Towards a mathematical Theory of the Madelung Equations

Even though the Madelung equations are central to many 'classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schr\"odinger equation is recovered without the imposition of an additional 'ad hoc quantization condition'--like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required.Comment: 85 pages, 1 figure; keywords: Madelung equations, Schr\"odinger equation, quantum potential, quantum vortices, stochastic mechanic

Reconciling Semiclassical and Bohmian Mechanics: IV. Multisurface Dynamics

In previous articles [J. Chem. Phys. 121 4501 (2004), J. Chem. Phys. 124 034115 (2006), J. Chem. Phys. 124 034116 (2006)] a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrodinger equation, such that the components Psi+- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications, and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+/-) transitions.Comment: 11 pages, 6 figure

Comment on "Born's rule for arbitrary Cauchy surfaces"

A recent article has raised the question of how to generalize the Born rule from non-relativistic quantum theory to curved spacetimes and claimed to answer it for the special-relativistic case (Lienert and Tumulka, Lett. Math. Phys. 110, 753 (2019)). The proposed generalization originated in prior works on `hypersurface Bohm-Dirac models' as well as approaches to relativistic quantum theory developed by Bohm and Hiley. In this comment, we raise three objections to the rule and the broader theory in which it is embedded. In particular, to address the underlying assertion that the Born rule is naturally formulated on a spacelike hypersurface, we provide an analytic example showing that a spacelike hypersurface need not remain spacelike under proper time evolution -- even in the absence of curvature. We finish by proposing an alternative `curved Born rule' for the one-body case on general spacetimes, which overcomes these objections, and in this instance indeed generalizes the one Lienert and Tumulka attempted to justify. The respective mathematical theory is almost analogous for the conservation of charge and mass, being two additional examples of physical quantities obtained from integrating a scalar field over particular hypersurfaces. Our approach can also be generalized to the many-body case, which shall be the subject of a future work.Comment: 12 pages, 3 figures; Keywords: Integral conservation laws, continuity equation, Born rule, detection probability, multi-time wave function, spacelike hypersurfac

Reconciling Semiclassical and Bohmian Mechanics: I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill-behaved at turning points. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by infinite forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.Comment: 17 pages, 5 figure

Andromeda: A Few-body Plane Wave Calculator

At TACCSTER last year, a novel method of ours to solve the 3-body lithium problem was presented. Without finishing, the computation plateaued at -7.3 (of -7.4) Hartree on an L = 67 ^ 9 grid running on a single TACC Lonestar5 node for three months. We have now released a new version of the Andromeda code capable of embarrassingly parallel operations. This improvement followed from a significant speedup of half the process, namely the free and exact creation of the Hamiltonian quantum operators and their operation in Sums of Products form. Even though this does not speed up the vector decomposition process, which is still the rate-limiting step, we can now distribute processing per term-state combination across numerous computational resources to overcome this problem. In particular, any 2-body interaction quantum operator is now a summation of processes defined by separate 1-body matrices for the 2-body diagonal, 1-body diagonal, and off-diagonal aspects of the quantum operation. Thus, every core in a parallel process can individually initialize the Coulombic quantum operator, which allows embarrassingly parallel operations across several state vectors. The current release has integrated the TACC/launcher as a vehicle to handle parallel operations. Digitize your wave function with the most local representation of the plane-wave basis. Tackle strongly correlated problems with a spatial component separated, but fully multi-body, Sums-of-Products representation. Compute 3-body quantum physics with a powerful scripting interface. Discover something