Quantum Theory from a Nonlinear Perspective by Dieter Schuch

Author:Dieter Schuch
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In the case of the damped HO, all three approaches discussed in detail in this subsection possess Gaussian WP solutions (see (4.40) for , (4.63) for and (2.​1) for ). The (Newton-type) equation of motion for the maximum of these WPs as well as the complex Riccati equations or equivalent real NL Ermakov equations and the corresponding Ermakov invariants (that exist in all these cases) might look different when expressed in the canonical variables of the respective approach (see Fig. 4.3). However, using the above-mentioned transformations they can all be transformed into the equation for the damped HO, (4.41), describing the motion of the maximum and thus the trajectory of the classical position and, for the dynamics of the width, the Riccati Eq. (4.123) or equivalent Ermakov Eq. (4.125) with the corresponding invariant 4.126) obtained on the physical level from the logarithmic NLSE (4.115). Therefore, this approach will be used in the next section to demonstrate the similarities and differences in comparison with the conservative case if the dynamics of the dissipative system is expressed with the help of the complex Riccati equation, like shown in Chap. 2.

On the physical level, the logarithmic NLSE combines the dissipative aspect of Kostin’s approach with the irreversible one of Doebner and Goldin and Beretta in a consistent manner and, apart from a purely TD contribution that does not affect the dynamics of the WP solutions, leads to identical results like the approach by Hasse that combines attempts by Süssmann and Albrecht (see Fig. 4.4). A closer comparison between Hasse’s form of the friction term and the logarithmic nonlinearity will be given in Chap. 6 where the dissipative version of TI NL quantum mechanics is discussed.

Fig. 4.4Interrelations amongst canonical and physical approaches for the description of dissipative systems

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