skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics

Abstract

We consider wave packets of free particles with a general energy-momentum dispersion relation E(p). The spreading of the wave packet is determined by the velocity v={partial_derivative}{sub p}E. The position-velocity uncertainty relation {delta}x{delta}v{>=}1/2 |<{partial_derivative}{sub p}{sup 2}E>| is saturated by minimal uncertainty wave packets {phi}(p)=Aexp(-{alpha}E(p)+{beta}p). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p)=p{sup 2}/2m, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the lattice dispersion relation E(p)=-cos(pa)/ma{sup 2} as well as for the relativistic dispersion relation E(p)={radical}(p{sup 2}+m{sup 2}). The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.

Authors:
 [1]
  1. Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, Bern University, Sidlerstrasse 5, CH-3012 Bern (Switzerland)
Publication Date:
OSTI Identifier:
21336078
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 324; Journal Issue: 12; Other Information: DOI: 10.1016/j.aop.2009.09.001; PII: S0003-4916(09)00166-3; Copyright (c) 2009 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DISPERSION RELATIONS; MASS; QUANTUM MECHANICS; RELATIVISTIC RANGE; UNIVERSE; VELOCITY; WAVE PACKETS

Citation Formats

Al-Hashimi, M.H., and Wiese, U -J. Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics. United States: N. p., 2009. Web. doi:10.1016/j.aop.2009.09.001.
Al-Hashimi, M.H., & Wiese, U -J. Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics. United States. doi:10.1016/j.aop.2009.09.001.
Al-Hashimi, M.H., and Wiese, U -J. Tue . "Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics". United States. doi:10.1016/j.aop.2009.09.001.
@article{osti_21336078,
title = {Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics},
author = {Al-Hashimi, M.H. and Wiese, U -J},
abstractNote = {We consider wave packets of free particles with a general energy-momentum dispersion relation E(p). The spreading of the wave packet is determined by the velocity v={partial_derivative}{sub p}E. The position-velocity uncertainty relation {delta}x{delta}v{>=}1/2 |<{partial_derivative}{sub p}{sup 2}E>| is saturated by minimal uncertainty wave packets {phi}(p)=Aexp(-{alpha}E(p)+{beta}p). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p)=p{sup 2}/2m, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the lattice dispersion relation E(p)=-cos(pa)/ma{sup 2} as well as for the relativistic dispersion relation E(p)={radical}(p{sup 2}+m{sup 2}). The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.},
doi = {10.1016/j.aop.2009.09.001},
journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 12,
volume = 324,
place = {United States},
year = {2009},
month = {12}
}