Home On integrals associated with the free particle wave packet
Article
Licensed
Unlicensed Requires Authentication

On integrals associated with the free particle wave packet

  • Alexander E. Patkowski EMAIL logo
Published/Copyright: June 19, 2020

Abstract

We discuss some properties of integrals associated with the free particle wave packet, ψ(x,t), which are solutions to the time-dependent Schrödinger equation for a free particle. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.

MSC 2010: 42A38; 35Q40; 11F20

Acknowledgements

We thank M. L. Glasser for providing a copy of his work [4], and also noting the hidden gem (4.1), and its proof.

References

[1] G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University, New York, 1999. 10.1017/CBO9781107325937Search in Google Scholar

[2] G. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part IV, Springer, New York, 2013. 10.1007/978-1-4614-4081-9Search in Google Scholar

[3] J. W. L. Glaisher, On the summation by definite integrals of geometric series of the second and higher order, Quart. J. Pure Appl. Math. (1871), 238–343. Search in Google Scholar

[4] M. L. Glasser, Time evolution of a non-Gaussian wave packet, Phys. Lett. A. 76 (1980), no. 3–4, 219–220. 10.1016/0375-9601(80)90471-5Search in Google Scholar

[5] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007. Search in Google Scholar

[6] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. Search in Google Scholar

[7] S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 (1915), 75–85. Search in Google Scholar

[8] R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples, 2nd ed., Oxford University, Oxford, 2006. 10.1093/oso/9780198530978.001.0001Search in Google Scholar

Received: 2018-09-05
Accepted: 2020-04-01
Published Online: 2020-06-19
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2020-2014/html
Scroll to top button