Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2013, Article ID 290216, 11 pages http://dx.doi.org/10.1155/2013/290216

Research Article Time Fractional Schrodinger Equation Revisited

B. N. Narahari Achar, Bradley T. Yale, and John W. Hanneken

Physics Department, University of Memphis, Memphis, TN 38152, USA

Correspondence should be addressed to B. N. Narahari Achar; [email protected]

Received 29 April 2013; Revised 1 July 2013; Accepted 2 July 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 B. N. Narahari Achar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle” obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have “fractional” 2 2 dimensions, chosen such that the “energy” has the correct dimension [𝑀𝐿 /𝑇 ]. The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a “fractional Planck constant” is introduced. The TFSE corresponds to a “subdiffusion” equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber’s work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a “free particle” is obtained. The total probability for a “free” particle is shown to go to zero in the limit of infinite time, in contrast with Naber’s result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.

1. Introduction The fractional derivative which is nonlocal by definition can be made “local” by a limiting process as shown by Kolwankar There has been an explosive research output in recent years and Gangal [41]. Highly irregular and nowhere differentiable in the application of methods of fractional calculus [1–13] functions can be analyzed locally using these local fractional to the study of quantum phenomena [14–42]. The well- derivatives. The Heisenberg principle in the fractional context known Schrodinger equation with a first-order derivative in has been investigated using local fractional Fourier analysis time and second-order derivatives in space coordinates was [42]. given by Schrodinger as an Ansatz. The Schrodinger equation The Schrodinger equation for a free particle has the has been generalized to (i) a space fractional Schrodinger appearance of a diffusion equation with an imaginary dif- equation involving noninteger order space derivatives but fusion coefficient. This suggests a method of deriving the retaining first-order time derivative [14–18], (ii) a time frac- Schrodinger equation as has been done using the Feynman tional Schrodinger equation involving non-integer order time path integral technique [43–45]basedontheGaussian derivative but retaining the second-order space derivatives probability distribution in the space of all possible paths. [19], or (iii) more general fractional Schrodinger equation In other words, the classical Brownian motion leads to where both time and space derivatives are of non-integer the Schrodinger equation in quantum mechanics. As far as order [20–26]. The fractional Schrodinger equation has also deriving the fractional Schrodinger equation is concerned, been obtained by using a fractional generalization of the the path integral approach for the Brownian-like paths Laplacian operator [20] and by using a fractional variation for the Levy stable processes which leads to the classical principle and a fractional Klein-Gordon equation [36]. In all space fractional diffusion equation has been extended to these cases the fractional derivatives employed have been the the Levy-like quantum paths leading to the space fractional regular fractional derivatives of the Riemann-Liouville type Schrodinger equation (SFSE) in the seminal papers of Laskin ortheCaputotype(generallyusedinphysicalapplications [14–18]. It may be noted that in this case, the time derivative is with initial conditions) which are both nonlocal in nature. still the integer first-order derivative; only the space part is of 2 Advances in Mathematical Physics fractional order. The SFSE still retains the Markovian charac- the space fractional part as the theory of the latter has been ter and other fundamental aspects such as the Hermiticity of well established in the works of Laskin [14–18]. Furthermore, the Hamilton operator. Parity conservation and the current the paper considers only the Caputo-type nonlocal fractional density have been explored in the space fractional quan- derivatives and not the local fractional derivatives discussed tum mechanics in terms of the Riesz fractional derivative. earlier. The paper starts from a generalization of the classical Applications of SFSE cover the dynamics of a free particle, dynamics into fractional dynamics of a free particle and particle in an infinite potential well, fractional Bohr atom, and then adapting the Feynman technique derives the correct thequantumfractionaloscillator.Thusthespacefractional equations for TFSE. It is demonstrated that Naber’s result of Schrodinger equation appears to have been well established probabilitybeinggreaterthanunityisspuriousandisaresult [18]. This theory has been further generalized recently within of the ad hoc raising of the imaginary number 𝑖 to a fractional the frame work of tempered ultradistributions [39]. Thus the power. The correct continuity equation for the probability theory of SFSE can be considered fully established from the density is also derived. The paper concludes with some new point of view of the Feynman path integral technique. results. The fractional time derivative was introduced into the Schrodinger equation by Naber [19] by simply replacing the 2. Feynman Path Integral Method first-order time derivative by a derivative of non-integer order and retaining the second-order space derivatives intact. The starting point for the Feynman method [43–46]isthe The resulting equation is referred to as the time fractional classical Lagrangian 𝐿=𝐿(𝑥,𝑥,̇ 𝑡) and the action 𝑆= Schrodinger equation (TFSE). He did not derive the TFSE ∫ 𝐿(𝑥, 𝑥,̇ 𝑡)𝑑𝑡 constructed from it. However, in view of the using the path integral or any other method. Naber carried generalization to fractional calculus methods to be carried out the time fractional modification to the Schrodinger equa- outlater,theequationsofmotionofaclassicalparticleinone tion in analogy with time fractional diffusion equation19 [ ] dimension in the usual notation are considered first: but included the imaginary number 𝑖 raised to a fractional 𝑑𝑥 power (the fractional degree being the same as the fractional 𝑚 =𝑝, order of the time derivative), implying a sort of the Wick 𝑑𝑡 (1) rotation. In Naber’s opinion [19], the TFSE is equivalent to 𝑑𝑝 =𝐹. the usual Schrodinger equation, but with a time-dependent 𝑑𝑡 Hamiltonian. He obtained the solutions for a free particle and a particle in a potential well. A lot of subsequent Integrating with respect to time yields workhasbeendoneontheTFSE,mostlybasedonNaber’s 𝑡 work [21–23, 28], including its generalization into space- 1 𝑥=𝑥0 + ∫ 𝑝 (𝜏) 𝑑𝜏, (2) time fractional quantum dynamics by including non-integer 𝑚 0 order derivatives in both time and space. Yet some basic 𝑡 questions have not been addressed. It has been observed that 𝑝=𝑝0 + ∫ 𝐹 (𝜏) 𝑑𝜏. (3) TFSE describes non-Markovian evolution and that the Green 0 function in the form of the Mittag-Leffler function does not satisfy Stone’s theorem on one-parameter unitary groups In the usual notation the Lagrangian is given by and the semiclassical approximation in terms of the classical 𝐿 (𝑥, 𝑥,̇ ) 𝑡 =𝑇−𝑉 (4) action is not defined [31]. There has been no derivation of TFSE on a basis similar to that of SFSE and it is the purpose and the action is given by of the present paper to rectify this lacuna. Since the path integral method of deriving the space fractional part of the 𝑡 𝑆=∫ 𝐿 (𝑥, 𝑥,̇ 𝜏) 𝑑𝜏. (5) Schrodinger equation is well established, the present paper 0 concentrates only on deriving the time fractional Schrodinger equation from the Feynman path integral approach, leading An outline of the Feynman path integral method is pre- to the time fractional Schrodinger equation as given by sented following very closely the account given by Feynman Naber. It may be pointed out that some results of Naber, and Hibbs [43]. The essence of the Feynman path integral suchasthetotalprobabilitybeinggreaterthanunity,are approach to quantum mechanics is in the probability ampli- difficult to understand physically. Moreover, several major tude (also known as the propagator or the Green function) 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 ) 𝑥 errors in Naber’s paper have gone unnoticed and in fact 𝑏 𝑏 𝑎 𝑎 for a particle starting from a position 𝑎 at 𝑡 𝑥 𝑡 have been repeated by workers who have followed his work. time 𝑎 to reach a position 𝑏 at a later time 𝑏,whicharises 𝑥 𝑥 Furthermore, some of these authors have introduced errors of from the contributions from all trajectories from 𝑎 to 𝑏: their own. Since many of the conclusions in Naber’s paper are 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 )= ∑ 𝜙 [𝑥 (𝑡)] , based on derivations which include these errors, it calls for a 𝑏 𝑏 𝑎 𝑎 (6) reexamination of Naber’s generalization to the time fractional all paths Schrodinger equation. where the contribution from each of the paths has the form The present paper derives the time fractional Schrodinger equation using the Feynman path integral technique. It 𝑖𝑆 𝜙 [𝑥 (𝑡)] = . [ ]. concentrates on the time fractional part only and not on const exp ℎ (7) Advances in Mathematical Physics 3

