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Limited entropic uncertainty as new principle of quantum physics
17 February 2000
Physics Letters B 474 Ž2000. 395–401
Limited entropic uncertainty as new principle of quantum physics M.L.D. Ion, D.B. Ion National Institute for Physics and Nuclear Engineering, IFIN-HH, P.O. Box MG-6, Bucharest, Romania Received 3 November 1999; accepted 7 January 2000 Editor: R. Gatto
Abstract In this paper the Principle of Limited Entropic Uncertainty (LEU-Principle) for the canonic conjugate variables in quantum physics is proved in a more general form by using Tsallis-like entropies for positive nonextensivities Ž q ) 0.. Results on experimental tests of this new principle for the angle-angular momentum Õariables in the quantum scattering are illustrated by using 49 sets of pion-nucleus phase shifts. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj; 12.40.Ee
The Uncertainty Principle ŽUP. of quantum mechanics discovered by Heisenberg w1x which constitute the corner-stone of quantum physics, asserts that : ŽUP. ‘‘there is an irreducible lower bound on the uncertainty in the result of a simultaneous measurement of non-commuting obserÕables’’. An well known example, is the Heisenberg inequality w1x for the usual position-momentum Ž " s c s 1.: 12 F D x D p, where D y denotes the standard deviation of the observable y. The extended version of this inequality for an arbitrary pair of quantum mechanical observable A and B is given by: 12 <²w A, B x:< F D A D B where the corresponding standard deviations D AŽ c . and D B Ž c . for a quantum system in a given normalized state < c : depends on the state < c :. In order to avoid this state-dependence many authors w2–7x proposed to use the information entropy as a measure of the uncertainty instead of above standard
quantitative formulation of the Heisenberg uncertainty principle. Indeed, the entropy of a quantum state describing physical system is a quantity expressing the uncertainty or randomness of the system. So, an alternative mathematical formulation of the uncertainty principle for position-momentum pair is given by the entropic inequality Žsee BialynickiBirula et al. w2x.
p e F Vx Vp ,
Ž 1.
where Vx ' exp S x 4 and Vp ' exp S p 4 are the statistical Õariances which are defined as the exponential of the differential entropies S x and S p corresponding to the Ž x and p .-observables. Recently, a real progress in the investigation of the quantum entropy it was achieved not only by proving new entropic uncertainty relations w2–7x for the standard additiÕe systems but also by generalization w7x of such results
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 0 3 6 - 8
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
396
to the nonextensiÕe statistics w8,9x. So, the state independent angle-angular momentum entropic lower bounds are proved w5,7x by using Tsallis-like entropies and Riesz theorem for the quantum scattering of the spinless particles. Moreover, the optimal entropic upper bounds for each Tsallis-like scattering entropy Su Ž q ., SLŽ q ., Su LŽ q ., were recently proved w6x in terms of optimal entropies derived from the principle of minimum distance in the space of states w10,11x. In this way, the entropic uncertainty band as a new concept in quantum physics is introduced. The generalized entropy as a measure of quantum uncertainty was also discussed in Ref. w12x in connection with the number-phase entropic uncertainty measure for the coherent states within the Pegg–Barnet quantum theory w13x. In this paper the Principle of Limited Entropic Uncertainty (LEU-Principle), as a new principle in quantum physics, is proved. Then, consistent experimental tests of the LEU-principle, obtained by using the available 49 sets of the pion-nucleus phase shifts w14x, are presented for both, extensiÕe Ž q s 1. and nonextensiÕe Ž q s 0.5 and q s 2.0., statistics cases.
Moreover, some results obtained by the application of LEU-Principle to the diffraction phenomena are discussed. LEU-Principle for quantum scattering. Here we present a short mathematical proof of the for the Principle of Limited Entropic Uncertainty in quantum scattering. So, we start with the scattering of two spinless particles for which all notation and definition from Ref. w5–7x will be adopted here. The angular distribution of probability P Ž x . as well as angular-momentum distribution p l ,l s 0,1,2, . . . 4 are presented in Table 1, where ddVs Ž x . s < f Ž x .< 2 is the differential cross section, x ' cos u , u-being the c.m. scattering angle, sel is the integrated elastic cross section, f l are the partial amplitudes while f Ž x . s ÝŽ2 l q 1. f l Pl Ž x . is the two-body scattering amplitude and Pl Ž x . are Legendre polynomials. Hence, using the results of Refs. w6,7x for the Tsallis-like scattering entropies, in this paper we prove the following standard quantitative formulation of the LEU-Principle. LEU-inequalities for scattering: Let sel and ddVs Ž y . be fixed from experiment for a fixed y g wy1,q 1x.
Table 1 The canonical conjugate variables Ž1., optimal distributions Ž2., Ž3., reproducing kernels Ž4., optimal variances Ž5. – Ž7., and Uncertainty Bands Ž8., corresponding to optimal scattering are compared with those of the diffraction on a single slit. No.
