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On a form of nonlinear dissipative wave mechanics valid in position‐ and momentum‐space
Author(s) -
Schuch Dieter
Publication year - 1994
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560520826
Subject(s) - dissipative system , momentum (technical analysis) , physics , position (finance) , nonlinear system , classical mechanics , position and momentum space , space (punctuation) , logarithm , mathematical analysis , mathematics , quantum mechanics , linguistics , philosophy , finance , economics
In wave mechanics an appropriate description of a system under the influence of a linearly velocity dependent frictional force can be given by a nonlinear Schrödinger equation ( NLSE ) with logarithmic nonlinearity. However, the particular logarithmic form of the dissipative nonlinear frictional term in the NLSE is connected with the definition of the momentum‐ or velocity‐operator in position‐space. Therefore, in momentum‐space, this form of the NLSE is no longer correct to describe the same physical sitution. This can be seen, e.g., from the fact that, in contrast to the linear case, the Fourier transform of the solution of the NLSE in position‐space does not fulfill anymore the logarithmic NLSE in momentum‐space. It will be shown, using results obtained from the theory in position‐space, that it is possible to find a form of the nonlinear dissipative frictional term which is valid in position‐ as well as in momentum‐space. Using this form, the NLSE looks like a diffusion equation with complex diffusion coefficient, i.e., a combination of a diffusion and a Schrödinger equation. The solution of this NLSE in momentum‐space will be discussed. © 1994 John Wiley & Sons, Inc.