Path integration via Hamilton-Jacobi coordinates and applications to potential barriers

Barut, A. O.; Duru, Ì. H.

doi:10.1103/physreva.38.5906

Cited by 25 publications

(32 citation statements)

References 4 publications

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“…On the one hand we know that the path integral representation (2) gives the usual one dimensional path integral in non-relativistic quantum mechanics 5 . On the other we find that only the (1,1) component in the Green functions remains finite, all others vanish. Furthermore we have…”

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confidence: 65%

delta '-function perturbations and Neumann boundary conditions by path integration

Grosche¹

1995

J. Phys. A: Math. Gen.

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δ ′ -function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual δ-function or a δ ′ -function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral.

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mentioning

confidence: 65%

delta '-function perturbations and Neumann boundary conditions by path integration

Grosche¹

1995

J. Phys. A: Math. Gen.

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“…The list of the exact solutions for this propagator is very short. For example, there is an exact solution for the spacetime propagator < x| exp(−iHt/ )|x ′ > of the Schrödinger equation in the one-dimensional square barrier case obtained in [8], but this solution is very complicated, implicit and not easy to analyze (see also [9,10,11]).…”

Section: Introductionmentioning

confidence: 99%

An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times

Los¹,

Los²

2016

Ukr. J. Phys.

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A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x, t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t → −i /kB T ), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. This density matrix provides information on the particles' density in the different spatial areas relative to the potential location and on the quantum coherences of the different particle spatial states. If one shifts to imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density |ψ(x, t)| 2 , the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures..

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“…The problem has been attacked many times. First and unsuccessful attempt to construct the Green function and propagator was done in [2], then the Green function of the problem was obtained by different methods in [4,6,7,9]. The explicit formula for the propagator between two points on the same side of a finite step barrier was correctly obtained in [4] for the first time, then the full problem has been analyzed in detail in [5] based on earlier developed path decomposition expansion method (PDX method) [1] and in "imaginary time" that corresponds in fact consideration of the diffusion rather than Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

Once again on quantum propagator for the step potential

Krylov

Belov

2012

AIP Conference Proceedings

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We present one another consideration of the propagator for a quantum particle in the presence of the step potential. This problem has been under treatment several times both as quantum and as stochastic ones but the explicit formulas derived in [12] for a problem in the "imaginary time representation" could hardly be used for computations in quantum case. Following mainly [5] the systematic exploration performed and the explicit expressions for the propagator in real time are constructed.

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Path integration via Hamilton-Jacobi coordinates and applications to potential barriers

Abstract: The path integral for the propagator is reduced to an ordinary integral in terms of the generators of a canonical transformation, and is evaluated exactly for square potential barriers in one dimension and for the radial square-well potential in two dimensions.

Cited by 25 publications

References 4 publications

delta '-function perturbations and Neumann boundary conditions by path integration

delta '-function perturbations and Neumann boundary conditions by path integration

An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times

Once again on quantum propagator for the step potential

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