A special class of solutions of the Schr�dinger equation for a free particle

Robert G. Cawley

V. V. Ul'yanov

И. Е. Сазонов, Н. И. Жирнов

Thomas Durt

We study certain non-linear generalisations of the Schr{ö}dinger equation which admit static solitonic 2 solutions in absence of external potential acting on the particle. We consider a class of solutions that can be written as a product of a solution of the linear Schr{ö}dinger equation with a peaked quantum soliton, in a regime where the size of the soliton is quite smaller than the typical scale of variation of the linear wave. In the non-relativistic limit, the solitons obey a generalized de Broglie-Bohm (dB-B) guidance equation. In first approximation, this guidance equation reduces to the dB-B guidance equation according to which they move at the so-called de Broglie-Bohm velocity along the hydrodynamical flow lines of the linear Schr{ö}dinger wave. If we consider a spinorial electronic wave function a la Dirac, its barycentre is predicted to move exactly in accordance with the dB-B guidance equation.

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

Luo Guan

Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.