Schuch D, Bonilla-Licea M. Uncertainty Relations in the Madelung Picture Including a Dissipative Environment.
Entropy (Basel) 2023;
25:312. [PMID:
36832678 PMCID:
PMC9955996 DOI:
10.3390/e25020312]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2022] [Revised: 01/31/2023] [Accepted: 02/02/2023] [Indexed: 06/18/2023]
Abstract
In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.
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