Optimal scattering
Ž1. Ž2.
l ' angular momentum and x s cos u plo1 s N1s ,l s 0, . . . , L0 , plo1 s 0, l G L 0 , Ns s ÝŽ2 l q 1. s Ž L o q 1. 2 , Ž L o q 1. 2 s s4 pel ddVs Ž1. Puo1 Ž x . s
Ž3.
Diffraction on single slit of width 2a PLo Ž lj
w K Ž x,1 . x 2
X ' kap, P o Ž p . s
K Ž1,1 .
s 12 P˙L oq1 Ž x . q P˙L oŽ x . , K Ž1,1. s 12 Ž L o q 1. 2 VLo1 s exp SLo1 4 s Ns
Ž5. Ž6.
Vuo1 s exp yHq1 y1 dx
4
w K Ž x,1 . x
2
ln
K Ž 1,1 .
½
s Ns exp yHq1 y1 dx Ž8.
ž
w K Ž x,1 . x K Ž 1,1 .
VLo s Na 2
/
w K Ž x,1 . x 2 K Ž 1,1 .
ln
ž
2 F Vu VL F NsVuo1
w K Ž x,1 . x 2 K Ž 1,1 .
2
q` Vpo s exp y p1 Hy` dX w sinX X x ln
Ž
ka p
w sinX X x 2 . 4
s w kap x expw2Ž1 y g .x Vpo VLo s exp SLo q S xo 4
Vuo1 VLo1 s exp SLo1 q Suo1 4
Ž7.
K aŽ0,0 .
2
w K a Ž X ,0 . x 2
s kap w sinX X x . sinX X , K Ž0,0. s ka X s kap, K Ž X,0. s Ž ka p p
K Ž x,1. s 21 ÝŽ2 l q 1. Pl Ž x .
Ž4.
lj ' k j , x ' p . s 1rNa , Na s 2 ka
/5
s 2p expw2Ž1 y C .x
p e F Vx Vp F 2p expw2Ž1 y C .x s NaVpo
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
Let Vu LŽ q . s exp Su LŽ q . be the statistical Õariances corresponding to the Tsallis-like scattering entropies Su LŽ q .. Then, the LEU-inequalities can be given in the following general form: exp Ž 1 y 2 1y q . r Ž q y 1 . F Vu LŽ q . F VuoLy Ž q . , for
q G 0,
Ž 2.
exp Ž 1 y 2 1y q . r Ž q y 1 . F VuoLy Ž q . F Vu LŽ q . , for
q - 0,
Ž 3.
where VuoLy Ž q . s exp SuoLy Ž q .
°1 ¢q y 1
~
s exp
Proof : Therefore, using the Tsallis-like Su Ž q .,SLŽ q .,Su LŽ q .4 scattering entropies, given in Ref. w6x by Eq. Ž6. – Ž11., for the canonical conjugate variables wcos u ,l x we define the statistical entropic Õariances VX Ž q . s exp S X Ž q . 4 , X ' u , L, u L, q g R ,
2 K Ž y, y .
maxrmin VX Ž q . 4 when sel and q
q1
Hy1
dx
K Ž x, y
2
q
K Ž y, y .
¶• ß
Ž 4.
K Ž x , y . s 12
s
L o y q1 PL o yq1Ž x . PL o yŽ y . y PL o yŽ x . PL o yq1Ž y . 2
xy y
,
Ž 5.
£ ' l0 VX q l1 q l2
ds dV
sel 4p
Ž 1 . y N Ý Ž 2 l q 1 . f l Pl Ž y . N 2 Ž 10 .
where l i s 0,1,2, are the corresponding Lagrange multipliers. Then, the singular solution l 0 s 0 of the Lagrange function Ž10. is obtained by solving the minimization problem
½Ý
Ý Ž 2 l q 1. Pl Ž y . Pl Ž y . ls0
s
Lo y q 1 2
P˙L o yq1 Ž y . PL o yŽ y .
yP˙L o yŽ y . PL o yq1 Ž y . ,
and the optimal angular momentum L o y can be obtained by solving the implicit algebraic equation ds dV
Ž y . s K Ž y, y .
sel 2p
Ž 2 l q 1. N f l N 2 q a
ds dV
yN Ý Ž 2 l q 1 . f l Pl Ž y . N 2
Ž 6.
Ž 7.
Ž . and P˙l Ž z . ' dPl z . The results on the LEU-Princidz ple wŽ2. – Ž7.x, for the particular case y s 1, are presented in Table 1.
Ž y . , are fixed
y Ý Ž 2 l q 1. N f l N 2
Lo y
K Ž y, y . s 12
dV
or equivalently
™ maxrmin,
Ý Ž2 l q1. Pl Ž x . Pl Ž y . ls 0
ds
Ž 9.
and the reproducing kernels K Ž x, y . and K Ž y, y . for two-body scattering Žsee Ref. w10x. are expressed as Lo y
Ž 8.
where the entropic index q / 1 controls the degree of nonextensiÕity of the Tsallis-like entropy reflected in the pseudoadditiÕity entropy rule: Su LŽ q . s Su Ž q . q SLŽ q . q Ž1 y q . Su Ž q . SLŽ q .. Now, an optimal upper bounds on the statistical Õariances VX Ž q ., X ' u , L, u L; q g R4 can be obtained, as in Refs. w6x, by solving the following constrained optimization problem
1 y Ý Ž 2 l q 1.
