Revival structures of coherent states for Xm exceptional orthogonal polynomials of the Scarf I potential within position-dependent effective mass
aa r X i v : . [ m a t h - ph ] J a n Revival structures of coherent states for X m exceptional orthogonal polynomials of the Scarf Ipotential within position-dependent effective mass Sid-Ahmed Yahiaoui and Mustapha Bentaiba ‡ LPTHIRM, D´epartement de physique, Facult´e des sciences, Universit´e SaˆadDAHLAB-Blida 1, B.P. 270 Route de Soumˆaa, 09 000 Blida, AlgeriaE-mail: s [email protected] , [email protected] Abstract.
The revival structures for the X m exceptional orthogonal polynomials ofthe Scarf I potential endowed with position-dependent effective mass is studied in thecontext of the generalized Gazeau-Klauder coherent states. It is shown that in the caseof the constant mass, the deduced coherent states mimic full and fractional revivalsphenomena. However in the case of position-dependent effective mass, although fullrevivals take place during their time evolution, there is no fractional revivals as definedin the common sense. These properties are illustrated numerically by means of somespecific profile mass functions, with and without singularities. We have also observeda close connection between the coherence time τ ( m )coh and the mass parameter λ .PACS numbers: 03.65.-w, 42.50.Ar, 42.50.Md
1. Introduction
It is well known that when working with quantum systems subjected to interact witha given interaction, usually considerations require to identify the mass-term with theconcept of effective mass. In a way, such quantum system becomes position-dependenteffective mass (PDEM). In recent years, the study of quantum system endowed withPDEM has become one of the active subjects of research due to its relevance in describingthe properties of a wide variety of physical problems, such as quantum wells, wires anddots [1], and semiconductor heterostructures [2]. We can also find their applicationsin many others fields, such as the effective interactions in nuclear physics [3], curvedspaces [4], P T -symmetry [5, 6], coherent states [7, 8, 9, 10, 11], and in the context ofthe Wigner’s distribution functions [12, 13, 14].Such systems stimulated a lot of work in mathematical physics for finding theexact solutions for the PDEM Schr¨odinger equation (PDEM SE). This quest has beenaddressed by many approaches and from different point of view; we may quote for ‡ Author to whom any correspondence should be addressed. m EOP, PDEM Scarf I potential and its coherent states revivals X m exceptional orthogonal polynomials ( X m EOP) by G´omez-Ullate et al. [22, 23] and Quesne [24] has been considered as a bigadvance in the understanding of mathematics and physics that can be brought aboutby such eigenfunctions. The X m EOP are the solutions of the second-order Sturm-Liouville eigenvalues problem with rational coefficients , obtained from the eigenfunctionsof exactly solvable systems which have a degree m eigen-polynomial deformation. Thusthey form complete and orthogonal polynomial sets generalizing COP of Hermite,Laguerre and Jacobi. The term exceptional is used to indicate that these polynomialsstart at degree m ( m ≥
1) called codimension , instead of the degree 0 constant term,thus avoiding restrictions of Bochner’s theorem. Recently, X m EOP have been studiedin a lot of works [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], including PDEMsystems [39].On the other hand quantum revivals [40, 41], which are the fundamental realizationof the time-dependent interference phenomena for bound-states with quantized energyspectra, arise when the wave-packet spreads inside the potential and reconstruct itselfduring a certain time T rev , called the revival time. The same also occurs at someinteger multiples of T rev , i.e. ( p/q ) T rev , when the evolving wave-packet will break upinto a set of mini-packets of its original form. However, aside from the revival dynamicsstudied for systems with constant mass which have been well understood in many works[42, 43, 44, 45, 46, 47, 48, 49, 50], comparatively we are aware of few papers that dealtwith revival dynamics for PDEM [51, 52] which did not receive much attention.It is the objective of this paper to fill this gap and to study the revival dynamicsof the generalized Gazeau-Klauder coherent states (GK CS) [53] for X m EOP of theScarf I potential in the specific case of PDEM which, as far as we know, have notbeen considered yet. So the quest for studying CS wave-packets confined in an effectivepotential, with a predetermined energy spectrum, is very interesting in hopes to seehow the mass function M ( x ) can affect their temporal evolution. Our analysis revealsthat, although that full revival still takes place during their time evolution T rev , thereis no trace of fractional revivals in opposition to the usual case of the constant mass(CM). We observe that not only full revivals are different but also depend closely on theprofile of the mass function used. These results are illustrated numerically by means oftwo profiles of the mass functions, with and without singularities, and agree with thoseobtained in [51]. We have also established the correspondence between the coherencetime τ ( m )coh and the mass parameter λ , defined in the sense that is emphasized in [54].The organization of our paper is as follows. In section 2 we generate the PDEMversion of exactly solvable potential already obtained in [55], in the case of constant m EOP, PDEM Scarf I potential and its coherent states revivals X m Jacobi polynomials via the PCT approach. In section 3 we showthat the obtained potentials are shape invariant in the back-ground of supersymmetricquantum mechanics. In section 4 we study the revival dynamics of Gazeau-Klauder CSwave-packets for the exceptional X m Scarf I potential in the cases where the mass isconstant and position-dependent. The last section is devoted to our conclusion.
