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

被引:37
作者
Carteret, HA [1 ]
Ismail, MEH
Richmond, B
机构
[1] Univ Waterloo, Dept Combinator & Optimizat, Waterloo, ON N2L 3G1, Canada
[2] Univ Cent Florida, Dept Math, Orlando, FL 32816 USA
来源
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL | 2003年 / 36卷 / 33期
关键词
D O I
10.1088/0305-4470/36/33/305
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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.
引用
收藏
页码:8775 / 8795
页数:21
相关论文
共 16 条
[1]  
Abramowitz M., 1974, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, DOI DOI 10.5555/1953048.2021068
[2]   QUANTUM RANDOM-WALKS [J].
AHARONOV, Y ;
DAVIDOVICH, L ;
ZAGURY, N .
PHYSICAL REVIEW A, 1993, 48 (02) :1687-1690
[3]  
Ambainis A., 2001, P 33 ACM S THEORY CO, P37, DOI 10.1145/380752.380757
[4]  
ANDREWS GE, 1999, SPECIAL FUNCTIONS
[5]  
Arfken G. B., 2000, MATH METHODS PHYSICI
[6]  
BRUN TA, 2002, IN PRESS PHYS REV A
[7]   ON ASYMPTOTICS OF JACOBI-POLYNOMIALS [J].
CHEN, LC ;
ISMAIL, MEH .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 1991, 22 (05) :1442-1449
[8]  
FEYNMAN R, 1965, QUANTUM MECH PATH IN
[9]  
GAWRONSKI W, 1991, STRONG ASYMPTOTICS L, P379
[10]   Justification of the stationary phase approximation in time-domain asymptotics [J].
McClure, JP ;
Wong, R .
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1997, 453 (1960) :1019-1031