On Hamilton-Jacobi theory: its geometry and relation to pilot-wave theory

Quantum Hamilton-Jacobi analysis of PT symmetric Hamiltonians

We apply the quantum Hamilton-Jacobi formalism, naturally defined in the complex domain, to a number of complex Hamiltonians, characterized by discrete parity and time reversal (PT) symmetries and obtain their eigenvalues and eigenfunctions. Examples of both quasi-exactly and exactly solvable potentials are analyzed and the subtle differences, in the singularity structures of their quantum momentum functions, are pointed out. The role of the PT symmetry in the complex domain is also illustrated.

Solutions of the Schrödinger equation given by solutions of the Hamilton--Jacobi equation

We find the form of the potential depending on the coordinates and the time such that a solution, $S$, of the Hamilton--Jacobi equation yields an exact solution, $\exp ({\rm i} S/\hbar)$, of the corresponding Schrödinger equation.

Moving Picture and Hamilton-Jacobi Theory in Quantum Mechanics

We propose a new picture, which we call the {\it moving picture}, in quantum mechanics. The Schrödinger equation in this picture is derived and its solution is examined. We also investigate the close relationship between the moving picture and the Hamilton-Jacobi theory in classical mechanics.

关于量子哈密顿-雅可比方程中李-哈密顿系统的一些特征

On some signatures of Lie-Hamilton System in Quantum Hamilton Jacobi Equation

本文研究了量子哈密顿-雅可比方程中李-哈密顿系统的特征，发现其可转换为凯利-克莱因里卡蒂方程，并探讨了李对称性和李积分。

The Generalized PT-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton-Jacobi Formalism

The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential are found for n=0,1 states.

The quantum Hamilton Jacobi equation and the link between classical and quantum mechanics

We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton-Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton-Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure.

Quantum Stationary Hamilton Jacobi Equation in 3-D for symmetrical potentials. Introduction of the Spin

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.

Canonical Transformations and the Hamilton-Jacobi Theory in Quantum Mechanics

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.

Novel formulation of Hamilton-Jacobi equation for higher derivative theory and quantum mechanical correspondence

For higher derivative theories, using the approach of Caratheodory's equivalent Lagrangian, we show that there exist novel formulations of Hamilton-Jacobi equations, which are different from the formulations derived from Hamilton's canonical approach. The quantum mechanical correspondences of these novel Hamilton-Jacobi equations lead to nonlinear quantum mechanics, which seem being able to avoid the unbounded negative energy problem in the quantum mechanics of higher derivative theories.

The Quantum Effective Mass Hamilton-Jacobi Problem

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.

Hamilton-Jacobi theory for Hamiltonian systems with non-canonical symplectic structures

A proposal for the Hamilton-Jacobi theory in the context of the covariant formulation of Hamiltonian systems is done. The current approach consists in applying Dirac's method to the corresponding action which implies the inclusion of second-class constraints in the formalism which are handled using the procedure of Rothe and Scholtz recently reported. The current method is applied to the nonrelativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported.

量子哈密顿-雅可比量子化与形状不变性

Quantum Hamilton-Jacobi Quantization and Shape Invariance

本文证明常规势的可加形状不变性与未破缺超对称性足以保证其在量子哈密顿-雅可比量子化框架中可解，从而扩展了该方法的适用范围。