Pl2 Ž y .
=
397
Ž 1.
5™
min.
Ž 11 .
Thus, the unique solution of the problem Ž11. exists and is given by f Ž x . s fo y Ž x . sf Ž y.
K Ž x, y. K Ž y, y .
, x , y g w y1,q 1 x ,
Ž 12 .
where f Ž x . is the two-body scattering amplitude and the reproducing kernels functions K Ž x, y . and K Ž y, y . are given by Eqs. Ž5. and Ž6.. Consequently,
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
398
one of the final result is not only the upper bound Ž2. for q G 0, but also the inequalities Vu Ž q . F Vuo y Ž q . , VLŽ q . F VLo y Ž q . , Vu LŽ q . F VuoLy Ž q . ,
for
q G 0,
Ž 13 .
and Vuo y Ž q . F Vu Ž q . , VLo y Ž q . F Vu Ž q . , VuoLy Ž q . 4 F Vu LŽ q . 4 ,
for
q F 0,
Ž 14 .
where Vuo y Ž q ., X s u , L, u L4 are the optimal statistical Õariances corresponding to the optimal state Ž12., and are given by 1
oy
Vu Ž q . s exp
qy1
~°1 y H1 dx
¢
y1
K Ž x, y
2
K Ž y, y .
qgR,
¶• ß, q
Ž 15 .
VLo y Ž q . s exp
½
=
exp 1 y Ž 2p .
1
1 y Ý Ž 2 l q 1.
qy1 Pl2
for
Ž y.
5
,
qgR,
Ž 16 .
and
PoyŽ x. s
Ž 19 . 1y q
. r Ž q y 1 . F VuoLŽ q . F Vu LŽ q . ,
q - 0,
Ž 20 .
°1 ¢q y 1
~
VuoLy Ž q . s exp
q`
Pl Ž y . p lo y s 0,
r Ž q y 1 . F Vu LŽ q . F VuoLŽ q . ,
where
while the optimal statistical variance VuoLy Ž q . is presented in formula Ž4.. We note that, the results Ž15., Ž16., are obtained with the aid of the optimal probability distributions
2 K Ž y, y .
for
1y q
q G 0,
exp 1 y Ž 2p .
q
2 K Ž y, y .
p lo y s
respectively. Then, by combining these two last inequalities we obtain the stated result: Ž1 y 2 1y q .rŽ q Ž q . F Vu LŽ q .. y 1. F Su LŽ q ., or, equivalently Vu min L LEU-inequalities for Diffraction on single slit. As an example we consider here the diffraction experiment of an incoming beam of monochromatic photons with momentum k, incident on a wall that contains an infinitely long slit of width 2a. Here we preserve all notations and definitions from Ref. w14x. Hence, by choosing the canonical conjugate variables lj s k j , and p, it is easy to prove the result from Table 1, which are analogous to the results obtained for the optimal states of the two body elastic scattering. Therefore, the LEU-inequalities for diffraction on single slit can be given in the following general form:
1r2
for
K Ž x, y. K Ž y, y .
,
for
=
Hy` dp
l s 0,1, . . . , L o y ,
l ) Lo y ,
Ž 17 .
1y
1 NSqy1
K Ž p, po
q
2
K Ž po , po .
¶• ß,
Ž 21 .
where qa
2
NS s
, K Ž y, y . / 0,
Hya d Ž k j . s 2 ka '‘‘number of states’’ Ž 22 .
y g w y1,q 1 x .
Ž 18 .
Therefore, we can conclude that by Eq. Ž13. for Vu LŽ q . we proved the upper bound part of the LEU-Principle. The proof of the lower bound Ž q . Ž2., for q G 0, can be obtained from Ž4. as Vumin L Ž . the limit of Vuo1 0. The lower bound L q when L o1 Ž q . of the LEU-Principle Ž3., for q - 0, can be Vumin L proved as follows. First we prove, via Lagrange multipliers, the bounds: w1 y 2 1y q xrŽ q y 1. F Su Ž q ., 0 F SLŽ q ., for q - 0. These bounds are equivalent to: 2 1y q F 1 q Ž1 y q . Su Ž q . and 1 F 1 q Ž1 y q . SLŽ q .,
™
and the reproducing kernel can be given by 1 qka K Ž p, po . s exp yilj p 4 exp ilj po 4 dlj 2p yka
H
s
ka sin ka Ž p y po .
p
ka Ž p y po .
,
lj ' Ž k j . ,
Ž 23 . where j is the coordinates in slit while p is the corresponding pattern coordinate as defined in Ref. w14x.
