348 research outputs found
The Hamilton--Jacobi Theory and the Analogy between Classical and Quantum Mechanics
We review here some conventional as well as less conventional aspects of the
time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its
connections with Quantum Mechanics. Less conventional aspects involve the HJ
theory on the tangent bundle of a configuration manifold, the quantum HJ
theory, HJ problems for general differential operators and the HJ problem for
Lie groups.Comment: 42 pages, LaTeX with AIMS clas
Geometric phase for mixed states: a differential geometric approach
A new definition and interpretation of geometric phase for mixed state cyclic
unitary evolution in quantum mechanics are presented. The pure state case is
formulated in a framework involving three selected Principal Fibre Bundles, and
the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint
orbits associated with Lie groups. It is shown that this framework generalises
in a natural and simple manner to the mixed state case. For simplicity, only
the case of rank two mixed state density matrices is considered in detail. The
extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure
to mixed states are also presented.Comment: 22 pages, revtex
Null Phase Curves and Manifolds in Geometric Phase Theory
Bargmann invariants and null phase curves are known to be important
ingredients in understanding the essential nature of the geometric phase in
quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are
submanifolds made up entirely of null phase curves, and so are equally
important for geometric phase considerations. It is shown that the complete
characterization of null phase manifolds involves both the Riemannian metric
structure and the symplectic structure of ray space in equal measure, which
thus brings together these two aspects in a natural manner.Comment: 10 pages, 1 figur
Wigner distributions for finite dimensional quantum systems: An algebraic approach
We discuss questions pertaining to the definition of `momentum', `momentum
space', `phase space', and `Wigner distributions'; for finite dimensional
quantum systems. For such systems, where traditional concepts of `momenta'
established for continuum situations offer little help, we propose a physically
reasonable and mathematically tangible definition and use it for the purpose of
setting up Wigner distributions in a purely algebraic manner. It is found that
the point of view adopted here is limited to odd dimensional systems only. The
mathematical reasons which force this situation are examined in detail.Comment: Latex, 13 page
Recursive parametrization of Quark flavour mixing matrices
We examine quark flavour mixing matrices for three and four generations using
the recursive parametrization of and matrices developed by some
of us in Refs.[2] and [3]. After a brief summary of the recursive
parametrization, we obtain expressions for the independent rephasing invariants
and also the constraints on them that arise from the requirement of mod
symmetry of the flavour mixing matrix
Wigner-Weyl isomorphism for quantum mechanics on Lie groups
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie
group is developed in detail. Several New features are shown to arise which
have no counterparts in the familiar Cartesian case. Notable among these is the
notion of a `semiquantised phase space', a structure on which the Weyl symbols
of operators turn out to be naturally defined and, figuratively speaking,
located midway between the classical phase space and the Hilbert space
of square integrable functions on . General expressions for the star product
for Weyl symbols are presented and explicitly worked out for the angle-angular
momentum case.Comment: 32 pages, Latex2
Phase-space descriptions of operators and the Wigner distribution in quantum mechanics II. The finite dimensional case
A complete solution to the problem of setting up Wigner distribution for
N-level quantum systems is presented. The scheme makes use of some of the ideas
introduced by Dirac in the course of defining functions of noncommuting
observables and works uniformly for all N. Further, the construction developed
here has the virtue of being essentially input-free in that it merely requires
finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task
which, as is shown, can always be accomplished analytically. As an
illustration, the case of a single qubit is considered in some detail and it is
shown that one recovers the result of Feynman and Wootters for this case
without recourse to any auxiliary constructs.Comment: 14 pages, typos corrected, para and references added in introduction,
submitted to Jour. Phys.
The Schwinger Representation of a Group: Concept and Applications
The concept of the Schwinger Representation of a finite or compact simple Lie
group is set up as a multiplicity-free direct sum of all the unitary
irreducible representations of the group. This is abstracted from the
properties of the Schwinger oscillator construction for SU(2), and its
relevance in several quantum mechanical contexts is highlighted. The Schwinger
representations for and SU(n) for all are constructed via
specific carrier spaces and group actions. In the SU(2) case connections to the
oscillator construction and to Majorana's theorem on pure states for any spin
are worked out. The role of the Schwinger Representation in setting up the
Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group is
brought out.Comment: Latex, 17 page
Conservation, Dissipation, and Ballistics: Mesoscopic Physics beyond the Landauer-Buettiker Theory
The standard physical model of contemporary mesoscopic noise and transport
consists in a phenomenologically based approach, proposed originally by
Landauer and since continued and amplified by Buettiker (and others).
Throughout all the years of its gestation and growth, it is surprising that the
Landauer-Buettiker approach to mesoscopics has matured with scant attention to
the conservation properties lying at its roots: that is, at the level of actual
microscopic principles. We systematically apply the conserving sum rules for
the electron gas to clarify this fundamental issue within the standard
phenomenology of mesoscopic conduction. Noise, as observed in quantum point
contacts, provides the vital clue.Comment: 10 pp 3 figs, RevTe
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