Information entropy of isospectral Pöschl-Teller potential

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1 Iia Joural of Pure & Applie Physics Vol. 43 December 5 pp Iformatio etropy of isospectral Pöschl-Teller potetial Ail Kumar Departmet of Physics Pajab Uiversity Chaigarh 6 4 Receive April 5; accepte September 5 The positio a mometum space iformatio etropies of the grou state a first excite state of the Pöschl-Teller potetial are exactly evaluate. Usig isospectral Hamiltoia approach a family of isospectral potetials which have the same eergy eigevalues as that of the origial potetial has bee costructe. The iformatio etropy cotet i the first few levels of the isospectral potetial has bee calculate a it is show that the iformatio etropy cotet i each level ca be re-arrage as a fuctio of eformatio parameter. Keywors: Iformatio etropy Isospectral Hamiltoia Itrouctio Boltzma-Shao Iformatio etropy is a fuametal quatity closely relate to thermoyamical etropy which measures the sprea or extet of the sigle particle esity i the cotext of esity fuctioal formalism use i may particle systems. The positio a mometum space etropies are give by the expressios S pos ρ( r) l ρ( r) r... () S mom ρ( p)l ρ( p) p... () where * * ρ ( r) ( r) ( r) a ρ ( p) ( p) ( p) are the probability esities i positio space a mometum space respectively a ( p) is the Fourier trasform of (r) i the mometum space. Iformatio etropy plays a crucial role i a stroger formulatio of the ucertaity relatios. A iterestig ucertaity relatio was iscovere by Bialyicki-Birula a Mycielski. It was prove that for wave fuctios ormalize to uity S pos + Smom ( + lπ ) where is the imesio. Though the S pos a S mom are iiviually uboue their sum is boue from below. The aalytical etermiatio of the positio a mometum space etropies has bee carrie out oly for a few quatum mechaical systems 3-8. The etropies of hyroge atom a simple harmoic oscillator were exactly calculate for the grou state i both positio a mometum space. The iformatio etropies i various cotexts e.g. iformatio theory mathematics mathematical physics chemical physics a other areas of physics have bee extesively aalyze i recet times 9-5. We use the isospectral Hamiltoia approach to stuy the isospectral wavefuctios a their etropies. Two Hamiltoias are sai to be strictly isospectral if they have exactly same eergy eigevalue spectrum a S-matrix 6-8 whereas the wave fuctios a their epeet quatities are ifferet. Though the iea of geeratig isospectral Hamiltoias usig the Gelfa-Levita approach 9 or the Darboux proceure were kow for some time the supersymmetric quatum mechaical techiques make the proceure look simpler -. Whe oe eletes a bou state of a give potetial V ( a re-itrouce the state it ivolves solvig a first orer ifferetial equatio which amits a free parameter. Thus a set of oe-imesioal family of potetials V ˆ( x ca be costructe which have the exactly same eergy spectrum as that of V (. I geeral for ay oe imesioal potetial (full lie or half-lie) with bou states oe ca costruct a - parameter family of strictly isospectral potetials i.e. potetials with eigevalues reflectio a trasmissio coefficiets ietical to those for origial potetial. This aspect has bee utilize profitably i may physical situatios which are of iterest to various fiels 3-5.

