Quantum tunneling in a dissipative environment

1992 ◽  
Vol 70 (9) ◽  
pp. 719-730 ◽  
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Wave Equation ◽  
Schrödinger Equation ◽  
Quantum Tunneling ◽  
Schrodinger Equation ◽  
Quantum Coherence ◽  
Harmonic Oscillators ◽  
Dissipative Systems ◽  
Many Body ◽  
Coupled Equations ◽  
Central Particle

A wave equation formulation of the problem of quantum tunneling in a dissipative medium is developed by considering a many-body system in which the central particle is subject to an arbitrary force law, and at the same time is coupled to a bath of noninteracting harmonic oscillators. For the motion of the central particle it is possible to obtain an effective Lagrangian and Hamiltonian by eliminating the degrees of freedom of the oscillators. However both of these operators are nonlocal, and it is difficult to derive a wave equation for this motion. As an alternative method one can write a many-body Schrödinger equation for the whole system, and then eliminate the wave functions of all of the oscillators. This result is a many-channel Schrödinger equation for the motion of the central particle. By truncating this set of coupled equations, one can solve the problem for different force laws. In particular, in this work, the cases of dissipative tunneling, hopping, and quantum coherence are studied in detail. It is also shown how this approach can be generalized to multidimensional dissipative systems.

Wave equation for dissipative systems derived from a quantized many-body problem

1980 ◽  
Vol 58 (7) ◽  
pp. 1019-1025 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Body Problem ◽  
Body Wave ◽  
Time Dependent ◽  
Dissipative Systems ◽  
Many Body ◽  
Dependent Part ◽  
The Many

A classical many-body problem composed of an infinite number of mass points coupled together by springs is quantized. The masses and the spring constants in this system are chosen in such a way that the motion of each particle is exponentially damped. Because of the quadratic form of the Hamiltonian, the many-body wave function of the system can be written as a product of two terms: a time-dependent phase factor which contains correlations between the classical motions of the particles, and a stationary state solution of the Schrödinger equation. By assuming a Hartree type wave function for the many-particle Schrödinger equation, the contribution of the time-dependent part to the single particle wave function is determined, and it is shown that the time-dependent wave function of each mass point satisfies the nonlinear Schrödinger–Langevin equation. The characteristic decay time of any part of the subsystem, in this model, is related to the stiffness of the springs, and is the same for all particles.

Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves

2019 ◽  
Vol 33 (24) ◽  
pp. 1950285
Author(s):  
Saviour Worlanyo Akuamoah ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Wave Equation ◽  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Momentum Operator ◽  
Matter Wave ◽  
Relativistic Wave Equation ◽  
Relativistic Energy ◽  
Substitution Method ◽  
Energy And Momentum

In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.

scholarly journals Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

2014 ◽  
Vol 15 (4) ◽  
pp. 853-894 ◽  
Author(s):  
Kin-Yiu Wong
Keyword(s):  
Schrödinger Equation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Stochastic Methods ◽  
Phosphoryl Transfer ◽  
Many Body ◽  
Molecular Systems ◽  
Pi Method ◽  
Variational Perturbation ◽  
Kp Theory

AbstractFeynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman’s path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert’s variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

scholarly journals Quantum Lattice-Gas Models for the Many-Body Schrödinger Equation

1997 ◽  
Vol 08 (04) ◽  
pp. 705-716 ◽  
Author(s):  
Bruce M. Boghosian ◽  
Washington Taylor
Keyword(s):  
Schrödinger Equation ◽  
Arbitrary Number ◽  
Schrodinger Equation ◽  
Lattice Gas ◽  
Quantum Computer ◽  
Many Body ◽  
Lattice Gas Models ◽  
Many Body Systems ◽  
The Many ◽  
Classical Computer

A general class of discrete unitary models are described whose behavior in the continuum limit corresponds to a many-body Schrödinger equation. On a quantum computer, these models could be used to simulate quantum many-body systems with an exponential speedup over analogous simulations on classical computers. On a classical computer, these models give an explicitly unitary and local prescription for discretizing the Schrödinger equation. It is shown that models of this type can be constructed for an arbitrary number of particles moving in an arbitrary number of dimensions with an arbitrary interparticle interaction.

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