Quantum Hamilton-Jacobi Theory Research Articles

The geometrical origin of dark energy

The geometrical formulation of the quantum Hamilton–Jacobi theory shows that the quantum potential is never trivial, so that it plays the rôle of intrinsic energy. Such a key property selects the Wheeler–DeWitt (WDW) quantum potential Q[g_{jk}] as the natural candidate for the dark energy. This leads to the WDW Hamilton–Jacobi equation with a vanishing kinetic term, and with the identification Λ=-κ2g¯Q[gjk].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Lambda =-\\frac{\\kappa ^2}{\\sqrt{{\\bar{g}}}}Q[g_{jk}]. \\end{aligned}$$\\end{document}This shows that the cosmological constant is a quantum correction of the Einstein tensor, reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas–Fermi theory. The quantum potential also defines the Madelung pressure tensor. The geometrical origin of the vacuum energy density, a strictly non-perturbative phenomenon, provides strong evidence that it is due to a graviton condensate. Time independence of the regularized WDW equation suggests that the ratio between the Planck length and the Hubble radius may be a time constant, providing an infrared/ultraviolet duality. We speculate that such a duality is related to the local to global geometry theorems for constant curvatures, showing that understanding the universe geometry is crucial for a formulation of Quantum Gravity.

Symmetric tops in combined electric fields: Conditional quasisolvability via the quantum Hamilton-Jacobi theory

We make use of the Quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasi-solvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasi-solvability of the time-independent Schroedinger equation as well as the corresponding finite sets of exact analytic solutions. We do so for this prototypical trigonometric system as well as for its anti-isospectral hyperbolic counterpart. An examination of the algebraic and numerical spectra of these two systems reveals mutually closely related patterns. The QHJ approach allows to retrieve the closed-form solutions for the spherical and planar pendula and the Razavy system that had been obtained in our earlier work via Supersymmetric Quantum Mechanics as well as to find a cornucopia of additional exact analytic solutions.

Quantum Forces in a Relativistic Entangled "Space-Time" State in Complex Hamiltonian Dynamics

Based on complex quantum Hamilton-Jacobi theory and its natural complex spacetime configuration, we realized new states for both positive and negative values of energy and momentum, and we discuss this complex configuration in a relativistic entangled "space-time" state and the resultant 12 extra wave functions than the four solutions of Dirac equation for a free particle. Accordingly, we calculate the quantum forces between particles and antiparticles in a relativistic entangled "space-time" state.

EPR & Klein Paradoxes in Complex Hamiltonian Dynamics and Krein Space Quantization

Negative energy states are applied in Krein space quantization approach to achieve a naturally renormalized theory. For example, this theory by taking the full set of Dirac solutions, could be able to remove the propagator Green function's divergences and automatically without any normal ordering, to vanish the expected value for vacuum state energy. However, since it is a purely mathematical theory, the results are under debate and some efforts are devoted to include more physics in the concept. Whereas Krein quantization is a pure mathematical approach, complex quantum Hamiltonian dynamics is based on strong foundations of Hamilton-Jacobi (H-J) equations and therefore on classical dynamics. Based on complex quantum Hamilton-Jacobi theory, complex spacetime is a natural consequence of including quantum effects in the relativistic mechanics, and is a bridge connecting the causality in special relativity and the non-locality in quantum mechanics, i.e. extending special relativity to the complex domain leads to relativistic quantum mechanics. So that, considering both relativistic and quantum effects, the Klein-Gordon equation could be derived as a special form of the Hamilton-Jacobi equation. Characterizing the complex time involved in an entangled energy state and writing the general form of energy considering quantum potential, two sets of positive and negative energies will be realized. The new states enable us to study the spacetime in a relativistic entangled “space-time” state leading to 12 extra wave functions than the four solutions of Dirac equation for a free particle. Arguing the entanglement of particle and antiparticle leads to a contradiction with experiments. So, in order to correct the results, along with a previous investigation [1], we realize particles and antiparticles as physical entities with positive energy instead of considering antiparticles with negative energy. As an application of modified descriptions for entangled (space-time) states, the original version of EPR paradox can be discussed and the correct answer can be verified based on the strong rooted complex quantum Hamilton-Jacobi theory [2-27] and as another example we can use the negative energy states, to remove the Klein's paradox without the need of any further explanations or justifications like backwardly moving electrons. Finally, comparing the two approaches, we can point out to the existence of a connection between quantum Hamiltonian dynamics, standard quantum field theory, and Krein space quantization [28-43].

