The hydrodynamical formulation of quantum mechanics and the two-dimensional parabolic potential barrier

Toshiki Shimbori

THE dissertation deals with the two-dimensional isotropic parabolic potential barrier as a solvable model of a two-dimensional unstable system in non-relativistic quantum mechanics. The time-independent Schrodinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigen-functions belonging to complex energy eigenvalues. There are stationary states with a real energy eigenvalues. There are statinary states with a real energy eigenvalue involved in those solutions, and they are infinitely degenerate. A physical picture of the generalized eigenstates of the unstable system is obtained from the hydrodynamical point of view. For the first few stationary states of the two-dimensional parabolic potential barrier, the complex velocity potentials in the hydrodynamical formulation of quantum mechanics express the two-dimensional irrotational flows round a right angle. We find that this result can be deduced from the general connection between complex velocity potentials and singular potentials in two-dimensional quantum systems.

Mario Schönberg

Sebastiano Sonego

Toshiki Shimbori, Tsunehiro Kobayashi

In the two-dimensional isotropic parabolic potential barrier V ( x , y ) = V 0 - m γ 2 ( x 2 + y 2 )/2, though it is a model of an unstable system in quantum mechanics, we can obtain the stationary states corresponding to the real energy eigenvalue V 0 . Further, they are infinitely degenerate. For the first few eigenstates, we will find the stationary flows round a right angle that are expressed by the complex velocity potentials W = ±γ z 2 /2. From the hydrodynamical point of view vortex structures for the general solutions are also studied.

Antonio Boulhosa Nassar

Antonio Boulhosa Nassar

Via the hydrodynamical formulation of quantum mechanics, an approach to the problem of tunneling through sharp-edged potential barriers is developed. Above all, it is shown how more general boundary conditions follow from the continuity of mass, momentum, and energy.