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arXiv:hep-th/0407059

Quantum mechanics in a cut Fock space

Maciej Trzetrzelewski

Published Jul 8, 2004• 6 citations
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Abstract

A recently introduced numerical approach to quantum systems is analyzed. The basis of a Fock space is restricted and represented in an algebraic program. Convergence with increasing size of basis is proved and the difference between discrete and continuous spectrum is stressed. In particular a new scaling low for nonlocalized states is obtained. Exact solutions for several cases as well as general properties of the method are given.

DOI
10.48550/arxiv.hep-th/0407059

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A recently introduced numerical approach to quantum systems is analyzed. The basis of a Fock space is restricted and represented in an algebraic program. Convergence with increasing size of basis is proved and the difference between discrete and continuous spectrum is stressed. In particular a new scaling low for nonlocalized states is obtained. Exact solutions for several cases as well as general properties of the method are given.
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The general problem of finding exactly soluble quantum systems is considered. It is argued that discrete space quantum mechanics emerges in a natural way as an avenue of approach. Discrete space quantum mechanics is formulated and applied to one-dimensional quantum systems with emphasis on single-channel models. It is found that a large variety of systems are exactly soluble in the sense that they only require the inversion of a finite-dimensional matrix. The interaction may in general be both nonlocal and non-time-reversal-invariant. The analytic structure of the resolvent is worked out in detail for a simple class of examples. It is shown that a slightly modified version of the usual continuous space Schrödinger equation may in principle be solved exactly for any finite range local potential by writing the solution in terms of corresponding discrete space solutions. It is also shown that from an algebraic viewpoint the models constructed are realizations of generalized versions of the Weyl relations.
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A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of Malliavin derivative on a projective Fock space, and their uniform continuity with respect to the inductive limite convergence is proved. A new form of QS calculus based on an inductive *-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and isometry and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the peseudounitary and *-homomorphism property of the QS integrable generators is proved.