Prog. Theor. Phys. Vol. 125 No. 2 (2011) pp. 205-224
Uncertainty Relation and Probability
— Numerical Illustration
1Institute of Quantum Science, College of Science and Technology,
Nihon University, Tokyo 101-8308, Japan
2Maskawa Institute for Science and Culture,
Kyoto Sangyo University, Kyoto 603-8555, Japan
(Received October 15, 2010; Revised December 5, 2010)
The uncertainty relation and the probability interpretation of quantum
mechanics are intrinsically connected, as is evidenced by the evaluation of
standard deviations. It is thus natural to ask if one can associate a very
small uncertainty product of suitably sampled events with a very small
probability. We have shown elsewhere that
some examples of the evasion of the uncertainty relation noted in the past are
in fact understood in this way. We here numerically illustrate that a very
small uncertainty product is realized if one performs a suitable sampling of
measured data that occur with a very small probability. We introduce a notion
of cyclic measurements. It is also shown that our analysis is consistent with
the Landau-Pollak-type uncertainty relation. It is suggested that the present
analysis may help reconcile the contradicting views about the “standard
quantum limit” in the detection of gravitational waves.
Subject Index :
002, 060, 064
DOI : 10.1143/PTP.125.205
- W. Heisenberg, Z. Phys. 43 (1927), 172.
- J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955).
- D. M. Appleby, Int. J. Theor. Phys. 37 (1998), 1491.
M. Ozawa, Phys. Lett. A 320 (2004), 367[CrossRef].
P. Busch, T. Heinonen and P. Lahti, Phys. Rep. 452 (2007), 155[CrossRef].
T. Miyadera and H. Imai, Phys. Rev. A 78 (2008), 052119[APS].
- E. H. Kennard, Z. Phys. 44 (1927), 326.
H. P. Robertson, Phys. Rev. 34 (1929), 163[APS].
L. E. Ballentine, Rev. Mod. Phys. 42 (1970), 358[APS].
M. Ozawa, Phys. Lett. A 318 (2003), 21[CrossRef].
- K. Fujikawa and K. Umetsu, Prog. Theor. Phys. 120 (2008), 797[PTP].
J. Maddox, Nature 331 (1988), 559[CrossRef].
T. Miyadera and H. Imai, Phys. Rev. A 76 (2007), 062108, [APS]and references therein.
- K. Fujikawa, Prog. Theor. Phys. 124 (2010), 747[PTP].
- V. B. Braginsky and Y. I. Vorontsov, Sov. Phys. -Usp. 17 (1975), 644.
C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg and M. Zimmermann, Rev. Mod. Phys. 52 (1980), 341[APS].
H. P. Yuen, Phys. Rev. Lett. 51 (1983), 719[APS].
C. M. Caves, Phys. Rev. Lett. 54 (1985), 2465[APS].
M. Ozawa, Phys. Rev. Lett. 60 (1988), 385[APS].