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

Arindam Chakraborty

Abstract

The general forms of Quantum Hamilton Jacobi Equation for a particle of constant mass, position dependent effective mass and non-Hermitian Effective mass Swanson model have been considered. It has been found that the said equations can be recast in the form of Cayley-Klein Riccati equations which admit a Lie-Hamilton structure. The possible expressions of Lie symmetry and Lie Integral have also been considered.

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

Abstract

The general forms of Quantum Hamilton Jacobi Equation for a particle of constant mass, position dependent effective mass and non-Hermitian Effective mass Swanson model have been considered. It has been found that the said equations can be recast in the form of Cayley-Klein Riccati equations which admit a Lie-Hamilton structure. The possible expressions of Lie symmetry and Lie Integral have also been considered.
Paper Structure (8 sections, 2 theorems, 57 equations)

This paper contains 8 sections, 2 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Quantum Hamilton Jacobi (QHJ) Equation and Cayley-Klein Riccati (CKR) System
  3. Lie-Hamilton System and QHJ equations
  4. Poisson Structure and Lie-Hamilton System
  5. Effective Mass, Potential Function, Lie Symmetry and Lie Integral
  6. Calculation of symmetry operator $Y$
  7. x-dependent Lie Integral
  8. Connection between Lie Integral and QHJ formalism

Key Result

Proposition 1

The vector fields $\{\chi_i : i=1, 2, 3\}$ called symplectic vector fields admit a family of Hamiltonian functions$\{\mathcal{H}_1, \mathcal{H}_2, \mathcal{H}_3\}=\{-p_1^{-1}, -p_2p_1^{-1}, -(p_1^2+p_2^2)p_1^{-1}\}$ relative to the symplectic form $\omega=p_1^{-2}dp_2\wedge dp_1$. The Poisson bracke

Theorems & Definitions (12)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Remark 2
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 2
  • ...and 2 more