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
399
Proof: The proof can be easy obtained as in preceding section following step by step the definitions and results from the Table 1. So, we can conclude that the diffraction is an optimal phenomenon which saturates the upper bound of the entropic uncertainty band given in the last row of the Table 1. Experimental tests of the LEU-Principle. For numerical investigation of our LEU-Principle is interesting to calculate the scattering variances Ž8. for extensiÕe statistics q s 1, as well as, for nonextensiÕe statistics case Že.g., q s 0.5 and q s 2. by reconstruction of the pion-nucleus scattering amplitudes using the experimental pion-nucleus phase-
Fig. 2. The experimental tests of the exp
½
1y
Ž L o q1 . 2 2
q y1
entropic bands:
5
r Ž q y1 . FVu Ž q . FVuo1 Ž q ., for the
extensiÕe statistics (q s 1) case, as well as for the nonextensiÕe statistics (q s 0.5 and 2) cases, calculated from pion-nucleus phase shifts analyses w15x by using Eq. Ž15. and Table 1.
Fig. 1. The experimental tests of the LEU-Principle: expwŽ1y Ž . ( 2 1y q . .rŽ q y1.x FVu LŽ q . FVuo1 L q , for the extensiÕe statistics q s 1) case, as well as for the nonextensiÕe statistics Ž q s 0.5 and 2. cases, calculated from pion-nucleus phase shifts analyses w15x by using Table 1.
shifts w13x. The results obtained in this way are represented in Figs. 1–3 as functions of the optimal angular momentum L o which is obtained from the same phase shifts by formula from Table 1. Therefore, in Figs. 1–3 we presented the results of the first experimental test of the Principle of Limited Uncertainty Ž3. in quantum scattering. The grey regions around the optimal variances VXo1 Žthe full curves in Figs. 1–3. are obtained by assuming an error of D L o s "1 in the estimation of the optimal angular momentum from the experimental data. From comparison of the results from Figs. 1–3, for different entropic index q, we see that the results correspond-
400
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
surement of non-commuting obserÕables for any extensiÕe and nonextensiÕe (q G 0. quantum systems. Žii. Two important concrete realizations of the LEU-Principle are explicitly obtained in this paper, namely, Ža. the LEU-inequalities Ž2. – Ž5. for the quantum scattering of spinless particles, and, Žb. the LEU-inequalities Ž19. – Ž23. for the diffraction on single slit of width 2a. In particular from the general results Ž2. – Ž5., in the limit y q1, we recovers in an exact form all the results previously reported in Refs. w5,6x. Here, an experimental illustration of the LEU-Principle is presented in Figs. 1–3 for the cases y s 1 and q s 0.5, 1, and 2. Žiii. For the nonextensive quantum systems with negative q we also proved the validity of the state independent entropic uncertainty relations: expŽ1– 2 1y q .rŽ q y 1.4 F Vu LŽ q .. Moreover, in this case we get wsee Eq. Ž14.x that the optimal Tsallis-like entropies Žif they exists for q - 0. provides only an important improvement of the above state independent entropic uncertainty relations.
™
References
Fig. 3. The experimental tests of the entropic bands: 1FVLŽ q . F VLo1 Ž q ., for the extensiÕe statistics (q s 1) case, as well as for the nonextensiÕe statistics (q s 0.5 and 2) cases, calculated from pion-nucleus phase shifts analyses w15x by using Table 1.
ing to VXo1 Ž q ., X ' u , L, u L, for all q, are in agreement to the LEU-Principle predictions. The main results and conclusions can be summarized as follows: Ži. In the present paper we introduced a new principle in quantum physics namely the Principle of Limited Entropic Uncertainty (LEU-Principle). This new principle includes in a more general and exact form not only the old Heisenberg uncertainty principle but also introduce an upper limit on the magnitude of the uncertainty in the quantum physics. The LEU-Principle asserts that : ’’there is an irreducible lower bound as well as an upper bound on the uncertainty in the result of a simultaneous mea-
w1x W. Heisenberg, Z. Phys. 43 Ž1927. 172; W. Heisenberg, The Physical Principles of the Quantum Theory, University of Chicago, 1930; E.H. Kennard, Z. Phys. 44 Ž1927. 326. w2x I. Bialynicki-Birula, J. Mycielski, Commun. Math. Phys. 44 Ž1975. 129; I. Bialynicki-Birula, J.L. Madajczuk, Phys. Lett. A 108 Ž1985. 384; D. Deutsch, Phys. Rev. Lett. 50 Ž1983. 631; M. Hosein Partovi, Phys. Rev. Lett. 50 Ž1983. 1883; K. Kraus, Phys. Rev. 35 Ž1987. 3070; H. Massen, J.B.M. Uffink, Phys. Rev. Lett. 60 Ž1988. 1103; J. Sanchez-Ruiz, Phys. Lett. A 181 Ž1993. 193. w3x M. Ohya, D. Petz, Quantum Entropy and Its Use, SpringerVerlag, Berlin-Heidelberg-New York-London-Paris-TokyoHong Kong-Barcelona-Budapest, pp. 279, 1993, and quoted therein references. w4x D.B. Ion, M.L. Ion, Phys. Lett. B 352 Ž1995. 155. w5x D.B. Ion, M.L.D. Ion, Phys. Rev. Lett. 81 Ž1998. 5714; Phys. Rev. E 60, Ž1999. Žto be published.. w6x M.L.D. Ion, D.B. Ion, Phys. Rev. Lett. 83 Ž1999. 463. w7x D.B. Ion, M.L.D. Ion, Phys. Lett. B Žto be published.. w8x C. Tsallis, J. Stat. Phys. 52 Ž1988. 479; C. Tsallis, Phys. Lett. A 195 Ž1994. 329. w9x M. Lyra, C. Tsallis, Phys. Rev. Lett. 80 Ž1998. 53 and quoted therein references. E.P. Borges, I. Roditi in Phys. Lett. A 246 Ž1998. 399. w10x D.B. Ion, Phys. Lett. B 376 Ž1996. 282. w11x D.B. Ion, International J. Theor. Phys. 24 Ž1985. 1217; D.B. Ion, International J. Theor. Phys. 25 Ž1986. 1257.