2. Generation of new PDEM potentials via PCT and their exceptional X m Jacobi polynomials
Taking the natural units ( ~ = m = 1) and using the ordering prescription adopted byBenDaniel and Duke [56], the one-dimensional PDEM SE can be expressed as (cid:18) −
12 d d x + M ′ ( x )2 M ( x ) dd x + M ( x ) V ( x ) (cid:19) ψ ( x ) = M ( x ) E n ψ ( x ) . (2.1)Then by applying the following PCT, ψ ( x ) = f ( x ) F ( g ( x )), to the eigenfunctions,it is not difficult to verify that (2.1) satisfies the second order differential equation [20]d F ( g )d g + Q ( g ) d F ( g )d g + R ( g ) F ( g ) = 0 , (2.2)where F ( g ) is some special function on g ( x ). The functions Q ( g ) and R ( g ) are given by Q ( g ) = g ′′ ( x ) g ′ ( x ) + 2 f ′ ( x ) f ( x ) g ′ ( x ) − M ′ ( x ) M ( x ) g ′ ( x ) , (2.3) R ( g ) = f ′′ ( x ) f ( x ) g ′ ( x ) − M ′ ( x ) f ′ ( x ) M ( x ) f ( x ) g ′ ( x ) + 2 M ( x ) g ′ ( x ) ( E n − V ( x )) . (2.4)Integrating (2.3), we arrive to express f ( x ) as f ( x ) = s M ( x ) g ′ ( x ) exp ( Z g ( x ) Q ( g ) d g ) , (2.5)and by inserting (2.5) into (2.4), one can see that we obtain a system where theassociated effective potential depends on the mass function E n − V eff ( x ) = g ′ ( x )2 M ( x ) (cid:18) R ( g ) −
12 d Q ( g )d g − Q ( g ) (cid:19) + 14 M ( x ) ( S ( g ′ ) − S ( M )) , (2.6)where S ( z ) = z ′′ /z − / z ′ /z ) is the Schwartz derivative of the function z ( x ) and theprime denotes the derivative with respect to x . It follows that the PDEM SE can besolved if the forms of Q and R are given for a mass function M ( x ). In order to obtainthe effective potential in the above equation, we impose that there must be a constanton the right-hand side of (2.6) representing the bound-state energy spectrum E n on theleft-hand side.From (2.5) the solution of the eigenfunctions ψ n ( x ) are given by ψ n ( x ) ∼ s M ( x ) g ′ ( x ) exp ( Z g ( x ) Q ( g ) d g ) F n ( g ( x )) , (2.7) m EOP, PDEM Scarf I potential and its coherent states revivals M ( x ) = 1.In the remainder of the paper, we choose to work under the special function F ( m ) n ( g )to be the PDEM X m Jacobi polynomials b P ( α,β,m ) n ( g ) studied in more details in [29, 55],where n ≥ m . This new family of orthogonal polynomials is orthonormal with respectto the weight function [55] c W ( m ) ( g ) = (1 − g ) α (1 + g ) β P ( − α − ,β − m ( g ) , ( g ≡ g ( x )) , where c W (0) ( g ) = W ( g ) is the weight function for the classical Jacobi polynomials and − ≤ g ( x ) ≤ +1 in order that L (cid:16) g, c W ( m ) ( g )d g (cid:17) -orthonormality holds.Moreover these polynomials are related to the classical Jacobi orthogonalpolynomials P ( α,β ) n ( g ) by the following relations b P ( α,β, n ( g ) = P ( α,β ) n ( g ) , (2.8) b P ( α,β,m ) n ( g ) = ( − m (cid:20) α + β + j + 12( α + j + 1) ( g − P ( − α − ,β − m ( g ) P ( α +2 ,β ) j − ( g )+ α − m + 1 α + j + 1 P ( − α − ,β ) m ( g ) P ( α +1 ,β − j ( g ) (cid:21) , ( j = n − m ≥
0) (2.9)if and only if the following restrictions hold simultaneously [22, 55]( R1 ) β = 0 , α, and α − β − m + 1
6∈ { , , , · · · , m − } , (2.10 a )( R2 ) α > m − , and sgn( α − β + 1) = sgn( β ) , (2.10 b )where sgn( · ) is the signum function. Under these conditions, the scalar product of theexceptional X m Jacobi polynomials leads to orthogonality relation, Z − (1 − g ) α (1 + g ) β h P ( − α − ,β − m ( g ) i b P ( α,β,m ) n ( g ) b P ( α,β,m ) l ( g ) d g = 2 s ( n − m + α + 1)Γ( n + β + 1)Γ( n − m + α + 2)(2 n − m + 2 s )( n − m + α + 1) Γ( n − m + 1)Γ( n − m + 2 s ) δ n,l , (2.11)where n, l ≥ m , 2 s = α + β + 1 and δ n,l is the Kr¨onecker’s symbol.Now for a fixed integer-parameter m ≥ α, β > −
1, the functions Q ( m ) ( g )and R ( m ) ( g ) can be generalized to the PDEM case and expressed in terms of classicalJacobi orthogonal polynomials P ( α,β ) n ( g ) through Q ( m ) ( g ) = ( α − β − m + 1) P ( − α,β ) m − ( g ) P ( − α − ,β − m ( g ) − α − β + ( α + β + 2) g − g , (2.12) R ( m ) ( g ) = β ( α − β − m + 1)1 + g P ( − α,β ) m − ( g ) P ( − α − ,β − m ( g ) + n + n ( α + β − m + 1) − βm − g , (2.13)and substituting (2.12) and (2.13) into (2.6), we get after lengthy but straightforwardcomputation E ( m ) n − V ( m )eff ( x ) = g ′ ( x )4 M ( x ) 2 n ( n − m + α + β + 1) + 2 m ( α − β − m + 1) + α + β + 21 − g ( x ) m EOP, PDEM Scarf I potential and its coherent states revivals − g ′ ( x )8 M ( x ) ( α − β + ( α + β + 2) g ( x )) ( α − β + ( α + β − g ( x ))(1 − g ( x )) + g ′ ( x )2 M ( x ) ( α − β − m + 1)( α + β + ( α − β + 1) g ( x ))1 − g ( x ) P ( − α,β ) m − ( g ( x )) P ( − α − ,β − m ( g ( x )) − g ′ ( x )4 M ( x ) ( α − β − m + 1) " P ( − α,β ) m − ( g ( x )) P ( − α − ,β − m ( g ( x )) + 14 M ( x ) (cid:20) g ′′′ ( x ) g ′ ( x ) − g ′′ ( x ) g ′ ( x ) (cid:21) − M ( x ) (cid:20) M ′′ ( x ) M ( x ) − M ′ ( x ) M ( x ) (cid:21) . (2.14)Equation (2.14) can be solved by choosing an appropriate g ( x ) in order to makethe right-hand side having a constant dependent on n , and considering that theeffective potential should be independent of n . Taking g ′ ( x ) / (1 − g ( x )) = cM ( x ),where c > g ( x ) = sin kµ ( x ) ( k = √ c ) leads to the construction of infinite families of new PDEMHermitian X m Scarf I potential whose effective potentials V ( m )eff ( x ), energy eigenvalues E ( m ) n and eigenfunctions ψ ( m ) n ( x ) are given by V ( m )eff ( x ) = k (cid:0) α + 2 β − (cid:1) sec ϑ ( x ) − k (cid:0) β − α (cid:1) sec ϑ ( x ) tan ϑ ( x ) − k α − β − m + 1) ( α + β + ( α − β + 1) sin ϑ ( x )) P ( − α,β ) m − (sin ϑ ( x )) P ( − α − ,β − m (sin ϑ ( x ))+ k α − β − m + 1) cos ϑ ( x ) " P ( − α,β ) m − (sin ϑ ( x )) P ( − α − ,β − m (sin ϑ ( x )) + 14 µ ′′′ ( x ) µ ′ ( x ) − µ ′′ ( x ) µ ′ ( x ) − k (cid:2) ( α + β + 1) + 4 m ( α − β − m + 1) (cid:3) , (2.15) E ( m ) n ≡ k e ( m ) n = k n ( n − m + α + β + 1) , (2.16) ψ ( m ) n ( x ) = N ( m ) n M / ( x ) (1 − sin ϑ ( x )) α + (1 + sin ϑ ( x )) β + P ( − α − ,β − m (sin ϑ ( x )) b P ( α,β,m ) n (sin ϑ ( x )) , (2.17)where ϑ ( x ) = kµ ( x ), N ( m ) n is the normalization constant and where we introduce theauxiliary mass function µ ′ ( x ) = p M ( x ). We expressly chose to deduce the effectivepotential in its shape given by (2.15) and its associated energy spectra (2.16), so thatthe ground-state energy E ( m )0 is chosen to be zero, in order to construct their associatedcoherent states `a la Gazeau-Klauder in the next section.It is worth noting that the bound-state wavefunctions (2.17) are physicallyacceptable if and only if the square-integrability condition fulfills the restriction | ψ ( m ) n ( x ) | / p M ( x ) → x and x of the interval of the effectivepotential (2.15).Then in order to normalize (2.17), as in the case of usual CM, we mustrequire that the auxiliary mass function µ ( x ) can be restricted to the region − ≤ g ( x ) ≤ +1 ⇒ − π k ≤ µ ( x ) ≤ π k , (2.18) m EOP, PDEM Scarf I potential and its coherent states revivals M ( x ). Thuswith the help of (2.11), the normalized constant N ( m ) n is given by N ( m ) n = k s − / s ( n − m + s ) ( n − m + α + 1) Γ( n − m + 1) Γ( n − m + 2 s )( n − m + α + 1) Γ( n − m + α + 2) Γ( n + β + 1) , where s = ( α + β + 1) / α, β > −
1. The new PDEM potentials (2.15) are infinite,since each m ≥ new exactly solvable PDEM potentials which are allsingular in the interval (2.18), due to the properties of the Jacobi polynomials.For m = 0, we recognize the well-known PDEM Scarf I potential associated to theclassical Jacobi polynomials [20] (see for instance, equation (23b) therein), namely V (0)eff ( x ) = k (cid:0) α + 2 β − (cid:1) sec ϑ ( x ) − k (cid:0) β − α (cid:1) sec ϑ ( x ) tan ϑ ( x ) − k α + β + 1) + 14 µ ′′′ ( x ) µ ′ ( x ) − µ ′′ ( x ) µ ′ ( x ) , (2.19)and all other potentials ( m ≥
1) are considered as extension of V (0)eff ( x ). Then it isobvious to interpret V ( m )eff ( x ) as the PDEM rationally extended
Scarf I potential family.In Figure 1 we have depicted the effective potential V ( m )eff ( x ) given in (2.15) plottedfor two profile of the mass functions, i.e. with and without singularities [59] M wos ( x ) = 11 + ( λx ) , and M ws ( x ) = 1(1 − ( λx ) ) , (2.20)inside the interval (2.18), for even and odd m up to 3 and for different values of themass parameter λ ∈ R .Due to the singularities of M ws ( x ), we observe that gradually as λ increases theshape of V ( m )eff ( x ) tend to gather near the classical turning points x ± = 1 /λ of the well.Contrary, the case M wos ( x ) reveals that V ( m )eff ( x ) are more extended along the whole realline R and become more sharper at the vicinity of x = 0, as λ increases.
3. Supersymmetry and shape-invariant approach
It is well-known that supersymmetric quantum mechanics (SUSY QM) has beensuccessfully applied to obtained exact solutions of Schr¨odinger equation, which dealswith pairs of Hamiltonians H ( m )1 , eff and H ( m )2 , eff that have the same energy spectra, butdifferent eigenfunctions. It has been shown that such pairs of Hamiltonians can beobtained through the concept of shape invariance, which is the sufficient condition for exact solvability and satisfying [16] V ( m )2 , eff ( x |{ a } ) = V ( m )1 , eff ( x |{ a } ) + R ( m ) ( { a } ) , (3.1)where { a i } , ( i = 1 , , · · · ), is a set of parameters, { a i +1 } = f ( { a i } ) is an arbitraryfunction describing the change of parameters and R ( m ) ( { a i } ) is independent of x .If these conditions are fulfilled, then the energy spectra of V ( m )1 , eff ( x ) can be obtainedalgebraically E ( m )1 ,n = n X i =1 R ( m ) ( { a i } ) . (3.2) m EOP, PDEM Scarf I potential and its coherent states revivals Figure 1.
