dorsal/arxiv
View SchemaThree routes to the exact asymptotics for the one-dimensional quantum walk
| Authors | Hilary A. Carteret, Mourad E. H. Ismail, Bruce Richmond |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0303105 |
| URL | https://arxiv.org/abs/quant-ph/0303105 |
| DOI | 10.1088/0305-4470/36/33/305 |
| Journal | J. Phys. A., vol. 36, no. 33, pp 8775-8795 (2003) |
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. This 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 vs. Schr\"{o}dinger 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 discuss how and why our approach is related to the two methods that have already been used to analyse these systems.
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"abstract": "We demonstrate an alternative method for calculating the asymptotic behaviour\nof the discrete one-coin quantum walk on the infinite line, via the Jacobi\npolynomials that arise in the path integral representation. This is\nsignificantly easier to use than the Darboux method. It also provides a single\nintegral representation for the wavefunction that works over the full range of\npositions, $n,$ including throughout the transitional range where the behaviour\nchanges from oscillatory to exponential. Previous analyses of this system have\nrun into difficulties in the transitional range, because the approximations on\nwhich they were based break down here. The fact that there are two different\nkinds of approach to this problem (Path Integral vs. Schr\\\"{o}dinger wave\nmechanics) is ultimately a manifestation of the equivalence between the\npath-integral formulation of quantum mechanics and the original formulation\ndeveloped in the 1920s. We discuss how and why our approach is related to the\ntwo methods that have already been used to analyse these systems.",
"arxiv_id": "quant-ph/0303105",
"authors": [
"Hilary A. Carteret",
"Mourad E. H. Ismail",
"Bruce Richmond"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/36/33/305",
"journal_ref": "J. Phys. A., vol. 36, no. 33, pp 8775-8795 (2003)",
"title": "Three routes to the exact asymptotics for the one-dimensional quantum walk",
"url": "https://arxiv.org/abs/quant-ph/0303105"
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