Here 𝑆 istheactiondefinedin(5)andℎ is Planck’s constant, and (5) yields for the action the quantum of action. The time integral of the Lagrangian 2 is to be taken along the path in question. Restricting to one 𝑚(𝑥 −𝑥 ) 𝑆=𝜀𝐿= 0 . (16) dimension,theprobabilityamplitudecanbewrittenas 2𝜀 𝑏 𝑡 𝑖 𝑏 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 )=∫ [ ∫ 𝐿𝑑𝑡] D𝑥 (𝑡) . As a consequence, (9)becomes 𝑏 𝑏 𝑎 𝑎 exp ℎ (8) 𝑎 𝑡𝑎 ∞ 2 1 𝑖𝑚(𝑥0 −𝑥 ) The symbol D indicates the fact that the operation of 𝜓 (𝑥, 𝑡) +𝜀 = ∫ exp [ ]𝜓(𝑥0,𝑡)𝑑𝑥0. −∞ 𝐴 2𝜀 integration is carried over all paths from 𝑎 to 𝑏. The wave function 𝜓(𝑥𝑏,𝑡𝑏) gives the total probability (17) amplitude to arrive at 𝑥𝑏 at 𝑡𝑏 satisfying (9), where the integral If 𝑥 differs appreciably from 𝑥0, the exponential in (17) is taken over all possible values of 𝑥𝑎 oscillates very rapidly and the integral over 𝑥0 contributes a ∞ very small value and only those paths which are very close to 𝜓(𝑥𝑏,𝑡𝑏)=∫ 𝐾(𝑥𝑏,𝑡𝑏;𝑥𝑎,𝑡𝑎)𝜓(𝑥𝑎,𝑡𝑎)𝑑𝑥𝑎. (9) 𝑥 −∞ give significant contributions. Changing the variable in the integral from 𝑥0 to 𝜂=𝑥−𝑥0 makes it 𝜓(𝑥0,𝑡)=𝜓(𝑥+𝜂,𝑡). The kernel 𝐾 can be computed by first carrying out a “time Since both 𝜀 and 𝜂 are small quantities, 𝜓(𝑥, 𝑡 +𝜀) may be slicing” operation by dividing the time interval from 𝑡𝑎 to 𝑡𝑏 expanded in Taylor’s series and only up to terms of order into 𝑁 segments of duration 𝜀 are retained. On the right-hand side, 𝜓(𝑥 + 𝜂, 𝑡) may be 𝜂 𝑡 −𝑡 expanded in Taylor’s series in powers of ,retainingterms 𝜀= 𝑏 𝑎 , up to second order in 𝜂 (the integral involving the first-order 𝑁 (10) term vanishes). Then (17)becomes where 𝜕𝜓 𝜓 (𝑥,) 𝑡 +𝜀 𝑡𝑎 =𝑡0 <𝑡1 <𝑡2 <⋅⋅⋅<𝑡𝑁−1 <𝑡𝑁 =𝑡𝑏; (11) 𝜕𝑡 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 ) ∞ 1 𝑚𝜂2 𝑏 𝑏 𝑎 𝑎 = ∫ exp [− ] −∞ 𝐴 2𝑖ℎ𝜀 (18) 1 𝑖𝑆 [𝑏,] 𝑎 = lim ∭ ⋅⋅⋅∫ exp [ ] 2 2 𝜀→0𝐴 ℎ (12) 𝜕𝜓 𝜂 𝜕 𝜓 ×(𝜓(𝑥,) 𝑡 +𝜂 + )𝑑𝜂. 2 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝜕𝑥 2 𝜕𝑥 × 1 2 3 ⋅⋅⋅ 𝑁−1 , 𝐴 𝐴 𝐴 𝐴 On the right-hand side the middle term vanishes on integra- where tion. It follows by equating the leading terms on both sides