Top (resp. bottom). Plot of the effective potential (2.15) plotted for M wos ( x ) (resp. M ws ( x )) for α = 1 and β = 2: m = 0 (solid line), m = 1 (dashed line), m = 2 (dotted line), and m = 3 (dot-dashed line). In the light of the last section, we introduce a pair of operators b Q m and b Q † m andthe associated superpotential W m ( x ) [59, 60] through b Q m = 1 √ (cid:18) M / ( x ) dd x M / ( x ) + W m ( x ) (cid:19) , (3.3 a ) b Q † m = 1 √ (cid:18) − M / ( x ) dd x M / ( x ) + W m ( x ) (cid:19) , (3.3 b )where the superpotential W m ( x ) is defined in terms of the ground-state wavefunctions ψ ( m ) m ( x ), for n = m , as ψ ( m ) m ( x ) ∼ p M ( x ) exp ( − Z µ ( x ) W m ( η ) d µ ( η ) ) . (3.4)The operators defined in (3.3 a ) and (3.3 b ) give rise to two effective partnerHamiltonians, namely, b H ( m )1 , eff ≡ b Q † m b Q m and b H ( m )2 , eff ≡ b Q m b Q † m , which are given by b H ( m )1 , eff = 12 " − (cid:18) M / ( x ) dd x M / ( x ) (cid:19) + W m ( x ) − W ′ m ( x ) p M ( x ) , (3.5 a ) b H ( m )2 , eff = 12 " − (cid:18) M / ( x ) dd x M / ( x ) (cid:19) + W m ( x ) + W ′ m ( x ) p M ( x ) , (3.5 b )where the relationship connecting the two effective partner potentials is given by V ( m )2 , eff ( x ) = V ( m )1 , eff ( x ) + 1 p M ( x ) W ′ m ( x ) , (3.6)which share the same energy spectrum, except the zero energy state, i.e. E ( m )1 ,n +1 = E ( m )2 ,n and E ( m )1 , = 0, for n = 0 , , , · · · and m = 1 , , , · · · . m EOP, PDEM Scarf I potential and its coherent states revivals V ( m )1 , eff ( x )and the ground-state wavefunction can be deduced straightforwardly from (2.17) ψ ( m ) m ( x ) = N ( m ) m M / ( x ) (1 − sin ϑ ( x )) α + (1 + sin ϑ ( x )) β + P ( − α − ,β − m (sin ϑ ( x )) b P ( α,β,m ) m (sin ϑ ( x )) , (3.7)where using (2.9), the exceptional X m Jacobi polynomials in (3.7) are reduced in termsof the classical Jacobi polynomials as b P ( α,β,m ) m (sin ϑ ( x )) = ( − m (cid:18) − mα + 1 (cid:19) P ( − α − ,β ) m (sin ϑ ( x )) . (3.8)Making use of (3.4), the superpotential W m ( x ) is given through W m ( x ) = 14 M ′ ( x ) M / ( x ) − p M ( x ) dd x ln ψ ( m ) m ( x )= k α − β ) sec ϑ ( x ) + k α + β + 1) sec ϑ ( x ) tan ϑ ( x ) − k α − β − m + 1) cos ϑ ( x ) P ( − α,β ) m − (sin ϑ ( x )) P ( − α − ,β − m (sin ϑ ( x )) − P ( − α − ,β − m − (sin ϑ ( x )) P ( − α − ,β ) m (sin ϑ ( x )) ! (3.9)where ϑ ( x ) = kµ ( x ). Inserting (3.9) into (3.5 a ) and (3.5 b ) and after some lengthy andalgebraic manipulations, we obtain the simplified expression of the effective partnerpotentials V ( m )1 , eff ( x | α, β ) = k (cid:0) α + 2 β − (cid:1) sec ϑ ( x ) − k (cid:0) β − α (cid:1) sec ϑ ( x ) tan ϑ ( x ) − k α − β − m + 1) ( α + β + ( α − β + 1) sin ϑ ( x )) P ( − α,β ) m − (sin ϑ ( x )) P ( − α − ,β − m (sin ϑ ( x ))+ k α − β − m + 1) cos ϑ ( x ) " P ( − α,β ) m − (sin ϑ ( x )) P ( − α − ,β − m (sin ϑ ( x )) − k (cid:2) ( α + β + 1) + 4 m ( α − β − m + 1) (cid:3) , (3.10 a ) V ( m )2 , eff ( x | α, β ) = k (cid:0) α + 1) + 2( β + 1) − (cid:1) sec ϑ ( x ) − k (cid:0) ( β + 1) − ( α + 1) (cid:1) sec ϑ ( x ) tan ϑ ( x ) − k α − β − m + 1) ( α + β + 2 + ( α − β + 1) sin ϑ ( x )) P ( − α − ,β +1) m − (sin ϑ ( x )) P ( − α − ,β ) m (sin ϑ ( x ))+ k α − β − m + 1) cos ϑ ( x ) " P ( − α − ,β +1) m − (sin ϑ ( x )) P ( − α − ,β ) m (sin ϑ ( x )) − k (cid:2) ( α + β + 1) + 4 m ( α − β − m + 1) (cid:3) . (3.10 b )We observe that the effective potential (3.10 a ) matches with (2.15), apart from theomission of the Schwartz’s derivative of the mass function due to our particular choiceof the operators b Q m and b Q † m in (3.3 a ) and (3.3 b ), respectively. Using (3.1), it is easy m EOP, PDEM Scarf I potential and its coherent states revivals a ) and (3.10 b ) are connected to each other through the translational shape invariant symmetry , namely V ( m )2 , eff ( x | α, β ) = V ( m )1 , eff ( x | α + 1 , β + 1) + R ( m ) ( α, β )= V ( m )1 , eff ( x | α + 1 , β + 1) + k α + β − m + 2) , (3.11)where here the set of parameters { a i } are defined by: { a } = ( α, β ), { a } = ( α +1 , β +1),and thus { a n } = ( α + n − , β + n − V ( m )1 , eff ( x ) are then given by E ( m )1 ,n = n X i =1 R ( m ) ( { a i } )= n X i =1 k α + β − m + 2 i )= k n ( n + α + β − m + 1) , (3.12)which are just the energy eigenvalues deduced in (2.16), with the fact that E ( m )1 , = 0 asit was expected.