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401 w12x M. Portesi, A. Plastino, Physica A 225 Ž1996. 412. w13x D.T. Pegg, S.M. Barnet, Europhys. Lett. 6 Ž1988. 483; Phys. Rev. A 39 Ž1989. 1665. w14x See the book: M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford. New York. Toronto. Sydney. Frankfurt, 1986.
401
w15x J. Frolich, H.G. Schlaile, L. Streit, H.F.K. Zingl, Z. Phys. A 302 Ž1981. 89; O. Dumbrais, J. Frolich, U. Klein, H.G. Schlaile, Phys. Rev. C 29 Ž1984. 581; J. Frolich, H. Pilkuhn, H.G. Schlaile, Nuclear Phys. A 415 Ž1984. 399; B. Brinkmoller, H.G. Schlaile, Phys. Rev. C 48 Ž1993. 1973; H.G. Schlaile, Phys. Rev. C 55 Ž1997. 2584.
Physics Letters B 474 Ž2000. 395–401
Limited entropic uncertainty as new principle of quantum physics M.L.D. Ion, D.B. Ion National Institute for Physics and Nuclear Engineering, IFIN-HH, P.O. Box MG-6, Bucharest, Romania Received 3 November 1999; accepted 7 January 2000 Editor: R. Gatto
Abstract In this paper the Principle of Limited Entropic Uncertainty (LEU-Principle) for the canonic conjugate variables in quantum physics is proved in a more general form by using Tsallis-like entropies for positive nonextensivities Ž q ) 0.. Results on experimental tests of this new principle for the angle-angular momentum Õariables in the quantum scattering are illustrated by using 49 sets of pion-nucleus phase shifts. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj; 12.40.Ee
The Uncertainty Principle ŽUP. of quantum mechanics discovered by Heisenberg w1x which constitute the corner-stone of quantum physics, asserts that : ŽUP. ‘‘there is an irreducible lower bound on the uncertainty in the result of a simultaneous measurement of non-commuting obserÕables’’. An well known example, is the Heisenberg inequality w1x for the usual position-momentum Ž " s c s 1.: 12 F D x D p, where D y denotes the standard deviation of the observable y. The extended version of this inequality for an arbitrary pair of quantum mechanical observable A and B is given by: 12 <²w A, B x:< F D A D B where the corresponding standard deviations D AŽ c . and D B Ž c . for a quantum system in a given normalized state < c : depends on the state < c :. In order to avoid this state-dependence many authors w2–7x proposed to use the information entropy as a measure of the uncertainty instead of above standard
quantitative formulation of the Heisenberg uncertainty principle. Indeed, the entropy of a quantum state describing physical system is a quantity expressing the uncertainty or randomness of the system. So, an alternative mathematical formulation of the uncertainty principle for position-momentum pair is given by the entropic inequality Žsee BialynickiBirula et al. w2x.
p e F Vx Vp ,
Ž 1.
where Vx ' exp S x 4 and Vp ' exp S p 4 are the statistical Õariances which are defined as the exponential of the differential entropies S x and S p corresponding to the Ž x and p .-observables. Recently, a real progress in the investigation of the quantum entropy it was achieved not only by proving new entropic uncertainty relations w2–7x for the standard additiÕe systems but also by generalization w7x of such results
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 0 3 6 - 8
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
396
to the nonextensiÕe statistics w8,9x. So, the state independent angle-angular momentum entropic lower bounds are proved w5,7x by using Tsallis-like entropies and Riesz theorem for the quantum scattering of the spinless particles. Moreover, the optimal entropic upper bounds for each Tsallis-like scattering entropy Su Ž q ., SLŽ q ., Su LŽ q ., were recently proved w6x in terms of optimal entropies derived from the principle of minimum distance in the space of states w10,11x. In this way, the entropic uncertainty band as a new concept in quantum physics is introduced. The generalized entropy as a measure of quantum uncertainty was also discussed in Ref. w12x in connection with the number-phase entropic uncertainty measure for the coherent states within the Pegg–Barnet quantum theory w13x. In this paper the Principle of Limited Entropic Uncertainty (LEU-Principle), as a new principle in quantum physics, is proved. Then, consistent experimental tests of the LEU-principle, obtained by using the available 49 sets of the pion-nucleus phase shifts w14x, are presented for both, extensiÕe Ž q s 1. and nonextensiÕe Ž q s 0.5 and q s 2.0., statistics cases.