𝑡𝑏 ∞ 2 𝑆 [𝑏,] 𝑎 = ∫ 𝐿 (𝑥, 𝑥,̇ ) 𝑡 𝑑𝑡 1 𝑚𝜂 (13) 𝜓 (𝑥,) 𝑡 = ∫ exp [− ]𝜓(𝑥,) 𝑡 𝑑𝜂 (19) 𝑡𝑎 −∞ 𝐴 2𝑖ℎ𝜀 is a line integral taken over the trajectory passing through the Hence point 𝑥(𝑡).Theconstant𝐴 is a normalizing factor. The Schrodinger equation for a free particle in one dimension is derived by considering a special case of (9), ∞ 𝑚𝜂2 2𝜋𝑖ℎ𝜀 which describes the evolution of the wave function from a 𝐴=∫ exp [− ]𝑑𝜂=√ , (20a) −∞ 2𝑖ℎ𝜀 𝑚 time 𝑡𝑎 to a time 𝑡𝑏,when𝑡𝑏 differs from 𝑡𝑎 by an infinitesimal amount 𝜀 and applied to the case of a free particle. This step is based on the fact that the semiclassical approximation is ∞ 1 𝑚𝜂2 𝜂2 𝜕2𝜓 𝑖ℎ 𝜕2𝜓 validnotonlyinthelimitofℎ→0but also in the limit ∫ [− ]( )𝑑𝜂=𝜀 . 𝐴 exp 2𝑖ℎ𝜀 2 𝜕𝑥2 2𝑚 𝜕𝑥2 (20b) ofsmalltimeinterval[45]. The Kernel is proportional to −∞ the exponential of (𝑖/ℎ) times the classical action for the Equating the remaining terms results in infinitesimal time interval 𝜀=𝑡𝑏 −𝑡𝑎.Withanobviouschange 𝑥 =𝑥 𝑥 =𝑥 𝑡 =𝑡 𝑡 =𝑡+𝜀 of notation 𝑏 , 𝑎 0, 𝑎 , 𝑏 and using the 𝜕𝜓 𝑖ℎ 𝜕2𝜓 fact that the particle is free, (2)yields = . (21) 𝜕𝑡 2𝑚 𝜕𝑥2 𝑚(𝑥−𝑥 ) 𝑝 = 0 (14) 0 𝜀 This can be recognized as the diffusion equation with an imaginary diffusion coefficient or the Schrodinger equation and (4)yields for a free particle in quantum mechanics. These considerations can be easily extended to the case 2 𝑚(𝑥−𝑥 ) ofaparticlemovinginapotentialfieldbyincorporatinga 𝐿=𝑇= 0 (15) 2𝜀2 potential term 𝑉(𝑥, 𝑡) in the Lagrangian 𝐿=𝑇−𝑉in (15). 4 Advances in Mathematical Physics

