRC Press eBooks•Oct 5, 2022•0 citations

- Path Integral Formulation

S. Rajasekar, R. Velusamy

The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.

CRC Press eBooks•Dec 10, 2014•1 citation
The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrödinger equation are obtained and the canonical commutation relations are satisfied. The latter can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional.
The use of white noise analysis is utilized in this study. We consider an open quantum system of coupled harmonic oscillator with uniform two dimensional electric field in a single harmonic oscillator environment. We obtain the quantum propagator for the said system by considering the Feynman path integral as a white noise functional. In addition, the resulting quantum propagator was consistent with the previous literature and also, this study gives more general form due to the inclusion of the electric field.
White noise analysis is used to derive the propagator of an open quantum system consisting of a harmonic oscillator which is coupled to an environment consisting of N multimode harmonic oscillators. The quantum propagators are obtained after solving for the normal modes of the system-environment interaction in order to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian is then used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator is then found to consist of a product of N simple harmonic oscillator propagators. N-2 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other 2 correspond to the non-degenerate normal mode frequencies. This method of deriving the propagators greatly simplifies the task of mathematically describing the dynamics of the open quantum system. The ease of use of this method suggests that, together with its inherent mathematical rigor, white noise analysis can be a powerful tool in analyzing the dynamics of an open quantum system.
Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
White noise analysis has gained considerable attention in the past years in the analysis of quantum systems, polymer entanglement, transport, and diffusion phenomena. However, there is dearth on the application of the analysis into open quantum systems specifically coupled harmonic oscillators in an environment. In this work, white noise analysis was used to study the dy- namics of the propagator of an open quantum system consisting of coupled harmonic oscillators that are coupled to an environment consisting of N-2 multimode harmonic oscillators. The quantum propagators were obtained after solving for the normal modes of the system-environment interaction to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian was used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator was found to consist of the product of N simple harmonic oscillator propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other three correspond to the non-degenerate normal mode frequen- cies, two of these nondegeneracies correspond to complex normal modes. This approach of describing the dynamics of the open quantum system greatly simplifies the task mathematically.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.