Superluminality and a curious phenomenon in the relativistic quantum Hamilton–Jacobi equation**Dedicated to Shawn Lane.

A basic problem in the relativistic quantum Hamilton–Jacobi theory is to understand whether it may admit superluminal solutions. Here we consider the averaging of the speed on a period of the oscillating term which is similar to Dirac's averaging of the oscillating part of the free electron's speed. Such an averaging solves the problem of getting the ℏ → 0 limit of the speed of the free particle, and leads to solutions that, depending on the integration constants, may be superluminal.

The quantum effective mass Hamilton–Jacobi problem

In this paper, 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 Pöschl–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 Pöschl–Teller potentials can be obtained by using the Riccati equation without solving a differential equation.

The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

We review here some conventional as well as less conventional as- pects of the time-independent and time-dependent Hamilton-Jacobi (HJ) the- ory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.

The research of the quantum Hamilton–Jacobi theory

The quantum Hamilton–Jacobi equation (QHJE) in the quantum Hamilton–Jacobi theory is analytically investigated. New exact analytical solutions for the quantum Hamilton–Jacobi equation have been obtained for the quantum generating function of quantum theory for the first time. We discover the exact analytical description for the general case for the quantum generating function in the quantum Hamilton–Jacobi theory.

Quantum Hamilton-Jacobi Theory

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 Schrödinger 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.

Shape invariance and the exactness of the quantum Hamilton–Jacobi formalism

Abstract Quantum Hamilton–Jacobi theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrodinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this Letter, we show that shape invariance also suffices to determine the eigenvalues in quantum Hamilton–Jacobi theory.

On modeling and visualizing single-electron spin motion

Abstract In this paper quantum Hamilton–Jacobi theory are exploited to model and visualize single-electron spin motion at zero-point energy state. Quantum Hamilton equations of motion are derived and solved analytically for an electron moving in a constant magnetic field. The resulting electron’s trajectories explain explicitly why the electron has quantized spin and orbital angular momenta and why the electron has an intrinsic spin ℏ /2 with a g factor of 2. This quantum spin model, unlike the usual one expressed by the abstract spin matrices, is fully based on the measurable motion trajectory of electron and may hopefully provide us a succinct guideline to visualize single-electron spin motion in laboratory.

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

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.

Solving quantum trajectories in Coulomb potential by quantum Hamilton–Jacobi theory

AbstractWe show that the quantum central‐force problems can be modeled and solved exactly by quantum Hamilton–Jacobi formulation, from which the quantum operators $\hat{L}$z, $\hat{L}$2, and $\hat{H}$ can be derived without using the quantization principle p → ($\hbar$/i)∂/∂x. Quantum conservation laws expressed by the Poisson bracket show that the eigenvalues of these quantum operators are just equal to the constants of motion along the eigen‐trajectories defined in a complex domain. The shell structure observed in bound systems, such as the hydrogen atom, is found to stem from the structure of the quantum potential, by which the quantum forces acting on the electron can be uniquely determined, the stability of atomic configuration can be justified, and the quantum trajectories of the electron can be obtained by integrating the related quantum Lagrange equations. On solving the quantum equations of motion, the solution of the Schrödinger equation serves as the first integration of the second‐order quantum Lagrange equations. The stable equilibrium points of the derived first‐order nonlinear quantum dynamics are shown to be identical to the positions with maximum probability predicted by standard quantum mechanics. The internal mechanism of how the quantum dynamics evolve continuously to classical dynamics and of how the quantum conservation laws transit continuously to the classical conservation laws as n → ∞ are analyzed in detail. The construction of the quantum scattering trajectory by searching for an unbound solution for the Schrödinger equation is investigated. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Exactly solvable systems and the quantum Hamilton–Jacobi formalism

We connect quantum Hamilton–Jacobi theory with supersymmetric quantum mechanics (SUSYQM). We show that the shape invariance, which is an integrability condition of SUSYQM, translates into fractional linear relations among the quantum momentum functions.

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 form 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.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

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 form 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.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

Hamilton-Jacobi Theory and the Quantum Action Variable

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.