72 research outputs found
Monopole Gauge Fields and Quantum Potentials Induced by the Geometry in Simple Dynamical Systems
A realistic analysis shows that constraining a quantomechanical system
produces the effective dynamics to be coupled with {\sl abelian/non-abelian
gauge fields} and {\sl quantum potentials} induced by the {\sl intrinsic} and
{\sl extrinsic geometrical properties} of the constraint's surface.
This phenomenon is observable in the effective rotational motion of some
simple polyatomic molecules. By considering specific examples it is shown that
the effective Hamiltonians for the nuclear rotation of linear and symmetric top
molecules are equivalent to that of a charged system moving in a background
magnetic-monopole field. For spherical top molecules an explicit analytical
expression of a non-abelian monopole-like field is found. Quantum potentials
are also relevant for the description of rotovibrational interactions.Comment: 24, LaTex, UPRF-94-40
Quantum Mechanics on Manifolds Embedded in Euclidean Space
Quantum particles confined to surfaces in higher dimensional spaces are acted
upon by forces that exist only as a result of the surface geometry and the
quantum mechanical nature of the system. The dynamics are particularly rich
when confinement is implemented by forces that act normal to the surface. We
review this confining potential formalism applied to the confinement of a
particle to an arbitrary manifold embedded in a higher dimensional Euclidean
space. We devote special attention to the geometrically induced gauge potential
that appears in the effective Hamiltonian for motion on the surface. We
emphasize that the gauge potential is only present when the space of states
describing the degrees of freedom normal to the surface is degenerate. We also
distinguish between the effects of the intrinsic and extrinsic geometry on the
effective Hamiltonian and provide simple expressions for the induced scalar
potential. We discuss examples including the case of a 3-dimensional manifold
embedded in a 5-dimensional Euclidean space.Comment: 12 pages, LaTe
Dynamics as Shadow of Phase Space Geometry
Starting with the generally well accepted opinion that quantizing an
arbitrary Hamiltonian system involves picking out some additional structure on
the classical phase space (the {\sl shadow} of quantum mechanics in the
classical theory), we describe classical as well as quantum dynamics as a
purely geometrical effect by introducing a {\sl phase space metric structure}.
This produces an modification of the classical equations of
motion reducing at the same time the quantization of an arbitrary Hamiltonian
system to standard procedures. Our analysis is carried out in analogy with the
adiabatic motion of a charged particle in a curved background (the additional
metric structure) under the influence of a universal magnetic field (the
classical symplectic structure). This allows one to picture dynamics in an
unusual way, and reveals a dynamical mechanism that produces the selection of
the right set of physical quantum states.Comment: LaTeX (epsfig macros), 30 pages, 1 figur
A Dynamical Mechanism for the Selection of Physical States in `Geometric Quantization Schemes'
Geometric quantization procedures go usually through an extension of the
original theory (pre-quantization) and a subsequent reduction (selection of the
physical states). In this context we describe a full geometrical mechanism
which provides dynamically the desired reduction.Comment: 6 page
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