𝑥 𝑝 This will necessitate [43] incorporating an additional factor where as usual 0, 𝑓0 refer to the initial position and initial {1 − (𝑖𝜀/ℎ)𝑉(𝑥, 𝑡)} in (18), which becomes value of the “fractional momentum” 𝑝𝑓,respectively.Itis 𝑥 𝑡 𝜕𝜓 ∞ 1 𝑚𝜂2 𝑖𝜀 to be noted that the variables , are still the space and 𝜓 (𝑥,) 𝑡 +𝜀 = ∫ [− ]{1− 𝑉 (𝑥,) 𝑡 } time variables and have the dimensions of length and time 𝜕𝑡 𝐴 exp 2𝑖ℎ𝜀 ℎ −∞ [𝐿] and [𝑇], respectively. However, the dynamical quantities 𝜕𝜓 𝜂2 𝜕2𝜓 obtained by the operation of fractional derivation have ×(𝜓(𝑥,) 𝑡 +𝜂 + )𝑑𝜂. different dimensions; for example, “fractional velocity” with 𝜕𝑥 2 𝜕𝑥2 𝐶 𝛼/2 𝛼/2 𝛼/2 the notation 0 𝐷𝑡 𝑥=𝑥̇ would have the units [𝐿/𝑇 ]. (22) The dimension for the parameter “𝑚𝑓”inthefractional 𝛼/2 Then (21)becomes momentum, 𝑝𝑓 =𝑚𝑓𝑥̇ ,isnolongerjust[𝑀] but has to be 2 𝜕𝜓 𝑖ℎ 𝜕2𝜓 𝑖 chosen [47] so that the fractional quantity 𝑝𝑓 /2𝑚𝑓 has the = − 𝑉𝜓. (23) 2 2 𝜕𝑡 2𝑚 𝜕𝑥2 ℎ dimensions of energy [𝑀𝐿 /𝑇 ]. Thus the dimension of the 2−𝛼 parameter “𝑚𝑓”is[𝑀𝑇 ] and the fractional momentum Multiplyingbothsidesby−ℎ/𝑖 results in the standard [𝑀𝐿/𝑇2−𝛼/2] Schrodinger equation of quantum mechanics: has the dimensions .TheLagrangianin(4)when generalized has the dimensions of energy. Of course, all 𝜕𝜓 ℎ2 𝜕2𝜓 𝛼→ 𝑖ℎ =− +𝑉𝜓. (24) quantities regain the standard dimensions in the limit 𝜕𝑡 2𝑚 𝜕𝑥2 2. It is to be noted that the imaginary number 𝑖 on the left- hand side is not arbitrary; it arises from the coefficient of the 4. Time Fractional Schrodinger Equation for a potential term 𝑉 in (23)butultimatelyfromthecoefficient Free Particle of 𝑆/ℎ in (7). This minor detail becomes important as will be discussed later in Section 8. There are two possible generalizations of the action integral These considerations will be generalized for a particle in (5) used in fractional dynamics [11]: obeying fractional dynamics. It can then be extended to the 𝑡 𝑆 = ∫ 𝐿(𝑥,𝑥̇𝛼/2,𝑡)𝑑𝑡, case of a particle in a potential field by including the potential I (28) term and making appropriate changes as will be described 0 later. 1 𝑡 𝑆 = ∫ 𝐿(𝑥,𝑥̇𝛼/2,𝜏)(𝑡−𝜏)𝛼/2−1𝑑𝜏. II (29) Γ (𝛼/2) 0 3. Fractional Dynamics of a Free Particle Since the Lagrangian has the dimensions of energy, the 𝑆 𝑆 The first step is to generalize the equations of motion, (1)– dimensions of action defined in (28)and(29), I and II,are (3), by replacing the integrals and derivatives by appropriate different. 𝑆 fractional integrals and derivatives. For physical problems The dimensions of I are the same as that of the regular [𝑀𝐿2/𝑇] 𝑆 [𝑀𝐿2/𝑇2−𝛼/2] with well-definable initial conditions the accepted practice is action, namely, ,butthatof II is . 𝛼→2𝑆 → to employ the Caputo fractional derivatives [5]. The Caputo In the Newtonian limit , II regular action. derivative of order 𝛽 is defined by [3]: These dimensional considerations have to be kept in mind in 𝑡 𝑛 generalizing the Feynman method. In particular, if the choice 𝐶 𝛽 1 𝑓 (𝜏) 𝑑𝜏 𝐷 𝑓 (𝑡) = ∫ (𝑛−1<𝛽<𝑛). from (29) is made, then a “fractional Planck constant” ℎ𝑓 0 𝑡 𝛽+1−𝑛 Γ(𝑛−𝛽) 0 (𝑡−𝜏) with appropriate dimensions must be introduced in order to (25) render the argument of the exponential in (7) dimensionless. For a “free particle” (27)yields In the limit 𝛽→𝑛, the Caputo derivative becomes the 𝑛 𝛼/2 ordinary th derivative of the function. 𝑝𝑓0𝑡 The fractional integral of order 𝛽 is defined by 𝑥=𝑥0 + , (30) 𝑚𝑓Γ (1+𝛼/2) 𝑡 𝛽 1 𝛽−1 0𝐼𝑡 𝑓 (𝑡) = ∫ 𝑓 (𝜏)(𝑡−𝜏) 𝑑𝜏. (26) 𝑝 =𝑝 . Γ(𝛽) 0 𝑓 𝑓0 (31) In generalizing the equations of motion, the second-order After carrying out the time slicing operation asin(11)and makingthesameapproximationtheevolutionofthewave time derivative in Newton’s law is replaced by a Caputo 𝜀 derivative of order 𝛼, and the first-order derivative is replaced function in an infinitesimal interval of time can now be by a Caputo derivative of order (𝛼/2) [47]. Then2 ( )and(3) obtained. Equation (30) yields become (𝑥 −0 𝑥 )𝑚𝑓Γ (1+𝛼/2) 𝑝 = (32) 1 𝑡 𝑓0 𝛼/2 𝑥=𝑥 + ∫ 𝑝 (𝜏)(𝑡−𝜏)𝛼/2−1𝑑𝜏, 𝜀 0 𝑚 Γ (𝛼/2) 𝑓 𝑓 0 and hence (27) 1 𝑡 𝑚 Γ (1+𝛼/2) (𝑥 − 𝑥 )2 𝑝 =𝑝 + ∫ 𝐹 (𝜏)(𝑡−𝜏)𝛼/2−1𝑑𝜏, 𝑓 0 𝑓 𝑓0 𝐿= . (33) Γ (𝛼/2) 0 2𝜀𝛼 Advances in Mathematical Physics 5