Moreover, some results obtained by the application of LEU-Principle to the diffraction phenomena are discussed. LEU-Principle for quantum scattering. Here we present a short mathematical proof of the for the Principle of Limited Entropic Uncertainty in quantum scattering. So, we start with the scattering of two spinless particles for which all notation and definition from Ref. w5–7x will be adopted here. The angular distribution of probability P Ž x . as well as angular-momentum distribution p l ,l s 0,1,2, . . . 4 are presented in Table 1, where ddVs Ž x . s < f Ž x .< 2 is the differential cross section, x ' cos u , u-being the c.m. scattering angle, sel is the integrated elastic cross section, f l are the partial amplitudes while f Ž x . s ÝŽ2 l q 1. f l Pl Ž x . is the two-body scattering amplitude and Pl Ž x . are Legendre polynomials. Hence, using the results of Refs. w6,7x for the Tsallis-like scattering entropies, in this paper we prove the following standard quantitative formulation of the LEU-Principle. LEU-inequalities for scattering: Let sel and ddVs Ž y . be fixed from experiment for a fixed y g wy1,q 1x.
Table 1 The canonical conjugate variables Ž1., optimal distributions Ž2., Ž3., reproducing kernels Ž4., optimal variances Ž5. – Ž7., and Uncertainty Bands Ž8., corresponding to optimal scattering are compared with those of the diffraction on a single slit. No.
Optimal scattering
Ž1. Ž2.
l ' angular momentum and x s cos u plo1 s N1s ,l s 0, . . . , L0 , plo1 s 0, l G L 0 , Ns s ÝŽ2 l q 1. s Ž L o q 1. 2 , Ž L o q 1. 2 s s4 pel ddVs Ž1. Puo1 Ž x . s
Ž3.
Diffraction on single slit of width 2a PLo Ž lj
w K Ž x,1 . x 2
X ' kap, P o Ž p . s
K Ž1,1 .
s 12 P˙L oq1 Ž x . q P˙L oŽ x . , K Ž1,1. s 12 Ž L o q 1. 2 VLo1 s exp SLo1 4 s Ns
Ž5. Ž6.
Vuo1 s exp yHq1 y1 dx
4
w K Ž x,1 . x
2
ln
K Ž 1,1 .
½
s Ns exp yHq1 y1 dx Ž8.
ž
w K Ž x,1 . x K Ž 1,1 .
VLo s Na 2
/
w K Ž x,1 . x 2 K Ž 1,1 .
ln
ž
2 F Vu VL F NsVuo1
w K Ž x,1 . x 2 K Ž 1,1 .
2
q` Vpo s exp y p1 Hy` dX w sinX X x ln
Ž
ka p
w sinX X x 2 . 4
s w kap x expw2Ž1 y g .x Vpo VLo s exp SLo q S xo 4
Vuo1 VLo1 s exp SLo1 q Suo1 4
Ž7.
K aŽ0,0 .
2
w K a Ž X ,0 . x 2
s kap w sinX X x . sinX X , K Ž0,0. s ka X s kap, K Ž X,0. s Ž ka p p
K Ž x,1. s 21 ÝŽ2 l q 1. Pl Ž x .
Ž4.
lj ' k j , x ' p . s 1rNa , Na s 2 ka
/5
s 2p expw2Ž1 y C .x
p e F Vx Vp F 2p expw2Ž1 y C .x s NaVpo
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
Let Vu LŽ q . s exp Su LŽ q . be the statistical Õariances corresponding to the Tsallis-like scattering entropies Su LŽ q .. Then, the LEU-inequalities can be given in the following general form: exp Ž 1 y 2 1y q . r Ž q y 1 . F Vu LŽ q . F VuoLy Ž q . , for
q G 0,
Ž 2.
exp Ž 1 y 2 1y q . r Ž q y 1 . F VuoLy Ž q . F Vu LŽ q . , for
q - 0,
Ž 3.
where VuoLy Ž q . s exp SuoLy Ž q .
°1 ¢q y 1
~
s exp
Proof : Therefore, using the Tsallis-like Su Ž q .,SLŽ q .,Su LŽ q .4 scattering entropies, given in Ref. w6x by Eq. Ž6. – Ž11., for the canonical conjugate variables wcos u ,l x we define the statistical entropic Õariances VX Ž q . s exp S X Ž q . 4 , X ' u , L, u L, q g R ,
2 K Ž y, y .
maxrmin VX Ž q . 4 when sel and q
q1
Hy1
dx
K Ž x, y
2
q
K Ž y, y .
¶• ß
Ž 4.
K Ž x , y . s 12
s
L o y q1 PL o yq1Ž x . PL o yŽ y . y PL o yŽ x . PL o yq1Ž y . 2
xy y
,
Ž 5.