4. Revival dynamics of Gazeau-Klauder CS for the extended PDEM Scarf Ipotential
Let us now adapt the material developed above to construct the Gazeau-Klauder CS(GK CS) [53] for the extended Hermitian PDEM Scarf I potential given in (2.15). Suchcoherent states are parameterized by two real parameters J and γ , and are defined by | ξ ( m ) ( x ; J, γ ) i = 1 N m ( J ) ∞ X n =0 J n/ exp n − i e ( m ) n γ oq ρ ( m ) n | ψ ( m ) n ( x ) i , (4.1)where γ = ωt . Here e ( m ) n are the dimensionless non-degenerate energy eigenvalues(2.16), satisfying e ( m ) n +1 > e ( m ) n > e ( m ) n − > · · · > e ( m )0 (2.16) = 0 , ( m ≥ ρ ( m ) n denotes the moments of probability distribution defined by ρ ( m ) n = Q ni =1 e ( m ) n , with ρ ( m )0 = 1. The last parameter appearing in (4.1) is the normalization constant given by N m ( J ) = ∞ X n =0 J n ρ ( m ) n ! / , (4.2)which is deduced from the normalization condition h ξ ( m ) ( x ; J, γ ) | ξ ( m ) ( x ; J, γ ) i = 1, where0 < J < R = lim n → + ∞ sup n q ρ ( m ) n and R denotes the radius of convergence. Under theseconsiderations, the moments ρ ( m ) n and the squared of normalization constant N m ( J ) aregiven by ρ ( m ) n = n !2 n Γ( n + 2 σ + 1)Γ(2 σ + 1) , (4.3) N m ( J ) = (2 J ) − σ Γ(2 σ + 1) I σ (2 √ J ) , (4.4) m EOP, PDEM Scarf I potential and its coherent states revivals R = ∞ . Here R ∋ σ = s − m and I σ ( · ) are the modified Bessel functions of the first kind [57].So that, the Gazeau-Klauder CS (4.1) are reduced to | ξ ( m ) ( x ; J, t ) i = (2 J ) σ/ q I σ (2 √ J ) ∞ X n =0 (2 J ) n/ exp n − i ω n ( n +2 σ )2 t op n ! Γ( n + 2 σ + 1) | ψ ( m ) n ( x ) i , (4.5)where ψ ( m ) n ( x ) are given by (2.17). It is well known that the concept of quantum revivalsarises from the weighting probabilities | c ( m ) n | for the general wave-packet, i.e., | Ψ ( m ) n ( x, t ) i = ∞ X n =0 c ( m ) n | ψ ( m ) n ( x ) i , (4.6)where P ∞ n =0 | c ( m ) n | = 1. So, when | Ψ ( m ) n ( x, t ) i in (4.6) play the role of our Gazeau-Klauder CS (4.5), then the weighting distribution depends on J as | c ( m ) n ( J ) | ≡ J n N m ( J ) ρ ( m ) n = (2 J ) n + σ n ! Γ( n + 2 σ + 1) I σ (2 √ J ) . (4.7) Figure 2.
Plot of the weighting distribution | c ( m ) n ( J ) | against n for α = 1 and β = 2,where m = 0 (solid line), m = 1 (dashed line), and m = 2 (dotted line). In Figure 2 we display the curves of | c ( m ) n ( J ) | as a function of quantum number n for various values of J and m . It is clear that all frames show a Gaussian-shapedfunction for the weighting distribution. We can observe that gradually as J increases,the weighting distributions become more and more stretched and are less peaked witha slight shift to the right localized around a mean value n ( m ) ≃ h n ( m ) i .On the other hand, the mean and the variance values of the number operator b N m are used to characterize the statistical features of the quantum system, whichcan be evaluated by means of the moments of probability. By making use of (4.7),a straightforward analytical calculation yields h n i ≡ ∞ X n =0 n | c ( m ) n ( J ) | = √ J I σ +1 (2 √ J ) I σ (2 √ J ) , (4.8 a ) h n i ≡ ∞ X n =0 n | c ( m ) n ( J ) | = 2 J I σ +2 (2 √ J ) I σ (2 √ J ) + √ J I σ +1 (2 √ J ) I σ (2 √ J ) , (4.8 b )in order to display the Mandel parameter Q ( m )M ( J ) Q ( m )M ( J ) ≡ h n i − h n i h n i − √ J I σ +2 (2 √ J ) I σ +1 (2 √ J ) − I σ +1 (2 √ J ) I σ (2 √ J ) ! . (4.9) m EOP, PDEM Scarf I potential and its coherent states revivals Q ( m )M ( J ) = 0 coincides with the definition of CS, while for Q ( m )M ( J ) < Q ( m )M ( J ) > Figure 3.
The Mandel parameter Q ( m )M ( J ) given by (4.9) against J for fixed value for β and varying α satisfying the conditions (4.10). The parameter m refers to: m = 0(solid line) and m = 1 (dashed line). In Figure 3 we display the behavior of the Mandel parameter (4.9) against J interms of different values of the parameters α , β and m = 0 ,
1. Solving simultaneously(2.10 a ) and (2.10 b ) for m = 1 give rise to four different cases depicted in Figure 3 byletters (A), (B), (C), and (D), respectively, i.e.,( A ) : − < β < , α > , and α < β (4.10 a )( B ) : − < β < , α ≤ , and α > β (4.10 b )( C ) : β > , α < β, and α ≥ c )( D ) : β > , α > β (4.10 d )keeping in mind that α, β > − m = 0 (solid lines), representing the classical Jacobi m EOP, PDEM Scarf I potential and its coherent states revivals α and β . However the case m = 1 (dashed lines),corresponding to the exceptional X Jacobi polynomials, has a completely differentbehavior compared to that of the case m = 0. At this stage, a few remarks are worthmentioning:(i) For the fixed value β = − , the state in A(a) starts with a super-Poissonianbehavior for a short range in J and becomes sub-Poissonian for J ≃ .
2. Graduallyas α decreases the trend reverses, and a kind of transition takes place so that thestates in A(c) (resp. A(d)) start at slightly sub-Poissonian, increase to super-Poissonian at J ≃ .
05 (resp. J ≃ .
10) and then decrease very fast to becomesub-Poissonian at J ≃ .