But for action, there are two choices: thetwocaseswillbeconsideredseparately.Thus,theequation 𝜓 (𝑥, 𝑡) for I becomes 2 2 𝛾 𝑚𝑓Γ (1+𝛼/2) (𝑥 −0 𝑥 ) 𝜀 𝑆 = , (34) 𝜓 (𝑥,) 𝑡 + 𝐶𝐷𝛾𝜓 (𝑥,) 𝑡 I 2𝜀𝛼−1 I 0 𝑡 I Γ(𝛾+1) 2 ∞ 2 𝑚𝑓Γ (1+𝛼/2) (𝑥 −0 𝑥 ) 1 𝑎 𝜂 𝑆 = . (35) = ∫ [− I ] II 2𝜀𝛼/2 𝐴 exp 𝜀𝛼−1 (42) −∞ I 𝜕𝜓 𝜂2 𝜕2𝜓 Making the appropriate changes, the equations for the ×{𝜓 (𝑥,) 𝑡 +𝜂 I + I }𝑑𝜂. evolution of the wave function in the two cases are I 𝜕𝑥 2 𝜕𝑥2

𝜓 (𝑥, 𝑡) +𝜀 On evaluating the integrals on the right-hand side of (42), the I middle term with the first power of 𝜂 vanishes. Equating the 2 ∞ 𝑖𝑚 Γ2 (1+𝛼/2) (𝑥−𝑥 ) leading terms on both sides of (42)yields 1 [ 𝑓 0 ] = ∫ exp (36) 𝜓 (𝑥, 𝑡) = (1/𝐴 )√𝜋𝜀𝛼−1/𝑎 𝜓 (𝑥, 𝑡) 𝐴 = 𝐴 2ℎ𝜀𝛼−1 I I I I ,requiringthat I −∞ I [ ] √𝜋𝜀𝛼−1/𝑎 I.Equation(42)reducesto ×𝜓 (𝑥 ,𝑡)𝑑𝑥 , I 0 0 𝜀𝛾 𝜓 (𝑥,) 𝑡 + 𝐶𝐷𝛾𝜓 (𝑥,) 𝑡 𝜓 (𝑥, 𝑡) +𝜀 I 0 𝑡 I II Γ(𝛾+1) (43) ∞ 2 𝛼−1 2 1 𝑖𝑚𝑓Γ (1+𝛼/2) (𝑥−𝑥0) 1 𝜀 𝜕 𝜓 (𝑥,) 𝑡 = ∫ [ ] (37) ={𝜓 (𝑥,) 𝑡 + I }. 𝐴 exp 2ℎ 𝜀𝛼/2 I 4 𝑎 𝜕𝑥2 −∞ II 𝑓 I 𝜀 ×𝜓 (𝑥0,𝑡)𝑑𝑥0. Equating the remaining terms requires that the powers of II mustbethesameonbothsides;thatis,𝛾=𝛼−1. Inserting 𝑎 the value of I and simplifying (43)yield Changing the variable in the integrals from 𝑥0 to 𝜂=𝑥−𝑥0 as before and introducing two constants 𝑖ℎ Γ (𝛼) 𝜕2𝜓 (𝑥,) 𝑡 𝐶𝐷𝛼−1𝜓 (𝑥,) 𝑡 = I . 0 𝑡 I 2 2 (44) 2𝑚𝑓 Γ (1+𝛼/2) 𝜕𝑥 2 𝑚𝑓Γ (1+𝛼/2) 𝑚𝑓Γ (1+𝛼/2) 𝑎 = ,𝑎= . (38) Similarly, by expanding (40)theequationfor𝜓 (𝑥, 𝑡) I 2𝑖ℎ II 2𝑖ℎ II 𝑓 becomes 𝜀𝛾 𝜓 (𝑥,) 𝑡 + 𝐶𝐷𝛾𝜓 (𝑥,) 𝑡 Equations (36)and(37)canbewrittenas II 0 𝑡 II Γ(𝛾+1)

