Abstract
We revisit the analogy suggested by Madelung between a non-relativistic time-dependent quantum particle, to a fluid system which is pseudo-barotropic, irrotational and inviscid. We first discuss the hydrodynamical properties of the Madelung description in general, and extract a pressure like term from the Bohm potential. We show that the existence of a pressure gradient force in the fluid description, does not violate Ehrenfest’s theorem since its expectation value is zero. We also point out that incompressibility of the fluid implies conservation of density along a fluid parcel trajectory and in 1D this immediately results in the non-spreading property of wave packets, as the sum of Bohm potential and an exterior potential must be either constant or linear in space. Next we relate to the hydrodynamic description a thermodynamic counterpart, taking the classical behavior of an adiabatic barotopric flow as a reference. We show that while the Bohm potential is not a positive definite quantity, as is expected from internal energy, its expectation value is proportional to the Fisher information whose integrand is positive definite. Moreover, this integrand is exactly equal to half of the square of the imaginary part of the momentum, as the integrand of the kinetic energy is equal to half of the square of the real part of the momentum. This suggests a relation between the Fisher information and the thermodynamic like internal energy of the Madelung fluid. Furthermore, it provides a physical linkage between the inverse of the Fisher information and the measure of disorder in quantum systems—in spontaneous adiabatic gas expansion the amount of disorder increases while the internal energy decreases.
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Notes
Indeed, more complex interpretations have been suggested with regards to the quantum pressure [3, 5], however the proposed pressure like term still seems to us the most straightforward one as we are seeking for the simplest analogy. A second rank tensor of the Cauchy stress tensor, embedding shear terms, as well as the introduction of turbulence to the ME, or the representation of it in terms of nonlinear diffusion are interesting interpretations, however they involve an amount of complexity that we try to avoid when considering the dynamics of a single quantum particle.
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Acknowledgments
The authors are grateful to Sonja Rosenlund Ahl for sharing her original thoughts. E.H. is grateful to Rachel Heifetz and Nili Harnik for their constructive comments. E.C. was supported in part by the Israel Science Foundation Grant No. 1311/14.
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Heifetz, E., Cohen, E. Toward a Thermo-hydrodynamic Like Description of Schrödinger Equation via the Madelung Formulation and Fisher Information. Found Phys 45, 1514–1525 (2015). https://doi.org/10.1007/s10701-015-9926-1
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DOI: https://doi.org/10.1007/s10701-015-9926-1