1,130 research outputs found

### Symplectic Structures and Quantum Mechanics

Canonical coordinates for the Schr\"odinger equation are introduced, making
more transparent its Hamiltonian structure. It is shown that the Schr\"odinger
equation, considered as a classical field theory, shares with Liouville
completely integrable field theories the existence of a {\sl recursion
operator} which allows for the infinitely many conserved functionals pairwise
commuting with respect to the corresponding Poisson bracket. The approach may
provide a good starting point to get a clear interpretation of Quantum
Mechanics in the general setting, provided by Stone-von Neumann theorem, of
Symplectic Mechanics. It may give new tools to solve in the general case the
inverse problem of quantum mechanics whose solution is given up to now only for
one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.Comment: 11 pages, LaTex fil

### Quantum Systems and Alternative Unitary Descriptions

Motivated by the existence of bi-Hamiltonian classical systems and the
correspondence principle, in this paper we analyze the problem of finding
Hermitian scalar products which turn a given flow on a Hilbert space into a
unitary one. We show how different invariant Hermitian scalar products give
rise to different descriptions of a quantum system in the Ehrenfest and
Heisenberg picture.Comment: 18 page

### Alternative linear structures for classical and quantum systems

The possibility of deforming the (associative or Lie) product to obtain
alternative descriptions for a given classical or quantum system has been
considered in many papers. Here we discuss the possibility of obtaining some
novel alternative descriptions by changing the linear structure instead. In
particular we show how it is possible to construct alternative linear
structures on the tangent bundle TQ of some classical configuration space Q
that can be considered as "adapted" to the given dynamical system. This fact
opens the possibility to use the Weyl scheme to quantize the system in
different non equivalent ways, "evading", so to speak, the von Neumann
uniqueness theorem.Comment: 32 pages, two figures, to be published in IJMP

### 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

### The space of density states in geometrical quantum mechanics

We present a geometrical description of the space of density states of a
quantum system of finite dimension. After presenting a brief summary of the
geometrical formulation of Quantum Mechanics, we proceed to describe the space
of density states \D(\Hil) from a geometrical perspective identifying the
stratification associated to the natural GL(\Hil)--action on \D(\Hil) and
some of its properties. We apply this construction to the cases of quantum
systems of two and three levels.Comment: Amslatex, 18 pages, 4 figure

### Alternative structures and bi-Hamiltonian systems on a Hilbert space

We discuss transformations generated by dynamical quantum systems which are
bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an
infinite-dimensional complex Hilbert space. We introduce the notion of
Hermitian structures in generic relative position. We provide few necessary and
sufficient conditions for two Hermitian structures to be in generic relative
position to better illustrate the relevance of this notion. The group of
bi-unitary transformations is considered in both the generic and non-generic
case. Finally, we generalize the analysis to real Hilbert spaces and extend to
infinite dimensions results already available in the framework of
finite-dimensional linear bi-Hamiltonian systems.Comment: 11 page

### Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

In this paper we present a survey of the use of differential geometric
formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework
from this perspective and provide a description of the Weyl-Wigner
construction. Finally, after reviewing the basics of the geometric formulation
of quantum mechanics, we apply the methods presented to the most interesting
cases of finite dimensional Hilbert spaces: those of two, three and four level
systems (one qubit, one qutrit and two qubit systems). As a more practical
application, we discuss the advantages that the geometric formulation of
quantum mechanics can provide us with in the study of situations as the
functional independence of entanglement witnesses.Comment: AmsLaTeX, 37 pages, 8 figures. This paper is an expanded version of
some lectures delivered by one of us (G. M.) at the ``Advanced Winter School
on the Mathematical Foundation of Quantum Control and Quantum Information''
which took place at Castro Urdiales (Spain), February 11-15, 200

### Tensorial description of quantum mechanics

Relevant algebraic structures for the description of Quantum Mechanics in the
Heisenberg picture are replaced by tensorfields on the space of states. This
replacement introduces a differential geometric point of view which allows for
a covariant formulation of quantum mechanics under the full diffeomorphism
group.Comment: 8 page

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