𝜓 (𝑥, 𝑡) +𝜀 ∞ 1 𝑎 𝜂2 I = ∫ [− II ] exp 𝛼/2 (45) −∞ 𝐴 𝜀 ∞ 1 𝑎 𝜂2 (39) II = ∫ [− I ]𝜓 (𝑥 + 𝜂, 𝑡) 𝑑𝜂, 𝐴 exp 𝜀𝛼−1 I 𝜕𝜓 𝜂2 𝜕2𝜓 −∞ I ×{𝜓 (𝑥,) 𝑡 +𝜂 II + II }𝑑𝜂. II 𝜕𝑥 2 𝜕𝑥2 𝜓 (𝑥, 𝑡) +𝜀 II On evaluating the integrals on the right-hand side of (45) ∞ 1 𝑎 𝜂2 (40) = ∫ [− II ]𝜓 (𝑥 + 𝜂, 𝑡) 𝑑𝜂. themiddletermvanishes.Equatingtheleadingtermson 𝐴 exp 𝜀𝛼/2 II −∞ II both sides of (45) as before yields for the normalizing factor 𝐴 = √𝜋𝜀𝛼/2/𝑎 II II.Equation(45)reducesto The left-hand sides of(39)and(40) can be expanded 𝜀𝛾 in fractional Taylor’s series [48]intime,withfractional 𝐶 𝛾 𝜓 (𝑥,) 𝑡 + 0 𝐷𝑡 𝜓 (𝑥,) 𝑡 derivative of order 𝛾, and keeping only the lowest-order term II II Γ(𝛾+1) in 𝛾 yields (46) 1 𝜀𝛼/2 𝜕2𝜓 (𝑥,) 𝑡 ={𝜓 (𝑥,) 𝑡 + II }. 2 𝜓 (𝑥, 𝑡) +𝜀 =𝜓 (𝑥,) 𝑡 II 4 𝑎 𝜕𝑥 I,II I,II II 𝜀 𝜀𝛾 (41) In the remaining terms, the powers of must be the same. This + 𝐶𝐷𝛾𝜓 (𝑥,) 𝑡 +⋅⋅⋅. 𝛾=𝛼/2 𝑎 0 𝑡 I,II requires . Inserting the value of II and simplifying Γ(𝛾+1) 𝜓 (𝑥, 𝑡) (46)yieldtheequationforthewavefunction II : 𝑖ℎ 𝜕2𝜓 (𝑥,) 𝑡 In the right-hand side a Taylor expansion with terms up to 𝐶𝐷𝛼/2𝜓 (𝑥,) 𝑡 = 𝑓 II . 0 𝑡 II 2 (47) second order in 𝜂 with respect to space can be carried out and 2𝑚𝑓 𝜕𝑥 6 Advances in Mathematical Physics

