Minimal Position-Velocity Uncertainty Wave Packets in Relativistic and Non-relativistic Quantum Mechanics

Wiese, U.-J.; Al-Hashimi, M.H.

doi:10.1016/j.aop.2009.09.001

Minimal Position-Velocity Uncertainty Wave Packets in Relativistic and Non-relativistic Quantum Mechanics

U.-J. Wiese
(
- Bern U.
)
,
M.H. Al-Hashimi
(
- Bern U.
)

Jul, 2009

32 pages

Published in:

Annals Phys. 324 (2009) 2599-2621

e-Print:

0907.5178 [quant-ph]

DOI:

10.1016/j.aop.2009.09.001

View in:

pdf

reference search27 citations

Citations per year

Abstract: (arXiv)

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 = \p_p E. The position-velocity uncertainty relation \Delta x \Delta v \geq {1/2} |< \p_p^2 E >| is saturated by minimal uncertainty wave packets

\Phi(p) = A \exp(- \alpha E(p) + \beta p)

. In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation

E(p) = p^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(p a)/m a^2

as well as for the relativistic dispersion relation

E(p) = \sqrt{p^2 + m^2}

. The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.

References(27)

Figures(14)

- Rept.Prog.Phys. 58 365

- Am.J.Phys. 76 1102

Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik

Paul Ehrenfest

- Z.Phys. 45 (1927) 7-8, 455-457
•
DOI:
- 10.1007/BF01329203

Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern

- Nuovo Cim. 22 (1961) 5, 1051-1068
•
DOI:
- 10.1007/BF02787889

- Phys.Rev. 139 963

Remark on Causality and Particle Localization

Gerhard C. Hegerfeldt
(
- Heidelberg U.
)

- Phys.Rev.D 10 (1974) 3320
•
DOI:
- 10.1103/PhysRevD.10.3320

Remarks on Causality, Localization and Spreading of Wave Packets

- Phys.Rev.D 22 (1980) 377
•
DOI:
- 10.1103/PhysRevD.22.377

VIOLATION OF CAUSALITY IN RELATIVISTIC QUANTUM THEORY?

Gerhard C. Hegerfeldt
(
- Gottingen U. and
- Caltech
)

- Phys.Rev.Lett. 54 (1985) 2395
•
DOI:
- 10.1103/PhysRevLett.54.2395

- Phys.Rev.D 36 2381

Causality and the scalar field energy

S.N. Mosley
(
- Nottingham U.
)
,
J.E.G. Farina
(
- Nottingham U.
)

- J.Phys.A 23 (1990) 3991-3996

- Phys.Lett.A 308 110

Causality and localization operators

F. Buscemi
(
- INFM, Palermo and
- Palermo U.
)
,
G. Compagno
(
- INFM, Palermo and
- Palermo U.
)

- Phys.Lett.A 334 (2005) 357-362
•
e-Print:
- quant-ph/0409139
•
DOI:
- 10.1016/j.physleta.2004.11.045

- Adv.Atom.Molec.Phys. 39 695

- Nuovo Cim. 37 543

Relativistic invariance and Hamiltonian theories of interacting particles

- Rev.Mod.Phys. 35 (1963) 350-375
•
DOI:
- 10.1103/RevModPhys.35.350

A POSITIVE ENERGY DYNAMICS AND SCATTERING THEORY FOR DIRECTLY INTERACTING RELATIVISTIC PARTICLES

S.N.M. Ruijsenaars
(
- Princeton U.
)

- Annals Phys. 126 (1980) 399-449,
- Ann.Phys.(N.y.) 126 (1980) 399-449 and Preprint - RUIJSENAARS, S. N. M. (REC.FEB 80) 77p
•
DOI:
- 10.1016/0003-4916(80)90182-7

Complete Integrability of Relativistic Calogero-moser Systems and Elliptic Function Identities

S.N.M. Ruijsenaars
(
- Munich, Max Planck Inst.
)

- Commun.Math.Phys. 110 (1987) 191
•
DOI:
- 10.1007/BF01207363

When do neutrinos really oscillate?: Quantum mechanics of neutrino oscillations

C. Giunti
(
- Johns Hopkins U.
)
,
C.W. Kim
(
- Johns Hopkins U. and
- Seoul Natl. U.
)
,
U.W. Lee
(
- Johns Hopkins U.
)

- Phys.Rev.D 44 (1991) 3635-3640
•
DOI:
- 10.1103/PhysRevD.44.3635

- NOONE S 1 103

- Am.J.Phys. 40 327

- NOONE A 6 5

SPREADING OF A RELATIVISTIC WAVE PACKET

Carlos Almeida
(
- Paraiba U.
)
,
Arthur Jabs
(
- Paraiba U.
)

- Am.J.Phys. 52 (1984) 921-925,
- Joao Pessoa Paraiba Fed. Univ. - UFPB-DF 83-08 (83,REC.MAY 84) 18p
•
DOI:
- 10.1119/1.13795

- Phys.Rev.A 73 069908

Localized States for Elementary Systems

T.D. Newton
(
- Princeton U.
)
,
Eugene P. Wigner
(
- Princeton U.
)

- Rev.Mod.Phys. 21 (1949) 400-406,
- Reprinted in *Noz, M.E. (ed.), Kim, Y.S. (ed.): Special relativity and quantum theory* 395-401
•
DOI:
- 10.1103/RevModPhys.21.400

On Newton-wigner Localization and Superluminal Propagation Speeds

S.N.M. Ruijsenaars
(
- Munich, Max Planck Inst.
)

- Annals Phys. 137 (1981) 33
•
DOI:
- 10.1016/0003-4916(81)90059-2

1-25 of 27
1
2
25 / page