Quantum hydrodynamics with complex quantities

Ciann‐Dong Yang

Moise Bonilla–Licea, Dieter Schuch, M. Bonilla

In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics.

R. Eisenschitz

Dieter Schuch, Moise Bonilla–Licea

Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space.

A. I. Laptukhov

Andika Kusuma Wijaya, Seramika Ariwahjoedi, Muhammad Farchani Rosyid

This research aims to formulate conserved quantities (classical observables) in Noether’s theorem for quantum hydrodynamics within the framework of symplectic geometry. The symplectic one-form or symplectic potential is identical to the Noether current. 31 The Noether current plays an important role in determining classical observables and conserved quantities, namely, [Formula: see text] and [Formula: see text] associated with the above symmetry in quantum hydrodynamics (nonrelativistic and relativistic) associated with Killing vector field and rotational symmetry group [Formula: see text].