Ultradistributions and The Fractionary Schrödinger Equation
aa r X i v : . [ m a t h - ph ] M a r Ultradistributions and TheFractionary Schr¨odinger Equation ∗ A. L. De Paoli and M. C. RoccaDepartamento de F´ısica, Fac. de Ciencias Exactas,Universidad Nacional de La Plata.C.C. 67 (1900) La Plata. Argentina.December 15, 2009
Abstract
In this work, we generalize the results of Naber about the Frac-tionary Schr¨odinger Equation with the use of the theory of TemperedUltradistributions. Several examples of the use of this theory are ∗ This work was partially supported by Consejo Nacional de Investigaciones Cient´ıficas; Argentina. iven. In particular we evaluate the Green’s function for a free parti-cle in the general case.PACS: 03.65.-w, 03.65.Bz, 03.65.Ca, 03.65.Db. Introduction
The properties of ultradistributions (ref.[6, 7]) are well adapted for their usein fractional calculus. In this respect we have shown that it is possible (ref.[2])to define a general fractional calculus with the use of them.Ultradistributions have the advantage of being representable by means ofanalytic functions. So that, in general, they are easier to work with them.They have interesting properties.One of those properties is that Schwartztempered distributions are canonical and continuously injected into tem-pered ultradistributions and as a consequence the Rigged Hilbert Space withtempered distributions is canonical and continuously included in the RiggedHilbert Space with tempered ultradistributions.Fractional calculus has found motivations in a growing area concerninggeneral stochastic phenomena.These include the appearance of alternativediffusion mechanisms other than Brownian, as well as classical and quantummechanics formalisms including dissipative forces, and therefore allowing anextension of the quantization schemes for non-conservative systems [3]. Inparticular it is interesting to study the fractional Schr¨odinger equation. Ouraim is to extend a previous study [1] about this equation. Using an analyticaldefinition of fractional derivative [2] we show here that it is possible to obtain3 general solution for the time fractional equation, for any complex value ofthe derivative index. Furthermore the associated Green functions can beevaluated in a straightforward way.This paper is organized as follow:In section 2 we define the fractional Schr¨odinger equation for all ν complexwith the use of the fractional derivative defined via the theory of temperedultradistributions. In section 3 we solve this equation for the free particle andgive three examples: ν = = and ν = . In section 4 we realize thetreatment of the potential well and we analyze the cases ν = = and ν = . In section 5 we study the Green fractional functions for the free parti-cle in three cases: the retarded Green function, the advanced Green functionand the Wheeler-Green function. As an example we prove that for ν = these functions coincide with the Green functions of usual Quantum Me-chanics.In section 6 we discuss the results obtained in the previous sections.Finally we have included three appendixes: a first appendix on distributionsof exponential type, a second appendix on tempered ultradistributions anda third appendix on fractional calculus using ultradistributions.4 The Fractional Schr¨odinger Equation