25 (resp. J ≃ . β = − , it is found that the states B(a) (resp. B(b)) acquire at the beginning asub-Poissonian behavior for a short range of J , become super-Poissonian at J ≃ . J ≃ . J ≃ . J ≃ . J >
0, and α, β > − α and β .However one has no rigorous answer to this remark.(v) We noticed that the transition quoted at the first point (i) occurs if and only if therestriction | α + β | = 1 holds, as shown in the frames A(b) and B(c).(vi) Finally, we observed in all frames that the behavior of the Mandel parameter reaches − as J → ∞ .The last observation can be mathematically explained using the asymptotic forms,as J → ∞ , of the modified Bessel functions I ν ( z ) (see the identity 14.143, pp. 693 of[57]) I ν ( z ) = e z √ πz ( P ν (i z ) − i Q ν (i z )) , (4.11)valid for − π/ < arg z < π/
2, where P ν (i z ) and Q ν (i z ) are defined through P ν (i z ) ∼ − (4 ν − ν − z ) + (4 ν − ν − ν − ν − z ) − · · · , Q ν (i z ) ∼ ν − z ) − (4 ν − ν − ν − z ) + · · · . For larger z ≡ √ J , i.e. as J → ∞ , the first terms in P ν (i z ) and Q ν (i z ) dominate,and thus it is convenient to rewrite (4.11) as I ν (2 √ J ) ∼ e √ J p π √ J (cid:18) − ν − √ J (cid:19) , m EOP, PDEM Scarf I potential and its coherent states revivals ν = (2 σ, σ + 1 , σ + 2) in the last identity, (4.9) becomes Q ( m )M ( J ) ∼ − − (4 σ + 1)(4 σ + 3)64 r J , ( σ = s − m ) , where in the limit J → ∞ , the second term in the last expression can be neglected andone is left with the Mandel parameter which tends to − , for ∀ σ ∈ R , as illustrated inFigure 3.As a prerequisite for obtaining quantum revivals, it has been shown that a coherentstate wave-packet of the form of (4.6) mimics quantum revivals, T ( m )rev = 4 π/ | e ′′ ( m ) n | ,and fractional revivals τ = ( p/q ) T rev , in which p and q are coprime integers, besidesthe classical timescale T ( m )cl = 2 π/ | e ′ ( m ) n | if and only if they are strongly well localizedaround a mean value n ≃ h n i . This means that we can expand the energy eigenvalues(2.16) in Taylor series in n as e ( m ) n = 12 " n ( n + 2 σ ) + 4 πT ( m )cl ( n − n ) + 2 π ( n + σ ) T ( m )cl ( n − n ) , (4.12)with k = 1 and the timescales are given by T ( m )cl = 2 π/ ( n + σ ) and T ( m )rev = 4 π, ∀ m ≥ | ξ ( m ) ( x ; J, t ) i = (2 J ) σ/ q I σ (2 √ J ) e − i ω n (cid:16) π + σ T ( m )cl (cid:17) t × ∞ X n =0 (2 J ) n/ exp (cid:8) − i ω π ( n − n ) (cid:0) n − nn + σ (cid:1) t (cid:9)p n ! Γ( n + 2 σ + 1) | ψ ( m ) n ( x ) i , (4.13)where t = t/T ( m )cl .Generally, the autocorrelation function, A ( m ) ( t ) = h ξ ( m ) ( x ; J, | ξ ( m ) ( x ; J, t ) i , andthe probability density of the time-evolved coherent state wave-packet are considered aswidely used techniques for describing and reproducing revival structures. To this endusing the product series relation of [58], the absolute square of A ( m ) ( t ) and theprobability density of (4.13) yield | A ( m ) ( t ) | = (cid:18) (2 J ) σ I σ (2 √ J ) (cid:19) ∞ X n =0 n X l =0 (2 J ) n exp n − i ω π ( n − l )( n +2 σ ) n + σ t o l ! ( n − l )! Γ( l + 2 σ + 1) Γ( n − l + 2 σ + 1) , (4.14) | ξ ( m ) ( x ; J, t ) | = (2 J ) σ I σ (2 √ J ) × ∞ X n =0 n X l =0 (2 J ) n/ exp n − i ω π ( n − l )( n +2 σ ) n + σ t op l ! ( n − l )! Γ( l + 2 σ + 1) Γ( n − l + 2 σ + 1) (cid:16) ψ ( m ) l ( x ) (cid:17) ∗ ψ ( m ) n − l ( x ) , (4.15)where the eigenfunctions ψ ( m ) n ( x ) are given in (2.17).Both expressions (4.14) and (4.15) are attributed to the exceptional X m PDEMScarf I potential and we use them to study their revival dynamics. However, it isobvious that all these results will reduce to those of the usual constant mass (CM) if weset M ( x ) = 1, i.e., µ ( x ) = x . At this stage our strategy in the remainder of this paper m EOP, PDEM Scarf I potential and its coherent states revivals X m PDEM system, with an appropriate choice of the mass function M ( x ),to see what general conclusions can be made.Throughout our results, we work in atomic units (a.u.), i.e., ~ = e = m = ω = 1,for which the conversions give 1 a . u . ≃ . × − m for lengths and 1 a . u . ≃ . × − sec for times. For convenience we also set k = 1. According to the discussion above, coherent state wave-packet of the Scarf I potentialhave perfect full revivals since their energy spectrum is quadratic in the quantum number n . We illustrate this by plotting the absolute square of A ( m ) ( t ) as a function of t = t/T ( m )cl and the modulus square of ξ ( m ) ( x ; J, t ) against x . Figure 4.
The modulus square of autocorrelation function A ( m ) ( t ) against t = t/T ( m )cl plotted for α = 3 / β = 5 / n max = 50. Here the parameter m refers to: m = 0 (redline), m = 1 (blue line) and m = 2 (green line). m EOP, PDEM Scarf I potential and its coherent states revivals | A ( m ) ( t ) | for J = 10, 20, 40, and 80, with α = , β = , n max = 50 (i.e., 50th excited state) for the classical ( m = 0) andexceptional ( m = 1 ,
2) PDEM Scarf I potential. The timescales are T ( m )rev = 4 π and T ( m )cl = 2 π/ ( n + σ ). As we can see, the usual CM involves a perfect full revivals as it wasexpected above and the sharp peaks arise due to the fractional revivals which becomemore apparent as J increases. However some permanent peaks still exist, no matter thevalues attributed to J and m , like those of T rev / T rev /
3, for all m = 0 , ,
2, witha change in the width and magnitude. We also observe that as J increases all curvesmerge into a single one, as can be seen in Figure 4(d). Figure 5.
Plot of | ξ ( m ) ( x ; J, t ) | against x for J = 20, plotted for α = 1, β = 2, and n max = 50. As for Figure 4, the parameter m refers to: m = 0 (red line) and m = 1(blue line). On the other hand we plot in Figure 5 the probability density of a coherent statewave-packet | ξ ( m ) ( x ; J, t ) | for J = 20, α = 1 , β = 2 and m = 0 ,
1. It is evident thatthe probability density is reconstructed after the time revival T ( m )rev as it was illustratedin the frame (d) compared to (a). In the first frame, t = 0, we observe two principaland symmetrical peaks for both cases, ( m = 0 , x ≃ .
