Three routes to the exact asymptotics for the one-dimensional quantum walk

    Authors

    H. A. Carteret; M. E. H. Ismail;B. Richmond

    Comments

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    Abbreviated Journal Title

    J. Phys. A-Math. Gen.

    Keywords

    Physics, Multidisciplinary; Physics, Mathematical

    Abstract

    We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path-integral representation. We calculate the asymptotics using a method that is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, n, including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (path integral versus Schrodinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We also discuss how and why our approach is related to the two methods that have already been used to analyse these systems.

    Journal Title

    Journal of Physics a-Mathematical and General

    Volume

    213

    Issue/Number

    33

    Publication Date

    1-1-2003

    Document Type

    Article

    Language

    English

    First Page

    8775

    Last Page

    8795

    WOS Identifier

    WOS:000186461900001

    ISSN

    0305-4470

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