Our starting point in the study of the fractional Schr¨odinger equation is thecurrent known Schr¨odinger equation: i ~ ∂ t ψ ( t, x ) = − ~
2m ∂ ψ ( t, x ) + V ( x ) ψ ( t, x ) (2.1)According to ref.[1], (2.1) can be writen as: iT p ∂ t ψ ( t, x ) = − L M p
2m ∂ ψ ( t, x ) + V ( x ) E p ψ ( t, x ) (2.2)where L P = p G ~ /c , T p = p G ~ /c , M p = p ~ c/G and E p = M p c .If we define N m = m/M p and N v = V/E p we obtain for (2.2) iT p ∂ t ψ ( t, x ) = − L m ∂ ψ ( t, x ) + N v ψ ( t, x ) (2.3)By analogy with ref.[1] we define the fractional Schr¨odinger equation for all ν complex as: ( iT p ) ν ∂ νt ψ ( t, x ) = − L m ∂ ψ ( t, x ) + N v ψ ( t, x ) (2.4)where the temporal fractionary derivative is defined following ref.[2] (seeAppendix III) 5 The Free Particle
From (2.4) for the free particle the fractionary equation is: ( i∂ t ) ν ψ ( t, x ) + L νp N m ∂ ψ ( t, x ) = (3.1)By the use of the Fourier transform (complex in the temporal variable andreal as usual in the spatial variable) the corresponding equation is (see Ap-pendix II and ref.[2]) k ν0 − L νp N m k ! ^ ψ ( k , k ) = b ( k , k ) (3.2)whose solution is: ^ ψ ( k , k ) = b ( k , k ) k ν0 − L νp N m k (3.3)and in the configuration space (anti-transforming) ψ ( t, x ) = I Γ ∞ Z − ∞ a ( k , k ) k ν0 − L νp N m k e − i ( k t + kx ) dk dk (3.4)where: a ( k , k ) = b ( k , k ) We proceed to analyze solutions of (3.4) for some typical cases in the followingsection. 6 xamples
As a first example we consider the case ν = Let α be given by: α = L p N m (3.5)From (3.4) we obtain ψ ( t, x ) = I Γ ∞ Z − ∞ a ( k , k ) k − αk e − i ( k t + kx ) dk dk (3.6)or equivalently: ψ ( t, x ) = ∞ Z − ∞ a ( k ) e − i ( α k t + kx ) dk + Z − ∞ ∞ Z − ∞ a ( k , k ) " ( k + i0 ) − αk − ( k − i0 ) − αk e − i ( k t + kx ) dk dk (3.7)where: a ( k ) = − a ( α k , k ) With some of algebraic calculus we obtain for (3.7): ψ ( t, x ) = ∞ Z − ∞ a ( k ) e − i ( ω t + kx ) dk + ∞ Z ∞ Z − ∞ a ( k , k ) k + ω e i ( k t − kx ) dk dk (3.8)7ith: ω = αk and where we have made the re-scaling: − a (− k , k ) → a ( k , k ) The first term in (3.8) represent free particle on-shell propagation and thesecond term describes the contribution of off-shell modes.As a second example we consider the case ν = .In this case (3.4) takes the form: ψ ( t, x ) = I Γ ∞ Z − ∞ a ( k , k ) k − ω e − i ( k t + kx ) dk dk (3.9)Evaluating the integral in the variable k we have: ψ ( t, x ) = ∞ Z − ∞ a ( k ) e − i ( ωt + kx ) dk (3.10)where a ( k ) = − ( ω, k ) . Thus we recover the usual expression for thefree-particle wave function.Finally we consider the case ν = . For it we have ψ ( t, x ) = I Γ ∞ Z − ∞ a ( k , k ) k − ω e − i ( k t + kx ) dk dk (3.11)After to perform the integral in the variable k we obtain from (3.11): ψ ( t, x ) = ∞ Z − ∞ a ( k ) e − i ( ωt + kx ) + b + ( k ) e i ( ωt + kx ) dk (3.12)8ith a ( k ) = − ( ω, k ) and b + ( k ) = − (− ω, − k ) We consider in this section the potential well. The fractionary equation fora particle confined to move within interval ≤ x ≤ a is: ( iT p ) ν ∂ νt ψ ( t, x ) = − L m ∂ ψ ( t, x ) (4.1)To solve this equation we use the method of separation of variables. Thus ifwe write: ψ ( t, x ) = ψ ( t ) ψ ( x ) (4.2)As is usual we obtain: ( iT p ) ν ∂ νt ψ ( t ) ψ ( t ) = − L m ∂ ψ ( x ) ψ ( x ) = λ (4.3)Then we conclude that ψ ( x ) satisfies: ∂ ψ ( x ) + m L ψ ( x ) = (4.4)The solution of (4.4) is