86 research outputs found
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
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