A Mixed-Entropic Uncertainty Relation

Transcription

1 A Mied-Entropic Uncertainty Relation Kamal Bhattacharyya* and Karabi Halder Department of Chemistry, University of Calcutta, Kolkata , India Abstract We highlight the advantages of using simultaneously the hannon and isher information measures in providing a useful form of the uncertainty relation for the positionmomentum case. It does not require any ourier transformation. The sensitivity is also noteworthy. PAC Numbers: Ge, Cf Keywords: hannon entropy, isher information, Uncertainty principle *Corresponding author (pchemkb@yahoo.com) 0

2 . Introduction Heisenberg s uncertainty relation (UR) involving standard deviations of position and reduced momentum k ( k = p/ ) is epressed as [] σ σ. () k The relation has later been etended to two arbitrary non-commuting observables []. A recent survey [3] has paid considerable attention to several points of this UR. However, it was also realized that σ is not always [4] a neat and physical measure of uncertainty associated with the mean value of. uch situations also include probability distributions (PD) like a Lorentzian for which σ = and distributions with multiple peaks. To combat the latter, an equivalent width concept [5] was put forward long back, but it did not gain much popularity. Instead, another form of the UR by involving hannon entropy has received sufficient curiosity. This UR is succinctly represented as [6 9] ( ) + ( k ) + ln( π ), () where () refers to the hannon entropy in position space and is defined by ( ) = ln( P( )), (3) with P() as the normalized PD. A similar definition for (k) follows with the corresponding PD Pk ( ) in k-space. If P ( ) =Ψ ( ), then Pk ( ) =Φ ( k) such that functions Ψ() and Φ(k) are ourier transforms (T) of each other: Ψ ( ) = (/ π ) Φ( k)ep[ ik] dk; Φ ( k) = (/ π ) Ψ( )ep[ ik] d. The entropic UR (EUR), relation (), has also been etended to general non-commuting observables [0]. The hannon entropy has received considerable popularity in a wide variety of contets (see, e.g., [ 6] and references quoted therein) starting from statistical mechanics [, ] to polymer chemistry [3], thermodynamics [4], quantum mechanics [5] and quantum chemistry [6]. o, EUR in the form () has received wide acceptance [4, 7]. As the net step, therefore, it has become natural to look for similar relations with other information measures. Thus, Rényi entropy has received some attention [8 0]; the relevance of Tsallis entropy in the contet of EUR has been eplored [ 3]; (4)

3 finally, role of the isher information measure has also been noted [4, 5]. till, one observes that it is EUR () that is the most popular one. The main difficulty with (), however, is the T part, unlike (). More often than not, the transform is obtained only after cumbersome eercises. o, we like to modify () in such a way that any T is avoided. We achieve this by recalling I (), the isher information in position space, a function of which replaces the (k) term in (). This makes a concomitant change at the right side of () as well. But, we shall see that, while a Gaussian yields the equality here too, our form is sometimes more sensitive than ().. The relation The isher information measure is defined by ( ) I ( ) = dln( P( )/ d. (5) or a real quantum-mechanical wave function, it is easy to check [5] that I ( ) 4 k = σ. (6) Coupled with (), (6) yields the well-known Cramer-Rao inequality [4]. In this contet, it is essential to point out that Hall [4] discussed at some length on the genesis of following two inequalities: πeσ ep[ ( )]; (7a) ep[ ( )] πe/ I ( ). (7b) Of these, the first one [(7a)] has received considerable concern because it shows directly that () offers a tighter inequality than (). Indeed, this feature alone provides a strong motivation in switching over to () from (). Need less to mention, the eact equality in either case follows for a Gaussian distribution. We shall, on the other hand, concentrate on the second inequality 7(b) on which only scanty attention was paid. A slight rearrangement leads us to ( ) + ln( I ( )) ln( πe). (8) If we compare (8) with (), it becomes apparent that the second term at the left of (8) accounts, in effect, for (k). This identification has a nice physical appeal [5]. As the isher information in position space increases, the same in k-space decreases and hence the entropy increases. Effectively, thus, (8) becomes the new EUR. Here, the T part is avoided naturally. One can put (8) more succinctly as

4 where ( ) + ( k) ln( π e) (9) ( k) ( k) = ln( I ( )). (0) Note that we have used here ( k) to denote a derived isher entropy. One might, for eample, define a true isher entropy in k-space as ( k) = ln( I ( k)), () but that will again invite the T, and hence is of little use here. 3. ome advantages Let us choose a few test cases to see how quickly one can compute the desired quantities: Case (i): or P L n L ( ) = (/ )sin ( π / ) with in (0, L), we find ( ) + ( k) ln() + ln( π ) + ln( n).53 + ln( n). () This PD refers to the quantum-mechanical particle in a -d bo model. Here, we notice that the entropy sum increases linearly with the logarithm of the quantum number. At large n, this is virtually equal to the number of zeroes of the PD. Indeed, we may appreciate that, while () is eact, such a dependence of the left side on the logarithm of the number of zeroes is a general semiclassical result for bound stationary states of any - dimensional potential. Actually, on employing the Wilson-ommerfeld quantization rule, one finds [5] that where and it satisfies (see also ec. IV of ref. [5]) ( ) + ( k) = ln( π ) + ln( n), (3) ( ) ( ) = ln P( ) (4) ( ) ( ). (5) We thus note two more aspects of the problem. irst, the appearance of ( k) in the present contet is a natural one. econdly, (3) and (5) yield ( ) + ( k) ln( π ) (6) 3