£ ' l0 VX q l1 q l2
ds dV
sel 4p
Ž 1 . y N Ý Ž 2 l q 1 . f l Pl Ž y . N 2 Ž 10 .
where l i s 0,1,2, are the corresponding Lagrange multipliers. Then, the singular solution l 0 s 0 of the Lagrange function Ž10. is obtained by solving the minimization problem
½Ý
Ý Ž 2 l q 1. Pl Ž y . Pl Ž y . ls0
s
Lo y q 1 2
P˙L o yq1 Ž y . PL o yŽ y .
yP˙L o yŽ y . PL o yq1 Ž y . ,
and the optimal angular momentum L o y can be obtained by solving the implicit algebraic equation ds dV
Ž y . s K Ž y, y .
sel 2p
Ž 2 l q 1. N f l N 2 q a
ds dV
yN Ý Ž 2 l q 1 . f l Pl Ž y . N 2
Ž 6.
Ž 7.
Ž . and P˙l Ž z . ' dPl z . The results on the LEU-Princidz ple wŽ2. – Ž7.x, for the particular case y s 1, are presented in Table 1.
Ž y . , are fixed
y Ý Ž 2 l q 1. N f l N 2
Lo y
K Ž y, y . s 12
dV
or equivalently
™ maxrmin,
Ý Ž2 l q1. Pl Ž x . Pl Ž y . ls 0
ds
Ž 9.
and the reproducing kernels K Ž x, y . and K Ž y, y . for two-body scattering Žsee Ref. w10x. are expressed as Lo y
Ž 8.
where the entropic index q / 1 controls the degree of nonextensiÕity of the Tsallis-like entropy reflected in the pseudoadditiÕity entropy rule: Su LŽ q . s Su Ž q . q SLŽ q . q Ž1 y q . Su Ž q . SLŽ q .. Now, an optimal upper bounds on the statistical Õariances VX Ž q ., X ' u , L, u L; q g R4 can be obtained, as in Refs. w6x, by solving the following constrained optimization problem
1 y Ý Ž 2 l q 1.
Pl2 Ž y .
=
397
Ž 1.
5™
min.
Ž 11 .
Thus, the unique solution of the problem Ž11. exists and is given by f Ž x . s fo y Ž x . sf Ž y.
K Ž x, y. K Ž y, y .
, x , y g w y1,q 1 x ,
Ž 12 .
where f Ž x . is the two-body scattering amplitude and the reproducing kernels functions K Ž x, y . and K Ž y, y . are given by Eqs. Ž5. and Ž6.. Consequently,
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
398
one of the final result is not only the upper bound Ž2. for q G 0, but also the inequalities Vu Ž q . F Vuo y Ž q . , VLŽ q . F VLo y Ž q . , Vu LŽ q . F VuoLy Ž q . ,
for
q G 0,
Ž 13 .
and Vuo y Ž q . F Vu Ž q . , VLo y Ž q . F Vu Ž q . , VuoLy Ž q . 4 F Vu LŽ q . 4 ,
for
q F 0,
Ž 14 .
where Vuo y Ž q ., X s u , L, u L4 are the optimal statistical Õariances corresponding to the optimal state Ž12., and are given by 1
oy
Vu Ž q . s exp
qy1
~°1 y H1 dx
¢
y1
K Ž x, y
2
K Ž y, y .
qgR,
¶• ß, q
Ž 15 .
VLo y Ž q . s exp
½
=
exp 1 y Ž 2p .
1
1 y Ý Ž 2 l q 1.
qy1 Pl2
for
Ž y.
5
,
qgR,
Ž 16 .
and
PoyŽ x. s
Ž 19 . 1y q
. r Ž q y 1 . F VuoLŽ q . F Vu LŽ q . ,
q - 0,
Ž 20 .
°1 ¢q y 1
~
VuoLy Ž q . s exp
q`
Pl Ž y . p lo y s 0,
r Ž q y 1 . F Vu LŽ q . F VuoLŽ q . ,
where
while the optimal statistical variance VuoLy Ž q . is presented in formula Ž4.. We note that, the results Ž15., Ž16., are obtained with the aid of the optimal probability distributions
2 K Ž y, y .
for
1y q
q G 0,
exp 1 y Ž 2p .
q
2 K Ž y, y .
p lo y s
respectively. Then, by combining these two last inequalities we obtain the stated result: Ž1 y 2 1y q .rŽ q Ž q . F Vu LŽ q .. y 1. F Su LŽ q ., or, equivalently Vu min L LEU-inequalities for Diffraction on single slit. As an example we consider here the diffraction experiment of an incoming beam of monochromatic photons with momentum k, incident on a wall that contains an infinitely long slit of width 2a. Here we preserve all notations and definitions from Ref. w14x. Hence, by choosing the canonical conjugate variables lj s k j , and p, it is easy to prove the result from Table 1, which are analogous to the results obtained for the optimal states of the two body elastic scattering. Therefore, the LEU-inequalities for diffraction on single slit can be given in the following general form:
1r2
for
K Ž x, y. K Ž y, y .