the habitual one: ψ ( x ) = b n sin (cid:16) nπa x (cid:17) (4.5)9ith: λ n = m (cid:18) nπL p a (cid:19) (4.6)and the boundary conditions satisfied by ψ ( x ) are: ψ ( ) = ψ ( a ) = As a consequence of (4.3),(4.5) and (4.6) the Fourier transform ^ ψ ( k ) of ψ ( t ) should be satisfy: ( k ν0 − λ n ) ^ ψ ( k ) = (4.7)whose solution is: ^ ψ ( k ) = c n ( k ) k ν0 − λ n (4.8)Therefore the final general solution for ψ ( t, x ) is: ψ ( t, x ) = ∞ X n = sin (cid:16) nπa x (cid:17) I Γ a n ( k ) e − ik t k ν0 − λ n dk (4.9)where we have defined: a n ( k ) = b n c n ( k ) which is an entire analytic function of k .10 xamples As a first example we consider the case ν = . For it the solution (4.9)takes the form: ψ ( t, x ) = ∞ X n = sin (cid:16) nπa x (cid:17) I Γ a n ( k ) e − ik t k − λ n dk (4.10)or equivalently: ψ ( t, x ) = ∞ X n = a n sin (cid:16) nπa x (cid:17) e − iλ t + ∞ X n = sin (cid:16) nπa x (cid:17) Z − ∞ " ( k + i0 ) − λ n − ( k − i0 ) − λ n × a n ( k ) e − ik t dk (4.11)After performing some algebra we have for (4.11) the expression: ψ ( t, x ) = ∞ X n = a n sin (cid:16) nπa x (cid:17) e − iλ t + ∞ X n = sin (cid:16) nπa x (cid:17) ∞ Z a n ( k ) k + λ e − ik t dk (4.12)Analogously as before, the second term in (4.12) represents of-shell stationarymodes.As a second example we consider ν = . In this case: ψ ( t, x ) = ∞ X n = sin (cid:16) nπa x (cid:17) I Γ a n ( k ) e − ik t k − λ n dk (4.13)11erforming the integral in the variable k we have: ψ ( t, x ) = ∞ X n = a n sin (cid:16) nπa x (cid:17) e − iλ n t (4.14)Which is the familiar general solution for the infinite well.Finally for ν = : ψ ( t, x ) = ∞ X n = sin (cid:16) nπa x (cid:17) I Γ a n ( k ) e − ik t k − λ n dk (4.15)and after to compute the integral: ψ ( t, x ) = ∞ X n = sin (cid:16) nπa x (cid:17) (cid:16) a n e − i √ λ n t + b n e + i √ λ n t (cid:17) (4.16)with a n = a n ( √ λ n ) and b + n = a n (− √ λ n ) As other application that shows the generality of the fractional calculusdefined with the use of ultradistributions, we give the evaluation of the Greenfunction corresponding to the free particle. Let β be defined as: β = L νp N m (5.1)Then G ( t − t ′ , x − x ′ ) should be satisfy the equation: ( i∂ t ) ν G ( t − t ′ , x − x ′ ) + β ∂ G ( t − t ′ , x − x ′ ) = δ ( t − t ′ ) δ ( x − x ′ ) (5.2)12s G is function of ( t − t ′ , x − x ′ ) it is sufficient to consider G as function of ( t, x ) : ( i∂ t ) ν G ( t, x ) + β ∂ G ( t, x ) = δ ( t ) δ ( x ) (5.3)For the Fourier transform ^ G of G we have: ( k ν0 − β k ) ^ G ( k , k ) = Sgn [ ℑ ( k )] + a ( k , k ) (5.4)where a ( k , k ) is as usual a rapidly decreasing analytic entire function of thevariable k . Selecting: a ( k , k ) = we obtain the equation for the retarded Green function: ( k ν0 − β k ) ^ G ret ( k , k ) = H [ ℑ ( k )] (5.5)and then: G ret ( t, x ) = I Γ ∞ Z − ∞ H [ ℑ ( k )] k ν0 − β k e − i ( k t + kx ) dk dk (5.6)If we take: a ( k , k ) = − we obtain the advanced Green function: G adv ( t, x ) = − I Γ ∞ Z − ∞ H [− ℑ ( k )] k ν0 − β k e − i ( k t + kx ) dk dk (5.7)13or the Wheeler Green function (half advanced plus half retarded): G W ( t, x ) = [ G adv ( t, x ) + G ret ( t, x )] (5.8)we have: G W ( t, x ) = I Γ ∞ Z − ∞ Sgn [ ℑ ( k )] k ν0 − β k e − i ( k t + kx ) dk dk (5.9) Example