25 and x ≃ m = 0 before and after dominant peaks. Allthese phenomena are periodic in the whole real line with a period δx = 2 π , being thebehavior for the case where the mass is constant. With time, our numerical simulationsshow us that these peaks oscillate back and forth between the walls of the well withrelative change in shape. Even as (4.14) and (4.15) are applicable for the exceptional X m PDEM Scarf I potential,two different profiles of the mass function, introduced in section 2, without and withsingularities were chosen M wos ( x ) = 11 + ( λx ) , and M ws ( x ) = 1(1 − ( λx ) ) , (4.16)respectively. These mass functions allow us to construct a coherent state wave-packetwith the proper behavior near the boundaries and have been used in many studies, seefor instance [59]. m EOP, PDEM Scarf I potential and its coherent states revivals M wos ( x ) without singularities. We take the profile of the massfunction to be of the form of M wos ( x ). This profile is without singularities andis a bounded function defined in the whole real line R , where its maximum value, M (max)wos ( x ) = 1, is reached at x = 0 and vanishing as | x | → ∞ . In this case the auxiliarymass function is calculated by a simple integration, which gives µ wos ( x ) = λ arcsinh( λx ).In Figure 6, we plot the probability density of a coherent state wave-packet endowedwith an effective mass function M wos ( x ) for different values of the mass parameter λ ,taking into account (2.18), i.e., | x | ≤ λ sinh (cid:0) πλ (cid:1) . The revival time is T ( m )rev = 4 π and we observe that all | ξ ( m ) ( x ; J, t ) | are restored after the time revival as they arepresented in the frames A-C(d). With the presence of the mass function, we can seethat the dependence of the mass on the position x affects wholly the behavior of coherentstate wave-packets inside the corresponding wells through two ways: firstly we observethat, although full revivals take place during time evolution T ( m )rev , there is no trace offractional revivals in the common sense on the opposition of the usual CM discussedabove. Secondly the symmetrical peaks of the Figure 5 do not occur in the case ofmass M wos ( x ), instead we observe only one peak, for each cases m = 0 ,
1, with the samewidth than the usual which is nearly zero at the left of the well and the formation ofripple before it. As time evolves we observe a well localized coherent state wave-packetoscillating back and forth between the walls, with the presence of two asymmetricalpeaks at T ( m )rev / λ increases, a coherent state wave-packet spreads on the wholereal line inside the region delimited by the walls of the potential well. This behavior isdue essentially to the features of the mass function. M ws ( x ) with two singularities. The second mass function ischosen to be of the form of M ws ( x ) with two singularities and defined in dom( M ws ) =( − /λ, +1 /λ ). The mass function rapidly grows near the classical turning points x ± = ± /λ and reaches its minimum at x = 0. The associated auxiliary mass functionis given by µ ws ( x ) = λ arctanh( λx ).In Figure 7 we display the probability density of a coherent state wave-packet withthe mass function M ws ( x ) for different values of λ , with the restriction (2.18) given by | x | ≤ λ tanh (cid:0) πλ (cid:1) . With this profile at hand, an analogous temporal evolution takesplace in the Figure 7(A) compared to those of Figure 6(A). This is essentially due to thefact that both hyperbolic functions ”sinh” and ”tanh” behave in the same manner as λ approaches zero and one can say that the same quantitative comments can be madein the first case with an exception that a new phenomenon is observed here. Contraryto the case of the mass M wos ( x ), a coherent state wave-packet in frames (B) and (C)becomes more peaked and tends to gather near the classical turning points x ± = ± /λ of the well due to singularities of the mass function and progressively becomes moresharper than the usual ones as λ increases. We see also that the amplitude of theprobability density of a coherent state wave-packet rapidly grows as one approaches theclassical turning points. It is clear that the presence of singularities in the mass function m EOP, PDEM Scarf I potential and its coherent states revivals Figure 6.
Plot of | ξ ( m ) ( x ; J, t ) | against x for the profile of the mass function M wos ( x ),with α = 1, β = 2, J = 20, n max = 50. The mass parameters λ are: (A) λ = 0 .
25, (B) λ = 1, and (C) λ = 2, and the parameter m refers to: m = 0 (red line) and m = 1(blue line). restrict the time evolution of a coherent state wave-packet inside the domain determinedby the turning points x ± , once again this is due to the features of the mass function.We end our analysis by showing the time evolution of the probability densityof a coherent state wave-packet | ξ ( m ) ( x ; J, t ) | for the exceptional X m PDEM Scarf Ipotential, characterized by not-equally spaced eigenenergies (2.16). It is well-known,as it was exposed in the pedagogical paper of Gutschick and Nieto [54], that for asystem with a such eigenenergies, coherent state wave-packets will dissipate and losetheir coherence in time. Thus the concept of coherence is discussed in our paper interms of classical period T ( m )cl = 2 π/ ( n ( m ) + σ ) and defined as follow [54]: the moreeigenstates ψ ( m ) n ( x ) have a significant overlap with the PDEM Gazeau-Klauder CS inbetween the walls of the potential V ( m )eff ( x ) , the longer will be the coherence time τ ( m )coh .To this end, Figure 8 and Figure 9 display the time evolution of probability densitiesof a PDEM coherent state wave-packet (4.15) for m = 0 and m = 1, respectively, forthe profile mass function M wos ( x ), given in (4.16), over one classical period T ( m )cl for λ = 0 . ,
2, and 4 in the case J = 20. All frames are taken as th of one classical periodand show: (i) | ξ ( m ) ( x ; J, t ) | (solid curves), (ii) the associated potential (dashed curves),and (iii) a vertical dotted line indicating the potential minimum. Case m = 0 . Numerical simulations show us that the exceptional X PDEM coherentstate wave-packet, associated to the classical Jacobi polynomials, starts theirmovement to the right of the potential minimum and oscillates back and forth inside m EOP, PDEM Scarf I potential and its coherent states revivals Figure 7.
Plot of | ξ ( m ) ( x ; J, t ) | against x for the profile of the mass function M ws ( x ),with α = 1, β = 2, J = 20, n max = 50. The mass parameters λ are: (A) λ = 0 .
25, (B) λ = 1, and (C) λ = 2, and m refers to: m = 0 (red line) and m = 1 (blue line). Figure 8.
