2,342 research outputs found
What is semiquantum mechanics?
Semiclassical approximations to quantum dynamics are almost as old as quantum
mechanics itself. In the approach pioneered by Wigner, the evolution of his
quasiprobability density function on phase space is expressed as an asymptotic
series in increasing powers of Planck's constant, with the classical
Liouvillean evolution as leading term. Successive semiclassical approximations
to quantum dynamics are defined by successive terms in the series. We consider
a complementary approach, which explores the quantum-clssical interface from
the other direction. Classical dynamics is formulated in Hilbert space, with
the Groenewold quasidensity operator as the image of the Liouville density on
phase space. The evolution of the Groenewold operator is then expressed as an
asymptotic series in increasing powers of Planck's constant. Successive
semiquantum approximations to classical dynamics are defined by successive
terms in this series, with the familiar quantum evolution as leading term.Comment: Talk presented at IVth International Symposium on Quantum Theory and
Symmetries, Varna, August, 2005. 13 pages, 4 figure
Differential equations of electrodiffusion: constant field solutions, uniqueness, and new formulas of Goldman-Hodgkin-Katz type
The equations governing one-dimensional, steady-state electrodiffusion are
considered when there are arbitrarily many mobile ionic species present, in any
number of valence classes, possibly also with a uniform distribution of fixed
charges. Exact constant field solutions and new formulas of
Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike
the usual approximate ones. Corresponding boundary conditions on the ionic
concentrations are identified. The question of uniqueness of constant field
solutions with such boundary conditions is considered, and is re-posed in terms
of an autonomous ordinary differential equation of order for the electric
field, where is the number of valence classes. When there are no fixed
charges, the equation can be integrated once to give the non-autonomous
equation of order considered previously in the literature including, in the
case , the form of Painlev\'e's second equation considered first in the
context of electrodiffusion by one of us. When , the new equation is a
form of Li\'enard's equation. Uniqueness of the constant field solution is
established in this case.Comment: 29 pages, 5 figure
Airy series solution of Painlev\'e II in electrodiffusion: conjectured convergence
A perturbation series solution is constructed in terms of Airy functions for
a nonlinear two-point boundary-value problem arising in an established model of
steady electrodiffusion in one dimension, for two ionic species carrying equal
and opposite charges. The solution includes a formal determination of the
associated electric field, which is known to satisfy a form of the Painlev\'e
II differential equation. Comparisons with the numerical solution of the
boundary-value problem show excellent agreement following termination of the
series after a sufficient number of terms, for a much wider range of values of
the parameters in the model than suggested by previously presented analysis, or
admitted by previously presented approximation schemes. These surprising
results suggest that for a wide variety of cases, a convergent series expansion
is obtained in terms of Airy functions for the Painlev\'e transcendent
describing the electric field. A suitable weighting of error measures for the
approximations to the field and its first derivative provides a monotonically
decreasing overall measure of the error in a subset of these cases. It is
conjectured that the series does converge for this subset.Comment: 30 pages, 9 figures. Typos corrected, figures modified, extra
references adde
Non-positivity of Groenewold operators
A central feature in the Hilbert space formulation of classical mechanics is
the quantisation of classical Liouville densities, leading to what may be
termed term Groenewold operators. We investigate the spectra of the Groenewold
operators that correspond to Gaussian and to certain uniform Liouville
densities. We show that when the classical coordinate-momentum uncertainty
product falls below Heisenberg's limit, the Groenewold operators in the
Gaussian case develop negative eigenvalues and eigenvalues larger than 1.
However, in the uniform case, negative eigenvalues are shown to persist for
arbitrarily large values of the classical uncertainty product.Comment: 9 pages, 1 figures, submitted to Europhysics Letter
Localizing the Relativistic Electron
A causally well-behaved solution of the localization problem for the free
electron is given, with natural space-time transformation properties, in terms
of Dirac's position operator. It is shown that, although this operator does not
represent an observable in the usual sense, and has no positive-energy
(generalized) eigenstates, the associated 4-vector density is observable, and
can be localized arbitrarily precisely about any point in space, at any instant
of time, using only positive-energy states. A suitable spin operator can be
diagonalized at the same time.Comment: 19 pages including 1 figure (1 LatTex2e file, 1 postscript file).
Uses package amssymb. Typos correcte
Quantum symmetries and the Weyl-Wigner product of group representations
In the usual formulation of quantum mechanics, groups of automorphisms of
quantum states have ray representations by unitary and antiunitary operators on
complex Hilbert space, in accordance with Wigner's Theorem. In the phase-space
formulation, they have real, true unitary representations in the space of
square-integrable functions on phase-space. Each such phase-space
representation is a Weyl-Wigner product of the corresponding Hilbert space
representation with its contragredient, and these can be recovered by
`factorising' the Weyl-Wigner product. However, not every real, unitary
representation on phase-space corresponds to a group of automorphisms, so not
every such representation is in the form of a Weyl-Wigner product and can be
factorised. The conditions under which this is possible are examined. Examples
are presented.Comment: Latex2e file, 37 page
Non-positivity of the Wigner function and bounds on associated integrals
The Wigner function shares several properties with classical distribution
functions on phase space, but is not positive-definite. The integral of the
Wigner function over a given region of phase space can therefore lie outside
the interval [0,1]. The problem of finding best-possible upper and lower bounds
for a given region is the problem of finding the greatest and least eigenvalues
of an associated Hermitian operator. Exactly solvable examples are described,
and possible extensions are indicated.Comment: 5 pages, Latex2e fil
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