Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Comment on: Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure [J. Math. Phys. 37, 4310–4319 (1996)]

Alexander Wurm, J. LaChapelle

Published May 1, 1997• 0 citations
Publisher

Abstract

The authors comment on the paper by J. LaChapelle, J. Math. Phys. 37, 4310 (1996), and give explicit expressions for the parametrization, its solution, and the Lie derivatives of the Schrödinger equation for the case of n-dimensional spherical coordinates.
DOI
10.1063/1.532015

Related papers

Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure

J. LaChapelle

A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.

Journal of Mathematical Physics•Sep 1, 1996•0 citations

Comment on path integral derivation of Schrödinger equation in spaces with curvature and torsion

P Fiziev, H Kleinert

We present a derivation of the Schrödinger equation for a path integral of a point particle in a space with curvature and torsion which is considerably shorter and more elegant than that commonly found in the literature.

arXiv:hep-th/9604172•Journal of Physics A Mathematical and General•Dec 7, 1996•4 citations

Comment on Path Integral Derivation of Schrödinger Equation in Spaces with Curvature and Torsion

P. Fiziev, H. Kleinert

We present a derivation of the Schr odinger equation for a path integral of a point particle in a space with curvature and torsion which is considerably shorter and more elegant than that commonly found in the literature. 1.

•Jan 1, 1996•3 citations

The quantum mechanical Hamiltonian in curvilinear coordinates: A simple derivation

Henrik Grum Kjaergaard, Ole Sonnich Mortensen

The quantum mechanical Hamiltonian in curvilinear coordinates is obtained in a straightforward manner without using results from tensor analysis.

American Journal of Physics•Apr 1, 1990•9 citations

Fractional Schrödinger equation for a particle moving in a potential well

Yuri Luchko

In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].

Journal of Mathematical Physics•Jan 1, 2013•123 citations

Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

Selçuk Ş. Bayın

Hawkins and Schwarz [J. Math. Phys. 54, 014101 (2013)]10.1063/1.4772533 claim that there is a flaw in the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrödinger equation [S. S. Bayin, J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268; S. S. Bayin, J. Math. Phys. 53, 084101 (2012)]10.1063/1.4739758. We give additional details about how these integrals are evaluated and show that there is no inconsistency for an infinite square well.

Journal of Mathematical Physics•Jul 1, 2013•11 citations
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.
In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].
Hawkins and Schwarz [J. Math. Phys. 54, 014101 (2013)]10.1063/1.4772533 claim that there is a flaw in the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrödinger equation [S. S. Bayin, J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268; S. S. Bayin, J. Math. Phys. 53, 084101 (2012)]10.1063/1.4739758. We give additional details about how these integrals are evaluated and show that there is no inconsistency for an infinite square well.