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

Physica Scripta

Quantum action-angle variable of the Hamilton–Jacobi theory in two dimensions

Gang Chen, Pei-cai Xuan, Jianli Wang

Published Mar 28, 2006• 6 citations
Publisher

Abstract

In this paper, a quantum action-angle variable in two dimensions is defined in the context of the quantum Hamilton–Jacobi theory, and then a quantization rule, which can give the exact bound state energy spectrum without solving the motion equation for the wavefunction, is derived from the singularities of the quantum momentum function. As illustrative examples, the Coulomb potential and the harmonic oscillator are considered.

DOI
10.1088/0031-8949/73/5/005

Related papers

Hamilton-Jacobi Theory and the Quantum Action Variable

Robert A. Leacock, Michael J. Padgett

A quantum action variable is defined in the context of a quantum Hamilton-Jacobi theory. The action variable can be used to find the exact bound-state energy levels of a quantum system without solving an equation of motion for the system wave functions.

Physical Review Letters•Jan 3, 1983•148 citations

Hamilton-Jacobi/action-angle quantum mechanics

Robert A. Leacock, Michael J. Padgett

A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1983•150 citations

Quantum Hamilton–Jacobi formalism and the bound state spectra

R. S. Bhalla, Ashok Kumar Kapoor, Prasanta Kumar Panigrahi

It is well known in classical mechanics that the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. Analogously, the equivalent quantum action variable appearing in the quantum Hamilton–Jacobi formalism can provide the energy eigenvalues of a bound state problem, without the necessity of solving the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well-known solvable potentials. It is also shown how this method provides an understanding as to why approximate quantization schemes such as ordinary and supersymmetric WKB can give exact answers for certain potentials.

arXiv:quant-ph/9512018•American Journal of Physics•Dec 1, 1997•71 citations

On the nodes of wave function and the quantum Hamilton-Jacobi solution

L. A. Poveda-Cuevas, Freddy Jackson Poveda-Cuevas

We present the analytic solution for the stationary quantum HamiltonJacobi equation. Knowing the strong relation between the Riccati and quantum Hamilton-Jacobi equations, we develop a simple method to obtain the exact solution. Then, in order to prove the validity of the proposed method, we use two central potentials: the three-dimensional harmonic oscillator and Coulomb potential, both with bound-states. Finally, we compute the action-angle variables in a entirely quantum version for to achieve connect with the nodes of the wave function.

arXiv:1609.01198•Sep 5, 2016•1 citation

Quantum Mechanics from the Hamilton-Jacobi Point of View

Alexander Jurisch

In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.

arXiv:quant-ph/0612217•Dec 27, 2006•0 citations

Classical and Quantum Behavior of Genaralized Oscillators - action variable, angle variable and quantum phase -

M. Omote, Shotaro Sakoda, S. Kamefuchi

The relation that exists in quantum mechanics among action variables, angle variables and the phases of quantum states is clarified, by referring to the system of a generalized oscillator. As a by-product, quantum-mechanical meaning of the classical Hamilton-Jacobi equation and related matters is clarified, where a new picture of quantum mechanics is introduced, to be called the Hamilton-Jacobi picture.

arXiv:hep-th/0106115•Jun 14, 2001•0 citations
In this paper, a quantum action-angle variable in two dimensions is defined in the context of the quantum Hamilton–Jacobi theory, and then a quantization rule, which can give the exact bound state energy spectrum without solving the motion equation for the wavefunction, is derived from the singularities of the quantum momentum function. As illustrative examples, the Coulomb potential and the harmonic oscillator are considered.
A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
It is well known in classical mechanics that the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. Analogously, the equivalent quantum action variable appearing in the quantum Hamilton–Jacobi formalism can provide the energy eigenvalues of a bound state problem, without the necessity of solving the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well-known solvable potentials. It is also shown how this method provides an understanding as to why approximate quantization schemes such as ordinary and supersymmetric WKB can give exact answers for certain potentials.
We present the analytic solution for the stationary quantum HamiltonJacobi equation. Knowing the strong relation between the Riccati and quantum Hamilton-Jacobi equations, we develop a simple method to obtain the exact solution. Then, in order to prove the validity of the proposed method, we use two central potentials: the three-dimensional harmonic oscillator and Coulomb potential, both with bound-states. Finally, we compute the action-angle variables in a entirely quantum version for to achieve connect with the nodes of the wave function.
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.