This completes the derivation based on the Feynman path 5. Probability Current and integral method and (44)and(47) constitute the time frac- the Continuity Equation tional Schrodinger equations for a free particle corresponding ∗ ∗ to two ways of defining the action integral for a fractional The probability density 𝜌 is defined by 𝜌=𝜓 𝜓=𝜓𝜓 .The dynamical system. In the limit 𝛼→2,thefractional complex conjugate wave function satisfies dynamicalsystemgoesovertotheregularNewtoniansystem 2 ∗ andinthiscaseboth(44)and(47)reduceto 𝛽 𝜕 𝜓 (𝑥,) 𝑡 𝜕 𝜓∗ (𝑥,) 𝑡 =−𝑖𝐷 . (50) 𝑡 𝑓 𝜕𝑥2 𝜕𝜓 (𝑥,) 𝑡 𝑖ℎ 𝜕2𝜓 (𝑥,) 𝑡 = (48) 𝜕𝑡 2𝑚 𝜕𝑥2 There is an identity satisfied by the Caputo derivative [5] 1−𝛽 𝛽 𝜕 𝜕 𝑓 (𝑡) =𝜕𝑓 (𝑡) , (51) given earlier, thus recovering the standard Schrodinger equa- 𝑡 𝑡 𝑡 tion for a free quantum particle [49]. where the right-hand side represents the regular first-order It should be noted that since 𝛼≤2,with𝛼=2being derivative. This identity can be used in studying the time the limiting case, the order of the time fractional derivative derivative of the probability density, given by is ≤1in both (44)and(47) and cannot exceed 1. This means the time fractional Schrodinger equation as derived ∗ ∗ ∗ 𝜕𝑡𝜌=𝜕𝑡 (𝜓𝜓 ) = (𝜕𝑡𝜓) 𝜓 +𝜓(𝜕𝑡𝜓 ) . (52) from the path integral method always corresponds to the “subdiffusion”caseincontrasttothecasewheretheTFSE Inserting from (51)gives is obtained by a simple replacement of the first-order time derivative by a fractional order derivative [19]. Furthermore, 𝜕 𝜌=(𝜕1−𝛽𝜕𝛽𝜓) 𝜓∗ +𝜓(𝜕1−𝛽𝜕𝛽𝜓∗) . the order 𝛼/2 of the fractional derivative corresponds to 𝑡 𝑡 𝑡 𝑡 𝑡 (53) the first-order regular derivative as has been used above in Section 3. Thus it appears that the second method of defining Substituting from (49)and(50), (53)yields the action leading to (47) is the natural way to generalize 𝜕2𝜓 (𝑥,) 𝑡 toTFSE.Inappearancealsoitisasiftheequationhas 𝜕 𝜌=(𝜕1−𝛽 (𝑖𝐷 )) 𝜓∗ been obtained by replacing all quantities in the Schrodinger 𝑡 𝑡 𝑓 𝜕𝑥2 equation by an equivalent fractional quantity, except the (54) 2 ∗ space derivative. Furthermore, (44) becomes a fractional 1−𝛽 𝜕 𝜓 (𝑥,) 𝑡 +𝜓(𝜕 (−𝑖𝐷 )) . order integro-differential equation when 𝛼<1and no longer 𝑡 𝑓 𝜕𝑥2 just a fractional order differential equation. Because of these reasons, it is considered the method of choice to use (47) Equation (54) can be rewritten after factoring out the constant as the TFSE derived from the path integral method and no and interchanging the order of space and time derivatives as further reference will be made to (44). 2 2 The coefficient of the space derivative term in(47)has 𝜕 1−𝛽 𝜕 1−𝛽 [𝐿2/𝑇𝛼/2] 𝜕 𝜌=𝑖𝐷 {( 𝜕 𝜓) 𝜓∗ −𝜓( 𝜕 𝜓∗)} the dimension , corresponding to the fractional 𝑡 𝑓 𝜕𝑥2 𝑡 𝜕𝑥2 𝑡 . (55) diffusion coefficient. Thus(47) can be considered a time fractional diffusion equation with an imaginary fractional 𝜓=𝜕̃ 1−𝛽𝜓 diffusion coefficient, just as(48), the regular Schrodinger Introducing a new function 𝑡 ,(55)canbewrittenas equation, can be considered to be a diffusion equation with 2 2 an imaginary diffusion coefficient. Thus all the mathematical 𝜕 ∗ 𝜕 ∗ 𝜕𝑡𝜌=𝑖𝐷𝑓 {( 𝜓)̃ 𝜓 −𝜓( 𝜓̃ )} . (56) machinery of time fractional diffusion theory5 [ , 50–59]can 𝜕𝑥2 𝜕𝑥2 be imported advantageously. Although it is possible to introduce additional parameters Defining a probability current density given by and cast (47) in a dimensionless form, it has not been done here. However, for convenience, the subscript II is dropped 𝜕𝜓 𝜕𝜓̃∗ 𝐽 =−𝑖𝐷 [𝜓∗̃ −𝜓 ] , from the wave function; a simplified notation for the Caputo 𝑥 𝑓 𝜕𝑥 𝜕𝑥 (57) 𝛽 derivative, 𝜕𝑡 𝜓,with𝛽=𝛼/2will be used. Naturally, 0<𝛽≤ 1,and𝛽→1yields the regular first-order time derivative, equation (56)canbewrittenas denoted by 𝜕𝑡. After incorporating these changes and defining anewconstant𝐷𝑓 =ℎ𝑓/2𝑚𝑓,(47)becomes 𝜕𝐽𝑥 𝜕 𝜌+ =0. (58) 𝑡 𝜕𝑥 𝜕2𝜓 (𝑥,) 𝑡 𝜕𝛽𝜓 (𝑥,) 𝑡 =𝑖𝐷 (0<𝛽≤1). This is the time fractional version of the continuity equation. 𝑡 𝑓 2 (49) ∗ ∗ 𝜕𝑥 In the limit 𝛽→1, 𝜓→𝜓̃ 𝜓̃ →𝜓 and (58)reproduce the continuity equation of standard quantum mechanics [49]. Equation (49) can be solved by a combination of the It may be noted that (58) differs from Naber’s result 24(( )in Fourier and Laplace transform methods [52–54]. [19]) and will be discussed later. Advances in Mathematical Physics 7

6. Solution for TFSE for a Free Particle 0.5 The solution for the TFSE for a free particle under the 0.4 𝛽 = 1/4

conditions )

𝜓(𝑥, 0)0 =𝜓 (𝑥); 𝜓(𝑥, 𝑡), →0 |𝑥| →, ∞ 𝑡>0is available 1/2 ) 𝛽 = 1/2 𝛽 t

in the literature but considered here for purposes of obtaining f the Green function for the TFSE. 0.3 By applying the combined Fourier and Laplace trans- (|x|/(D

𝛽 𝛽 = 3/4 forms defined by G 0.2 1/2 ∞ ∞ ) 𝛽=1 𝛽 ̂ 1 −𝑖𝑘𝑥 −𝑠𝑡 t 𝜓̃ (𝑘,) 𝑠 = ∫ 𝑒 [∫ 𝑒 𝜓 (𝑥,) 𝑡 𝑑𝑡] 𝑑𝑥 (59) f √2𝜋 −∞ 0 (D 0.1 equation (49)reducesto 0.0 𝛽 ̂ 𝛽−1 2 ̂ 𝑠 𝜓̃ (𝑘,) 𝑠 −𝑠 𝜓̃ (𝑘,) 0 =−𝑖𝐷𝑓𝑘 𝜓̃ (𝑘,) 𝑠 (60) 012345 𝛽 1/2 |x|/(Dft ) resulting in Figure 1: Reduced probability density function. 𝜓̃ (𝑘,) 0 𝑠𝛽−1 𝜓̃̂ (𝑘,) 𝑠 = . 𝛽 2 (61) 𝑠 +𝑖𝐷𝑓𝑘

Applying the inverse Laplace transform, (61)yields TheGreenfunctionin(65)isthengivenby