When we select ν = we obtain the usual Green functions of QuantumMechanics. For example for G ret we have: G ret ( t, x ) = I Γ ∞ Z − ∞ H [ ℑ ( k )] k − β k e − i ( k t + kx ) dk dk (5.10)or equivalently: G ret ( t, x ) = ∞ Z − ∞ ∞ Z − ∞ ( k + i0 ) − β k e − i ( k t + kx ) dk dk (5.11)After the evaluation of the integral in the variable k , G ret takes the form: G ret ( t, x ) = − i2π H ( t ) ∞ Z − ∞ e − i ( β k t + kx ) dk (5.12)With a square’s completion (5.12) transforms into: G ret ( t, x ) = − iH ( t ) √ t e ix24β2t ∞ Z − ∞ e is ds (5.13)14rom the result of ref.[4] ∞ Z − ∞ e is ds = √ πe − i π4 (5.14)we have G ret ( t, x ) = − iH ( t ) (cid:16) m2πi ~ t (cid:17) e imx22 ~ t (5.15)Taking into account that for ν = : β = ~ we obtain the usual form of G ret (see ref.[5]) G ret ( t − t ′ , x − x ′ ) = − iH ( t − t ′ ) (cid:18) m2πi ~ ( t − t ′ ) (cid:19) e im ( x − x ′ ) ~ ( t − t ′ ) (5.16)With a similar calculus we have for G adv : G adv ( t − t ′ , x − x ′ ) = iH ( t ′ − t ) (cid:18) m2πi ~ ( t ′ − t ) (cid:19) e im ( x − x ′ ) ~ ( t − t ′ ) (5.17)and for G W : G W ( t − t ′ , x − x ′ ) = − i2 Sgn ( t − t ′ ) (cid:18) m2πi ~ | t − t ′ | (cid:19) e im ( x − x ′ ) ~ ( t − t ′ ) (5.18) In a earlier paper (ref.[2] we have shown the existence of a general frac-tional calculus defined via tempered ultradistributions. All ultradistribu-tions provide integrands that are analytic functions along the integration15ath. These properties show that tempered ultradistributions provide anappropriate framework for applications to fractional calculus. With the useof this calculus we have generalized in the present work the results obtainedby Naber (ref.[1]). We have defined the fractionary Schr¨odinger equation forall values of the complex variable ν and treated the cases of the free particleand the potential well. For ν = the results obtained coincide with the usualQuantum Mechanics, and the cases ν = and ν = have shown the ap-pearance of extra terms, besides to those with the usual ( ν = ) framework.We have obtained a general expression for the Green function of the free par-ticle and shown that for ν = this Green function coincide with the obtainedin ref.[5].. For the benefit of the reader we give in this paper two Appendixeswith the main characteristics of n-dimensional tempered ultradistributionsand their Fourier anti-transformed distributions of the exponential type, anda third Appendix about the general fractional calculus defined via the use oftempered ultradistributions. 16 eferences [1] M. Naber: J. of Math. Phys , 3339 (2004).[2] D. G. Barci, G. Bollini, L. E. Oxman, M. C. Rocca: Int. J. of Theor.Phys. , 3015 (1998).[3] F. Riewe: Phys. Rev. E , 3581 (1997).[4] L. S. Gradshtein and I. M. Ryzhik : “Table of Integrals, Series, andProducts”. Sixth edition, 3.322, 333 Academic Press (2000).[5] L. Schiff:“Quantum Mechanics”, 65, McGraw-Hill Kogakusha, Ltd(1968).[6] J. Sebastiao e Silva : Math. Ann. , 38 (1958).[7] M. Hasumi: Tˆohoku Math. J. , 94 (1961).[8] I. M. Gel’fand and N. Ya. Vilenkin : “Generalized Functions” Vol. 4 .Academic Press (1964).[9] I. M. Gel’fand and G. E. Shilov : “Generalized Functions”
Vol. 2 .Academic Press (1968).[10] L. Schwartz : “Th´eorie des distributions”. Hermann, Paris (1966).17
Appendix I: Distributions of ExponentialType
For the sake of the reader we shall present a brief description of the principalproperties of Tempered Ultradistributions.
