3,752 research outputs found
Wigner distributions for finite state systems without redundant phase point operators
We set up Wigner distributions for state quantum systems following a
Dirac inspired approach. In contrast to much of the work on this case,
requiring a phase space, particularly when is even, our
approach is uniformly based on an phase space grid and thereby
avoids the necessity of having to invoke a `quadrupled' phase space and hence
the attendant redundance. Both odd and even cases are analysed in detail
and it is found that there are striking differences between the two. While the
odd case permits full implementation of the marginals property, the even
case does so only in a restricted sense. This has the consequence that in the
even case one is led to several equally good definitions of the Wigner
distributions as opposed to the odd case where the choice turns out to be
unique.Comment: Latex, 14 page
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
The Sampling Theorem and Coherent State Systems in Quantum Mechanics
The well known Poisson Summation Formula is analysed from the perspective of
the coherent state systems associated with the Heisenberg--Weyl group. In
particular, it is shown that the Poisson summation formula may be viewed
abstractly as a relation between two sets of bases (Zak bases) arising as
simultaneous eigenvectors of two commuting unitary operators in which geometric
phase plays a key role. The Zak bases are shown to be interpretable as
generalised coherent state systems of the Heisenberg--Weyl group and this, in
turn, prompts analysis of the sampling theorem (an important and useful
consequence of the Poisson Summation Formula) and its extension from a coherent
state point of view leading to interesting results on properties of von Neumann
and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late
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
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
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
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