Time evolution of the probability density of a coherent state wave-packetfor the mass function M wos ( x ) for m = 0, with α = 1, β = 2, J = 20, and n max = 50.The mass parameters λ are: (A) λ = 0 .
5, (B) λ = 2, and (C) λ = 4. the well. We observe in Figure 8(A) a shorter coherence time in terms of classicalperiod which means that, for λ = 0 .
5, the system loses its coherence quickly as itwas represented in the frame A(b). In Figure 8(B), gradually as λ increases, λ = 2,a coherent state wave-packet continues to lose its coherence but slowly comparedto the previous case. In both cases, we see that the coherence time is less than oneclassical period, i.e., τ (0)coh < T (0)cl ≃ . h n (0) i ∼ n (0) ≃ . m EOP, PDEM Scarf I potential and its coherent states revivals Figure 9.
Time evolution of the probability density of a coherent state wave-packetfor the mass function M wos ( x ) for m = 1, with α = 1, β = 2, J = 20, and n max = 50.The mass parameters λ are: (A) λ = 0 .
5, (B) λ = 2, and (C) λ = 4. However in Figure 8(C), corresponding to λ = 4, we observe that PDEM coherentstate wave-packets are more peaked at the vicinity of the potential minimum andflatten out very quickly to reach zero at the right of the potential well. Theeigenstates and PDEM CS wave-packets overlap very well, and these phenomena arethe signature of a longer coherence time, at least longer than one classical period,i.e., τ (0)coh > T (0)cl . Case m = 1 . In Figure 9(A), corresponding to λ = 0 .
5, the remark which should beemphasized is that we observe a strong loss of coherence since, due to the definitionherein above, eigenstates which are outside of the well do not overlap with thePDEM CS wave-packets. However, the slow loss of coherence in Figure 9(B) isvery similar to that of Figure 8(B), with τ (1)coh < T (1)cl ≃ . h n (1) i ∼ n (1) ≃ . λ and we finish before our conclusion with this observation: a larger massparameter λ contributes significantly to a longer coherence time τ ( m )coh .
5. Conclusion
In this paper our primary concern is to investigate how the mass function, representedhere by the parameter λ , can affect the revival structure of an arbitrary quantumsystem. To this end we have constructed the PDEM Gazeau-Klauder coherent statesfor the exceptional X m Scarf I potential endowed with PDEM, where their statisticaland dynamical properties have been studied. We have shown that these potentials are m EOP, PDEM Scarf I potential and its coherent states revivals m = 0). In particular, for the usual CM,we have constructed full and fractional revivals with the help of the autocorrelationfunction. However, in the case of PDEM, things are completely different from the usualcase and our results agree with those obtained by Schmidt in [51]. We have observedthat, although full revivals still take place during their time evolution T ( m )rev , there is notrace of fractional revivals in the common sense, on the opposition to the usual. Insteadof these effects, we have obtained bell-shaped coherent state wave-packets located inthe right of the well, oscillating back and forth between the walls. We have concludedthat not only quantum revivals are different and affected but also depend closely on theprofile of the mass function. In this context two profiles were chosen, with and withoutsingularities, to illustrate numerically the dynamic of their revival structures.We have also observed that for a longer coherence time τ ( m )coh , defined here in thesense of a slow loss of coherence, corresponds a larger mass parameter λ . Then, wesuspect the effect that λ affects considerably τ ( m )coh , which leads the state to lose itscoherence in time more (resp. less) rapidly as λ becomes smaller (resp. larger). References [1] Harrison P 2005
Quantum Wiles, Wires and Dots. Theoretical and Computational Physics ofSemiconductior Nanostructures (New York: John Wiley & Sons, LTD)[2] Bastard G 1998
Wave Mechanics Applied to Semiconductor Heterostructures (France: les ´Editionsde Physique, les Ullis)[3] Ring P and Schuck P 1980
The Nuclear Many Boby Problems (New York: Springer)[4] Quesne C and Tkachuk V M 2004
J. Phys. A: Math. Gen. Phys. Lett. A
J. Phys. A: Math. Theor. J. Math. Phys. J. Phys. A: Math. Theor. J. Math. Phys. J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. J. Math. Phys. (submitted)[13] Chen Z D and Chen G 2006
Phys. Scr. Phys. Scr. Factorization Methods in Quantum Mechanics (Dordrecht: Springer)[16] Cooper F, Khare A and Sukhtame U 2001
Supersymmetry and Quantum Mechanics (Singapore:World scientific)[17] Iachello F 2006
Lie Algebra and Applications , Lect. Notes Phys. 708 (Berlin: Springer)[18] Yahiaoui S-A and Bentaiba M 2009
Int. J. Theor. Phys. J. Phys. A: Math. Theor. Commun. Theor. Phys. Classical and Quantum Orthogonal Polynomials in OneVariable (Cambridge: Cambridge Univ. Press)[22] G´omez-Ullate D, Kamran N and Milson R 2009
J. Math. Anal. Appl.
J. Approx. Theor. m EOP, PDEM Scarf I potential and its coherent states revivals [24] Quesne C 2008 J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. SIGMA J. Math. Phys. SIGMA Contemp. Math.
SIGMA Found. Comput. Math. J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. J. Math. Phys. J. Phys. A: Math. Theor. J. Math. Phys. J. Phys.: Conf. Ser.
Phys. Lett. A
Phys. Rep.
Am.J. Phys. Phys. Rev. B Phys. Rev. B Phys. Rev. A New J. Phys. New J. Phys. J. Phys.: Condens. Matter Phys. Rev. A Phys. Lett. A
Phys. Rev. D Phys. Lett. A
Phys. Scr. Coherent states in Quantum Mechanics (Berlin: Wiley-VCH)[54] Gutschick V P and Nieto M M 1980
Phys. Rev. D J. Phys. A: Math. Theor. Phys. Rev. B
Mathematical Methods for Physicists. AComprehensive Guide , 7th. edition (New York: Academic Press)[58] Gradshteyn I S and Ryzhik I M 2007
Table of Integrals, Series and Products (New York: AcademicPress)[59] Cruz y Cruz S, Negro J and Nieto L M 2007
Phys. Lett. A
J. Phys. A: Math. Gen.38