Time-periodic solutions of multidimensional Hamilton systems

E. Caurier, Stanisław Drożdż, Marek Płoszajczak

Kazuo Takatsuka, Hiroshi Ushiyama

A class of complex solutions to the time-independent Hamilton-Jacobi equation in the real-valued configuration space that represent multidimensional nonclassical motions such as dynamical tunneling, namely, energetically allowed but dynamically forbidden transition, as well as the ordinary tunneling are shown. We introduce a quantity called ``parity of motion'' into each coordinate in configuration space for the Hamilton-Jacobi equation and thereby construct the solutions. Positive parity induces merely ordinary classical motion, while negative parity allows nonclassical motion such as tunneling. These solutions are classified by a given set of parities, each class of which forms a sheet in the entire solution space. New canonical equations of motion are derived with which nonclassical paths are generated in each sheet. Furthermore, it is shown that each sheet is associated with two kinds of action integrals: One, which is real valued, satisfies the principle of least action, thereby generating both ordinary and tunneling trajectories, but is not a solution to the Hamilton-Jacobi equation, while the other action is a solution to the time-independent Hamilton-Jacobi equation, and is complex valued in a tunneling region. Only in Newtonian mechanics, which forms an extreme sheet having all the parities positive, do these two actions happen to coincide with each other. Numerical examples for dynamical tunneling among tori in the H\'enon-Heiles system and ordinary potential tunneling in a three-dimensional system are presented.

Daniel R. Bes

Alexandru Popa

It is well known that the Schrödinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrödinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.

Daniel R. Bes

Michael Beyer, Markus Patzold, Wilfried Grecksch +1 more

Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton's equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.