2 KUMAR: INFORMATION ENTROPY OF PÖSCHL-TELLER POTENTIAL 959 I this paper we cosier the hyperbolic Pöschl- Teller (PT) potetial a calculate the positio a mometum space iformatio etropy exactly for grou state a first excite state. Usig isospectral Hamiltoia approach the eforme potetial a their wave fuctios are costructe a use to calculate the iformatio etropy for the isospectral potetial. Isospectral Hamiltoia Approach The coectio betwee the bou state wavefuctios a the potetial is oe of the key igreiets i solvig exactly for the spectrum of oeimesioal potetial problems. Oce we kow the grou state wavefuctio ( ) a choose its eergy to be zero we ca factorize the Hamiltoia as H A A where (i uits h m ) A + W a A + W are the supersymmetric operators a W x x x [l ] is calle the superpotetial. We have H A A ε... (3) Multiplyig both sies by A we get AA ( A ) ε ( A ) H ( A ) ε ( A ).... (4) Here H is the supersymmetric parter Hamiltoia of H with eigefuctios χ A. It is obvious that H has the same eigevalue spectrum as that of H but for the case A which is the case of supersymmetry broke. Explicitly the relatio betwee Hamiltoias reas E () () E + ; E ( ) () + () () [ E+ ] A [ E () ] A () + () The superpotetial relates the supersymmetric parter potetials V ( ) a V ( ) as x x V W W m.... (5) x It is well kow that for the potetial V ( x ) the origial potetial V ( x ) is ot uique. The argumet is as follows. Suppose H has aother factorizatio BB where B x + Wˆ ( x ) the H AA BB but H B B is ot A A rather it efies a certai ew Hamiltoia. For superpotetial W ˆ ( x ) the parter potetial V ( ) is x ˆ V W + Wˆ ( ).... (6) x Cosier the most geeral solutio as Wˆ W + φ( which emas that φ + W φ( + φ (.... (7) The solutio of the Eq. 7 is φ l[ + λ] x x where ( x ) x a λ is a costat. Therefore we obtai ˆ W W + l[ + λ].... (8) x The correspoig oe-parameter family of potetials V ( x is give as ˆ ˆ V ( x λ ) V (l( + )....(9) x The ormalize grou state wavefuctio correspoig to the potetial V ( x reas ˆ λ( + ˆ ( x... () + λ where λ / ( ). The excite state eigefuctios for the potetial V ( x are give by ˆ + ( x ˆ + x + E + + W I ( + λ ()

3 96 INDIAN J PURE & APPL PHYS VOL 43 DECEMBER 5 Usig the similar proceure the two-parameter grou state wavefuctio 6 ca be obtaie as ( ˆ ( x λ λ). φ ( ) ˆ x λ λ A ( λ) A ( + λ)... () The Eqs (9)-() represet the oe-parameter family of isospectral potetials a wavefuctios which shall be use to calculate the iformatio etropy. Geeralizatio to -parameter eforme potetials was oe elsewhere 6. 3 Iformatio etropy for Pöschl-Teller Potetial We start with Schröiger equatio for hyperbolic PT potetial which is reflectioless a amits bou states ( + ) sech x ( ) ( ) + E. m m m x... (3) The ormalize grou state is give by ( ) sech x... (4) ) where B ( ) is the beta fuctio. The positio space iformatio etropy for the grou state is obtaie as the aalytical expressio ( ) l + l B + [ Ψ() Ψ( )]... (5) S pos ( ) S pos A [ l A+ ( ) B A [l Ψ [ (( )) Ψ( )]] ( ) B [l [ Ψ( ) Ψ( )]] + B ( ) Ψ Ψ + B Ψ Ψ + ] (7) where A. The grou state a first ) ) excite state iformatio etropy as a fuctio of are plotte i Fig. which shows the ecrease i iformatio etropy with the icrease i the umber of bou states. The correspoig mometum space etropies ca be evaluate by obtaiig the mometum space wavefuctios which are the Fourier trasforms of the correspoig positio space wavefuctios. For grou state we obtai ( ) ( p) ) ip ip B + π... (8) () π πp For a we get sech( ) () 3π πp a cosech( ) respectively. The S mom ca be easily evaluate as ( lπ ) a.336 for above two cases respectively. where Ψ is the Psi fuctio. As the umber of bou states icreases the grou state etropy ecreases. For the potetial has oly oe bou state a the positio space etropy of that state is.3. The first excite state wavefuctio is give by ( ) sech x tah x ) )... (6) After legthy but straightforwar calculatios we obtai the aalytical expressio for positio space etropy i the first excite state as Fig. Grou state iformatio etropy (soli lie) a first excite state iformatio etropy (broke lie) i positio space as a fuctio of umber of bou states for Pöschl-Teller potetial