Notations . The notations are almost textually taken from ref[7]. Let R n (res. C n ) be the real (resp. complex) n-dimensional space whose pointsare denoted by x = ( x , x , ..., x n ) (resp z = ( z , z , ..., z n ) ). We shall use thenotations:(i) x + y = ( x + y , x + y , ..., x n + y n ) ; αx = ( αx , αx , ..., αx n ) (ii) x ≧ means x ≧
0, x ≧
0, ..., x n ≧ (iii) x · y = n P j = x j y j (iV) | x | = n P j = | x j | Let N n be the set of n-tuples of natural numbers. If p ∈ N n , then p =( p , p , ..., p n ) , and p j is a natural number, ≦ j ≦ n . p + q denote ( p + q , p + q , ..., p n + q n ) and p ≧ q means p ≧ q , p ≧ q , ..., p n ≧ q n . x p means x p x p ...x p n n . We shall denote by | p | = n P j = p j and by D p we denote thedifferential operator ∂ p + p + ... + p n /∂x ∂x ...∂x np n For any natural k we define x k = x k1 x k2 ...x kn and ∂ k /∂x k = ∂ nk /∂x k1 ∂x k2 ...∂x kn H of test functions such that e p | x | | D q φ ( x ) | is bounded for anyp and q is defined ( ref.[7] ) by means of the countably set of norms: k ^ φ k p = sup ≤ q ≤ p, x e p | x | (cid:12)(cid:12) D q ^ φ ( x ) (cid:12)(cid:12) , p =
0, 1, 2, ... (7.1)According to reference[9] H is a K { M p } space with: M p ( x ) = e ( p − ) | x | , p =
1, 2, ... (7.2) K { e ( p − ) | x | } satisfies condition ( N ) of Guelfand ( ref.[8] ). It is a countableHilbert and nuclear space: K { e ( p − ) | x | } = H = ∞ \ p = H p (7.3)where H p is obtained by completing H with the norm induced by the scalarproduct: < ^ φ, ^ ψ > p = ∞ Z − ∞ e ( p − ) | x | p X q = D q ^ φ ( x ) D q ^ ψ ( x ) dx ; p =
1, 2, ... (7.4)where dx = dx dx ...dx n If we take the usual scalar product: < ^ φ, ^ ψ > = ∞ Z − ∞ ^ φ ( x ) ^ ψ ( x ) dx (7.5)then H , completed with (7.5), is the Hilbert space H of square integrablefunctions. 19he space of continuous linear functionals defined on H is the space Λ ∞ of the distributions of the exponential type ( ref.[7] ).The “nested space” H = ( H , H , Λ ∞ ) (7.6)is a Guelfand’s triplet ( or a Rigged Hilbert space [8] ).In addition we have: H ⊂ S ⊂ H ⊂ S ′ ⊂ Λ ∞ , where S is the Schwartzspace of rapidly decreasing test functions (ref[10]).Any Guelfand’s triplet G = ( Φ , H , Φ ′ ) has the fundamental propertythat a linear and symmetric operator on Φ , admitting an extension to aself-adjoint operator in H , has a complete set of generalized eigen-functionsin Φ ′ with real eigenvalues. The Fourier transform of a function ^ φ ∈ H is φ ( z ) = ∞ Z − ∞ ^ φ ( x ) e iz · x dx (8.1) φ ( z ) is entire analytic and rapidly decreasing on straight lines parallel to thereal axis. We shall call H the set of all such functions. H = F { H } (8.2)20t is a Z { M p } space ( ref.[9] ), countably normed and complete, with: M p ( z ) = ( + | z | ) p (8.3) H is also a nuclear space with norms: k φ k pn = sup z ∈ V n ( + | z | ) p | φ ( z ) | (8.4)where V k = { z = ( z , z , ..., z n ) ∈ C n : | Imz j | ≦ k, 1 ≦ j ≦ n } We can define the usual scalar product: < φ ( z ) , ψ ( z ) > = ∞ Z − ∞ φ ( z ) ψ ( z ) dz = ∞ Z − ∞ ^ φ ( x ) ^ ψ ( x ) dx (8.5)where: ψ ( z ) = ∞ Z − ∞ ^ ψ ( x ) e − iz · x dx and dz = dz dz ...dz n By completing H with the norm induced by (8.5) we get the Hilbert spaceof square integrable functions.The dual of H is the space U of tempered ultradistributions ( ref.[7] ). Inother words, a tempered ultradistribution is a continuous linear functionaldefined on the space H of entire functions rapidly decreasing on straight linesparallel to the real axis.The set U = ( H , H, U ) is also a Guelfand’s triplet.21oreover, we have: H ⊂ S ⊂ H ⊂ S ′ ⊂ U . U can also be characterized in the following way ( ref.[7] ): let A ω bethe space of all functions F ( z ) such that: I - F ( z ) is analytic for { z ∈ C n : | Im ( z ) | > p, | Im ( z ) | > p, ..., | Im ( z n ) | >p } . II - F ( z ) /z p is bounded continuous in { z ∈ C n : | Im ( z ) | ≧ p, | Im ( z ) | ≧ p, ..., | Im ( z n ) | ≧ p } , where p =
