Flows of quasi-classical trajectories and asymptotics solutions of the Schr\"odinger equation

V. V. Khablov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 773-784

Résumé

The paper analyzes the asymptotics of solutions of the Schrödinger equation with respect to a small parameter $\hbar$. It is well known that short-wave asymptotics for solutions of this equation leads to a pair of equations — the Hamilton–Jacobi equation for the phase and the continuity equation. These equations coincide with the ones for the potential flows of an ideal fluid. The physical meaning of the wave function is invariant with respect to of the complex plane rotations group, and the asymptotics is constructed as a point-dependent action of this group on some function that is found by solving the transfer equation. It is shown that if the Heisenberg group is used instead of the rotation group, then the limit of the Schrödinger equations solutions with $\hbar$ tending to zero, lead to equations for vortex flows of an ideal fluid in a potential field of forces. If the original Schrödinger equation is nonlinear, then equations for barotropic processes in an ideal fluid are obtained.

Détail

Keywords: Schroödinger equation, short-wave asymptotics, quasi-classical approximation, quasi-classical limit.
Mots-clés : Euler equations