5 for the lowest (n = ) quantum state. While semiclassical results are usually valid in the n limit, (6) is not too far from the eact inequality (8). In view of the current interest in I for chrödinger energy eigenfunctions [6], such a result is worth mentioning. Case (ii): With P ( ) ( α / π)( α ) = + and in (-, ), it turns out that ( ) + ( k) ln() + ln( π ).8. (7) 3 We talked about this Lorentzian PD earlier where the UR () fails to perform. Here, however, we notice how quickly one obtains a neat result. Case (iii): We net choose here that P 3/ ( ) = 4 πα ( / π) ep[ α ] with in (0, ). One finds ( ) + ( k) ln(6 π) + γ.55, (8) where γ is the Euler constant. This PD corresponds to the classical Mawell speed distribution and, therefore, (8) shows the applicability of EUR to such a classical case too. Indeed, to incorporate such classical PD, we have deliberately avoided any reference to at the outset and used the variable k instead of p. Case (iv): We finally choose one to the result P 3 ( ) ( α / π)( α ) = + with in (-, ). This PD leads ( ) + ( k) ln() + ln( π ).57. (9) 7 This particular eample has a special appeal. Going over to Pk ( ), we obtain for this case ( ) + ( k) = 3ln() + ln( π ) ln( π ). (0) This result may be contrasted with (9). One notes that this non-gaussian PD reveals a deviation of about 0.7% in the EUR (8) from the result for a Gaussian while (0) shows that the value departs from the equality only by 3.7% when one opts for EUR (). uch an outcome brings to light that, in situations, form (8) may be more sensitive than () as well. Actually, this deviation should have a direct link with the overlap of a chosen function with a Gaussian. urther work along this direction may be useful. 4. Conclusion In summary, the use of mied entropies in constructing EUR has not been tried so far. We have found here a form of EUR [see (8)] by invoking simultaneously the hannon and isher information measures. It is simple and applicable to both classical 4

6 and quantum PD. It does not require any T. ometimes, it is more sensitive than the parent form (). Its semiclassical relevance in the contet of chrödinger eigenvalue problems is also notable. urther eploration with this form vis-à-vis other forms of EUR may be worthwhile. The derived isher entropy in k-space may also find other interesting applications. References. W. Heisenberg, Z. Phys. 43, 7 (97).. H. P. Robertson, Phys. Rev. 34, 63 (99). 3. P. Busch, T. Heinonen and P. Lahti, Phys. Rep. 45, 55 (007). 4. I. Bialynicki-Birula and L. Rudnicki, in tatistical Compleity, ed. K. D. en, pringer, Dordrecht (0), Ch.. 5. M. Bauer and P. A. Mello, Proc. Natl. Acad. ci. UA 73, 83 (976). 6. I. I. Hirschman, Am. J. Math. 79, 5 (957). 7. I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 9 (975). 8. D. Deutsch, Phys. Rev. Lett. 50, 63 (983). 9. M. H. Partovi, Phys. Rev. Lett. 50, 883 (983). 0. V. Majern ık and L. Richterek, Eur. J. Phys. 8, 79 (997).. R. D. Levine and M. Tribus, eds. The Maimum Entropy ormalism, MIT Press, Cambridge, MA (979).. B. Buck and V. A. Macaulay, eds. Maimum Entropy in Action, OUP, Oford (99). 3. G. La Penna, J. Chem. Phys. 9, 86 (003). 4. A. Antoniazzi, D. anelli, J. Barre, P-H. Chavanis, T. Dauois and. Ruffo, Phys. Rev. E 75, 0 (007). 5. C. Das and K. Bhattacharyya, Phys. Rev. A 79, 007 (009). 6.. Noorizadeh and E. hakerzadeh, Phys. Chem. Chem. Phys., 474 (00). 7.. Wehner and A. Winter, New J. Phys., (00). 8. I. Bialynicki-Birula and L. Rudnicki, Phys. Rev. A 74, 050 (006). 9.. Zozor, M. Portesi and C. Vignat, Physica A 387, 4800 (008). 0. A. E. Rastegin, J. Phys. A: Math. Theor. 43, 5530 (00).. A. K. Rajagopal, Phys. Lett. A 05, 3 (995). 5

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