0, 1, 2, ... depends on F ( z ) .Let Π be the set of all z -dependent pseudo-polynomials, z ∈ C n . Then U is the quotient space: III - U = A ω /Π By a pseudo-polynomial we understand a function of z of the form P s z sj G ( z , ..., z j − , z j + , ..., z n ) with G ( z , ..., z j − , z j + , ..., z n ) ∈ A ω Due to these properties it is possible to represent any ultradistributionas ( ref.[7] ): F ( φ ) = < F ( z ) , φ ( z ) > = I Γ F ( z ) φ ( z ) dz (8.6) Γ = Γ ∪ Γ ∪ ...Γ n where the path Γ j runs parallel to the real axis from − ∞ to ∞ for Im ( z j ) > ζ , ζ > p and back from ∞ to − ∞ for Im ( z j ) < − ζ , − ζ < − p . ( Γ surrounds all the singularities of F ( z ) ).Formula (8.6) will be our fundamental representation for a tempered ul-22radistribution. Sometimes use will be made of “Dirac formula” for ultradis-tributions ( ref.[6] ): F ( z ) = ( ) n ∞ Z − ∞ f ( t )( t − z )( t − z ) ... ( t n − z n ) dt (8.7)where the “density” f ( t ) is such that I Γ F ( z ) φ ( z ) dz = ∞ Z − ∞ f ( t ) φ ( t ) dt (8.8)While F ( z ) is analytic on Γ , the density f ( t ) is in general singular, so thatthe r.h.s. of (8.8) should be interpreted in the sense of distribution theory.Another important property of the analytic representation is the fact thaton Γ , F ( z ) is bounded by a power of z ( ref.[7] ): | F ( z ) | ≤ C | z | p (8.9)where C and p depend on F .The representation (8.6) implies that the addition of a pseudo-polynomial P ( z ) to F ( z ) do not alter the ultradistribution: I Γ { F ( z ) + P ( z ) } φ ( z ) dz = I Γ F ( z ) φ ( z ) dz + I Γ P ( z ) φ ( z ) dz But: I Γ P ( z ) φ ( z ) dz = P ( z ) φ ( z ) is entire analytic in some of the variables z j ( and rapidly de-creasing ), ∴ I Γ { F ( z ) + P ( z ) } φ ( z ) dz = I Γ F ( z ) φ ( z ) dz (8.10) The purpose of this sections is to introduce definition of fractional derivationand integration given in ref. [6]. This definition unifies the notion of integraland derivative in one only operation. Let ^ f ( x ) a distribution of exponentialtype and F ( Ω ) the complex Fourier transformed Tempered Ultradistribution.Then: F ( k ) = H [ ℑ ( k )] ∞ Z ^ f ( x ) e ikx dx − H [− ℑ ( k )] Z − ∞ ^ f ( x ) e ikx dx (9.1)( H ( x ) is the Heaviside step function) and ^ f ( x ) = I Γ F ( k ) e − ikx dk (9.2)where the contour Γ surround all singularities of F ( k ) and runs parallel toreal axis from − ∞ to ∞ above the real axis and from ∞ to − ∞ below thereal axis. According to [6] the fractional derivative of ^ f ( x ) is given by d λ ^ f ( x ) dx λ = I Γ (− ik ) λ F ( k ) e − ikx dk + I Γ (− ik ) λ a ( k ) e − ikx dk (9.3)24here a ( k ) is entire analytic and rapidly decreasing. If λ = − , d λ /dx λ isthe inverse of the derivative (an integration). In this case the second term ofthe right side of (9.3) gives a primitive of ^ f ( x ) . Using Cauchy’s theorem theadditional term is I a ( k ) k e − ikx dk = ( ) (9.4)Of course, an integration should give a primitive plus an arbitrary constant.Analogously when λ = − (a double iterated integration) we have I a ( k ) k e − ikx dk = γ + δx (9.5)where γ and δδ