Misc,
An approach to nonstandard quantum mechanics
(December 2006)
Abstract
We use nonstandard analysis to formulate quantum mechanics in
hyperfinite-dimensional spaces. Self-adjoint operators on
hyperfinite-dimensional spaces have complete eigensets, and bound states and
continuum states of a Hamiltonian can thus be treated on an equal footing. We
show that the formalism extends the standard formulation of quantum mechanics.
To this end we develop the Loeb-function calculus in nonstandard hulls. The
idea is to perform calculations in a hyperfinite-dimensional space, but to
interpret expectation values in the corresponding nonstandard hull. We further
apply the framework to non-relativistic quantum scattering theory. For
time-dependent scattering theory, we identify the starting time and the
finishing time of a scattering experiment, and we obtain a natural separation
of time scales on which the preparation process, the interaction process, and
the detection process take place. For time-independent scattering theory, we
derive rigorously explicit formulas for the Møller wave operators and the
S-Matrix.