4 KUMAR: INFORMATION ENTROPY OF PÖSCHL-TELLER POTENTIAL 96 4 Iformatio Etropy for Isospectral Pöschl- Teller Potetial Usig isospectral Hamiltoia approach we ca costruct -parameter family of isospectral potetials. For 3 the oe parameter a two parameter isospectral potetials are show i Fig. for some values of eformatio parameter λ. The potetial for λ is just the iitial potetial V ( whereas for other particular values of λ a λ we see separate wells for two parameter eformatio. For -level potetial the oe parameter isospectral grou state wavefuctio is obtaie as ˆ ( x where f λ( + ) sech sih x [ { sech x + f } ] ) k + λ k ( ( )( )... ( k)) (( 3)( 5)... ( k )) (k ) sech x + 3 x... (9) a the first excite state wave fuctio is give by ˆ ( x A[sech 3 A + x tah x + ( ) sech x tah x sech + λ x]... () where A ( {sih x sech 3 sech x 3 ( )! (3)!! sech}. The positio space iformatio etropy for 3 is calculate usig the isospectral wavefuctios. I the case of ueforme potetial it has values.66.6 a.64 for grou state first excite state a seco excite state respectively but whe it is evaluate usig ˆ ( x ˆ ( x a ˆ ( x it is fou that the iformatio etropy cotet is reuce substatially with the eformatio parameter. (3) For λ. we have S pos. 5 a this value icreases to.66 for large values of λ. Similar (3) results are also obtaie for S but for S the (3) pos pos iformatio etropy cotet first icreases from a smaller value a the ecreases to the ueforme value for large eformatio parameter value. The total iformatio etropy for grou state first excite state a seco excite state is plotte as a fuctio of eformatio parameter i Fig. 3 which shows that the total etropy is reuce by choosig smaller values of λ. We also compute the two parameter wavefuctios a calculate the iformatio etropy cotet for ifferet positive a egative values of eformatio parameter λ a λ. The results are show i Figs 4 a 5 for grou state a first excite state respectively. For large values of λ a Fig. 3-level Pöschl-Teller potetial (soli lie) oe-parameter eforme potetial (ashe lie λ.) a two-parameter eforme potetial (otte lie λ λ.5) Fig. 3 Oe-parameter eforme total iformatio etropy ŜŜ +Ŝ +Ŝ as a fuctio of λ. For λ Ŝ S +S +S

5 96 INDIAN J PURE & APPL PHYS VOL 43 DECEMBER 5 ecreases a the icreases a becomes costat at the value of ueforme iformatio etropy. Fig. 4 Two-parameter eforme grou state iformatio etropy as a fuctio of λ for ifferet (large itermeiate a small) values of λ. For soli lie λ. for otte lie λ 3. a for broke lie λ. Fig. 5 Two-parameter eforme first excite state iformatio etropy as a fuctio of λ ifferet (large itermeiate a small) values of λ. For soli lie λ. for otte lie λ 3. a for broke lie λ. λ the value of eforme iformatio etropy approaches the ueforme value. The variatio of Ŝ with λ is show for large itermeiate a small values of λ. I this case the value of iformatio etropy is reuce from.66 to. or eve less for smaller values of λ. I first excite state for small λ the value of Ŝ icreases with icrease i λ but it become costat below the ueforme value. For large λ the value of Ŝ first 5 Coclusio We have stuie the iformatio etropies of a class of systems which belog to the Pöschl-Teller family of potetials. For positio space iformatio etropies of grou state a first excite state of the hyperbolic potetial the exact results were obtaie for a rage of potetial stregths. The expressio for mometum space etropy was obtaie aalytically for the grou state. Usig isospectral Hamiltoia approach we costructe the family of isospectral potetials a calculate the iformatio etropy i positio space. It is fou that the iformatio etropy is reuce for the smaller values of eformatio parameter. For lower iformatio etropy the wave fuctio will be more cocetrate a the accuracy i preictig the localizatio of the particle will be higher. This approach ca also be applie i the reuctio of Heiseberg's ucertaity i positio space. The ucertaity i positio space is give by ( Δx ) x x which are calculate for each level i three level potetial. For the ueforme case it is straightforwar to check that x so ucertaity i positio space is equal to x. This is calculate for grou state first excite state a seco excite state to be..8 a.7 respectively. Whe we cosier the eforme wavefuctios ( Δ will get cotributios ot oly from x but also from x. Sice the wavefuctio o ot have particular symmetry properties hece ( Δ for fiite λ is always smaller tha the ( Δ calculate for the states of ueforme potetial. Ackowlegemet Author is thakful to Dr C N Kumar for useful iscussios a ackowleges the Coucil of Scietific a Iustrial Research (CSIR) New Delhi for assistace through SRF. Refereces Katz A Priciples of Statistical Mechaics The Iformatio Theory Approach (Freema Sa Fracisco) 967. Bialiicki-Birula I Mycielski J Commu Math Phys 44 (975) 9. 3 Gare S R Beale R D Phys Rev A 36 (987) Gare S R Beale R D Phys Rev A 3 (985) 6.

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