,
for
=
Hy` dp
l s 0,1, . . . , L o y ,
l ) Lo y ,
Ž 17 .
1y
1 NSqy1
K Ž p, po
q
2
K Ž po , po .
¶• ß,
Ž 21 .
where qa
2
NS s
, K Ž y, y . / 0,
Hya d Ž k j . s 2 ka '‘‘number of states’’ Ž 22 .
y g w y1,q 1 x .
Ž 18 .
Therefore, we can conclude that by Eq. Ž13. for Vu LŽ q . we proved the upper bound part of the LEU-Principle. The proof of the lower bound Ž q . Ž2., for q G 0, can be obtained from Ž4. as Vumin L Ž . the limit of Vuo1 0. The lower bound L q when L o1 Ž q . of the LEU-Principle Ž3., for q - 0, can be Vumin L proved as follows. First we prove, via Lagrange multipliers, the bounds: w1 y 2 1y q xrŽ q y 1. F Su Ž q ., 0 F SLŽ q ., for q - 0. These bounds are equivalent to: 2 1y q F 1 q Ž1 y q . Su Ž q . and 1 F 1 q Ž1 y q . SLŽ q .,
™
and the reproducing kernel can be given by 1 qka K Ž p, po . s exp yilj p 4 exp ilj po 4 dlj 2p yka
H
s
ka sin ka Ž p y po .
p
ka Ž p y po .
,
lj ' Ž k j . ,
Ž 23 . where j is the coordinates in slit while p is the corresponding pattern coordinate as defined in Ref. w14x.
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
399
Proof: The proof can be easy obtained as in preceding section following step by step the definitions and results from the Table 1. So, we can conclude that the diffraction is an optimal phenomenon which saturates the upper bound of the entropic uncertainty band given in the last row of the Table 1. Experimental tests of the LEU-Principle. For numerical investigation of our LEU-Principle is interesting to calculate the scattering variances Ž8. for extensiÕe statistics q s 1, as well as, for nonextensiÕe statistics case Že.g., q s 0.5 and q s 2. by reconstruction of the pion-nucleus scattering amplitudes using the experimental pion-nucleus phase-
Fig. 2. The experimental tests of the exp
½
1y
Ž L o q1 . 2 2
q y1
entropic bands:
5
r Ž q y1 . FVu Ž q . FVuo1 Ž q ., for the
extensiÕe statistics (q s 1) case, as well as for the nonextensiÕe statistics (q s 0.5 and 2) cases, calculated from pion-nucleus phase shifts analyses w15x by using Eq. Ž15. and Table 1.
Fig. 1. The experimental tests of the LEU-Principle: expwŽ1y Ž . ( 2 1y q . .rŽ q y1.x FVu LŽ q . FVuo1 L q , for the extensiÕe statistics q s 1) case, as well as for the nonextensiÕe statistics Ž q s 0.5 and 2. cases, calculated from pion-nucleus phase shifts analyses w15x by using Table 1.
shifts w13x. The results obtained in this way are represented in Figs. 1–3 as functions of the optimal angular momentum L o which is obtained from the same phase shifts by formula from Table 1. Therefore, in Figs. 1–3 we presented the results of the first experimental test of the Principle of Limited Uncertainty Ž3. in quantum scattering. The grey regions around the optimal variances VXo1 Žthe full curves in Figs. 1–3. are obtained by assuming an error of D L o s "1 in the estimation of the optimal angular momentum from the experimental data. From comparison of the results from Figs. 1–3, for different entropic index q, we see that the results correspond-
400
M.L.D. Ion, D.B. Ion r Physics Letters B 474 (2000) 395–401
surement of non-commuting obserÕables for any extensiÕe and nonextensiÕe (q G 0. quantum systems. Žii. Two important concrete realizations of the LEU-Principle are explicitly obtained in this paper, namely, Ža. the LEU-inequalities Ž2. – Ž5. for the quantum scattering of spinless particles, and, Žb. the LEU-inequalities Ž19. – Ž23. for the diffraction on single slit of width 2a. In particular from the general results Ž2. – Ž5., in the limit y q1, we recovers in an exact form all the results previously reported in Refs. w5,6x. Here, an experimental illustration of the LEU-Principle is presented in Figs. 1–3 for the cases y s 1 and q s 0.5, 1, and 2. Žiii. For the nonextensive quantum systems with negative q we also proved the validity of the state independent entropic uncertainty relations: expŽ1– 2 1y q .rŽ q y 1.4 F Vu LŽ q .. Moreover, in this case we get wsee Eq. Ž14.x that the optimal Tsallis-like entropies Žif they exists for q - 0. provides only an important improvement of the above state independent entropic uncertainty relations.
™
References
Fig. 3. The experimental tests of the entropic bands: 1FVLŽ q . F VLo1 Ž q ., for the extensiÕe statistics (q s 1) case, as well as for the nonextensiÕe statistics (q s 0.5 and 2) cases, calculated from pion-nucleus phase shifts analyses w15x by using Table 1.
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