3 research outputs found
Noether Symmetries, Dynamical Constants of Motion, and Spectrum Generating Algebras
When discussing consequences of symmetries of dynamical systems based on
Noether's first theorem, most standard textbooks on classical or quantum
mechanics present a conclusion stating that a global continuous Lie symmetry
implies the existence of a time independent conserved Noether charge which is
the generator of the action on phase space of that symmetry, and which
necessarily must as well commute with the Hamiltonian. However this need not be
so, nor does that statement do justice to the complete scope and reach of
Noether's first theorem. Rather a much less restrictive statement applies,
namely that the corresponding Noether charge as an observable over phase space
may in fact possess an explicit time dependency, and yet define a constant of
the motion by having a commutator with the Hamiltonian which is nonvanishing,
thus indeed defining a dynamical conserved quantity. Furthermore, and this
certainly within the Hamiltonian formulation, the converse statement is valid
as well, namely that any dynamical constant of motion is necessarily the
Noether charge of some symmetry leaving the system's action invariant up to
some total time derivative contribution. The present contribution revisits
these different points and their consequences, straightaway within the
Hamiltonian formulation which is the most appropriate for such issues. Explicit
illustrations are also provided through three general but simple enough classes
of systems.Comment: 1+27 pages, to appear in the Int. J. Mod. Phys.
Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials
Within the context of Supersymmetric Quantum Mechanics and its related
hierarchies of integrable quantum Hamiltonians and potentials, a general
programme is outlined and applied to its first two simplest illustrations.
Going beyond the usual restriction of shape invariance for intertwined
potentials, it is suggested to require a similar relation for Hamiltonians in
the hierarchy separated by an arbitrary number of levels, N. By requiring
further that these two Hamiltonians be in fact identical up to an overall shift
in energy, a periodic structure is installed in the hierarchy of quantum
systems which should allow for its solution. Specific classes of orthogonal
polynomials characteristic of such periodic hierarchies are thereby generated,
while the methods of Supersymmetric Quantum Mechanics then lead to generalised
Rodrigues formulae and recursion relations for such polynomials. The approach
also offers the practical prospect of quantum modelling through the engineering
of quantum potentials from experimental energy spectra. In this paper these
ideas are presented and solved explicitly for the cases N=1 and N=2. The latter
case is related to the generalised Laguerre polynomials, for which indeed new
results are thereby obtained. At the same time new classes of integrable
quantum potentials which generalise that of the harmonic oscillator and which
are characterised by two arbitrary energy gaps are identified, for which a
complete solution is achieved algebraically.Comment: 1+19 page