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The complete solution of the Hamilton-Jacobi equation in Quantum Mechanics  [PDF]
Rafael Ferraro
Physics , 1996,
Abstract: An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.
Derivation of the Schr?dinger equation from the Hamilton-Jacobi equation in Feynman's path integral formulation of quantum mechanics  [PDF]
J. H. Field
Physics , 2012,
Abstract: It is shown how the time-dependent Schr\"{o}dinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schr\"{o}dinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an {\it a priori} assumption equivalent to Feynman's dynamical postulate. De Broglie's concepts of 'matter waves' and their phase and group velocities are also critically discussed.
Quantum mechanics without spacetime IV : a noncommutative Hamilton-Jacobi equation  [PDF]
T. P. Singh
Physics , 2004,
Abstract: It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspondence rule for the commutation relations, it can be argued that this noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics.
Quantum Stationary Hamilton Jacobi Equation in 3-D for symmetrical potentials. Introduction of the Spin  [PDF]
T. Djama
Physics , 2001,
Abstract: We establish the quantum stationary Hamilton-Jacobi equation in 3-D and its solutions for three symmetrical potentials, Cartesian symmetry potential, spherical symmetry potential and cylindrical symmetry potential. For the two last potentials, a new interpretation of the Spin is proposed within the framework of trajectory representation.
The Relativistic Quantum Stationary Hamilton Jacobi Equation for Particle with Spin 1/2  [PDF]
T. Djama
Physics , 2003,
Abstract: For one dimensional motions, we derive from the Dirac Spinors Equation (DSE) the Quantum Stationary Hamilton-Jacobi Equation for particles with spin 1/2. Then, We give its solution. We demonstrate that the $QSHJES_{1\over2}$ have two explicit forms, which represent the two possible projection of the Spin 1/2.
Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation  [PDF]
R. Friedberg,T. D. Lee,W. Q. Zhao
Physics , 1999, DOI: 10.1007/BF03035922
Abstract: We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $\epsilon$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $\epsilon$, we show how the coefficients of $g^{-m}\epsilon^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.
Quantum Hamilton-Jacobi Theory  [PDF]
Marco Roncadelli,L. S. Schulman
Physics , 2007, DOI: 10.1103/PhysRevLett.99.170406
Abstract: Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schroedinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory.
The Quantum Effective Mass Hamilton-Jacobi Problem  [PDF]
Ozlem Yesiltas
Physics , 2010, DOI: 10.1088/1751-8113/43/9/095305
Abstract: In this article, the quantum Hamilton- Jacobi theory based on the position dependent mass model is studied. Two effective mass functions having different singularity structures are used to examine the Morse and Poschl- Teller potentials. The residue method is used to obtain the solutions of the quantum effective mass- Hamilton Jacobi equation. Further, it is shown that the eigenstates of the generalized non-Hermitian Swanson Hamiltonian for Morse and Poschl-Teller potentials can be obtained by using the Riccati equation without solving a differential equation.
Canonical Transformations and the Hamilton-Jacobi Theory in Quantum Mechanics  [PDF]
Jung-Hoon Kim,Hai-Woong Lee
Physics , 1999, DOI: 10.1139/cjp-77-6-411
Abstract: Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number forms of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.
Exponential Type Complex and non-Hermitian Potentials within Quantum Hamilton-Jacobi Formalism  [PDF]
Ozlem Yesiltas,Ramazan Sever
Mathematics , 2007,
Abstract: PT-/non-PT-symmetric and non-Hermitian deformed Morse and Poschl-Teller potentials are studied first time by quantum Hamilton-Jacobi approach. Energy eigenvalues and eigenfunctions are obtained by solving quantum Hamilton-Jacobi equation.
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