2 𝛽 𝜓̃ (𝑘,) 𝑡 = 𝜓̃ (𝑘,) 0 𝐸𝛽,1 (−𝑖𝐷𝑓𝑘 𝑡 ) (62) 1 1 |𝑥| 𝐺𝛽 (𝑥,) 𝑡 = 𝑀𝛽/2 ( ). (68) in terms of the Mittag-Leffler function defined by a series or 2 𝛽/2 𝛽/2 √𝑖𝐷𝑓𝑡 √𝑖𝐷𝑓𝑡 bytheinverseLapalcetransform[3]:

∞ 𝑛 𝛼−𝛽 𝑧 −1 𝑠 𝐸 (𝑧) = ∑ =𝐿 { } . In the context of fractional diffusion, the function 𝑀𝛽/2(𝑧) 𝛼,𝛽 Γ(𝛼𝑛+𝛽) 𝑠𝛼 −𝑧 (63) 𝑛=0 belongs to the Wright type of probability densities character- 𝛽/2 ized by the similarity variable 𝑧 = |𝑥|/√𝐷𝑓𝑡 ,where𝐷𝑓 is The Green function solution can be written as ∞ ∫ 𝑀 (𝑧)𝑑𝑧 =1 ∞ the fractional diffusion coefficient, with 0 𝛽/2 . 𝜓 (𝑥,) 𝑡 = ∫ 𝜓 (𝑥−𝜉) 𝐺𝛽 (𝜉,) 𝑡 𝑑𝜉, (64) Furthermore, the probability densities are non-Markovian −∞ and exhibit a variance consistent with slow anomalous dif- 𝜎2(𝑡) = (2/Γ(𝛽 + 1))𝐷 𝑡𝛽 where the Green function is given by the inverse Fourier fusion [49–51], 𝛽 𝑓 . 𝛽→1 transform of the Mittag-Leffler function In the limit of , the probability density function goes over to the Gaussian 1 ∞ 𝐺 (𝑥,) 𝑡 = ∫ 𝑒𝑖𝑘𝑥𝐸 (−𝑖𝐷 𝑘2𝑡𝛽)𝑑𝑘, 𝛽 √ 𝛽,1 𝑓 (65) 2𝜋 −∞ 1 2 𝑀 (𝑧) = 𝑒−𝑧 /4 1/2 √𝜋 (69) where the Mittag-Leffler function has been defined [3]before in (63). The Fourier inversion in (65) can be carried out [52–54] corresponding to regular diffusion. A plot of the reduced using the property that the Mittag-Leffler function is related probability density function is given in Figure 1. through the Laplace integral to another special function of The Green function for regular diffusion describes a prob- the Wright type denoted by ability density, whereas the corresponding Green function for 𝑀 (𝑧) = 𝑊 (−𝑧; −𝛽, 1−𝛽) the Schrodinger equation is the propagator, which describes 𝛽 theprobabilityamplitudefortheparticletopropagatefrom 𝑥 𝑡 𝑥 𝑡 ∞ (−𝑧)𝑛 (66) 𝑎 at 𝑎 to 𝑏 at 𝑏.Inexactlythesameway,theGreen = ∑ 0<𝛽<1, function for time fractional diffusion describes a probability 𝑛!Γ (−𝛽𝑛 + 1 −𝛽) 𝑛=0 density, whereas the Green function in (68)isthefractional propagator and gives the probability amplitude. Of particular where the Wright function is defined by [54] interest is the Fourier component of the wave function in (62)inconnectionwiththetotalprobabilityas𝑡→∞;the ∞ 𝑧𝑛 𝑊 (𝑧; 𝜆,) 𝜇 = ∑ 𝜆>−1,𝜇∈𝐶. (67) case discussed by Naber [19] and will be discussed in the next 𝑛=0 𝑛!Γ (𝜆𝑛 +𝜇) section. 8 Advances in Mathematical Physics

7. Comments about Some Results in by Naber, and the function under discussion is the Fourier Naber’s Work [19] component of the free particle wave function, in his notation

] This section draws attention to some errors in Naber’s Ψ=Ψ0𝐸] (𝜔(−𝑖𝑡) ) , (74) paper which have gone unnoticed and have been reproduced repeatedly. Naber’s equations will be referred to by their where 𝐸](𝑧) in Naber’s notation corresponds to the Mittag- number and a prefix N. Naber explicitly states that he uses Leffler function 𝐸],1(𝑧) definedin(63). the Caputo derivative, which has been defined earlier in25 ( ) The Mittag-Leffler function with the complex argument in this paper. Although Naber does not give the explicit has been separated into an oscillatory part and a part based definition of the Caputo fractional derivative in his paper [19], on the evaluation of the inverse Laplace transform itcanbeinferredfrom(NA.3)givenintheappendixtohis paper. 1 𝛾+𝑖∞ 𝑒𝑠𝑡𝑠]−1𝐴 𝑑𝑠 (a) However, Naber writes in (N16) for a Caputo deriva- 𝐴 (𝑡) = ∫ 0 ] ] (75) tive of order (1−]), reproduced here for convenience in (70): 2𝜋𝑖 𝛾−𝑖∞ 𝑠 −𝜎𝑖 1 𝐷1−]𝜓 (𝑡, 𝑥) = along a Hankel contour and considering the contribution 𝑡 Γ (1−]) 1/] from the residue of the pole 𝑠0 =𝜎 𝑖 together with the (70) 𝑡 𝑑𝜓 (𝜏,) 𝑥 𝑑𝜏 contribution from the integral along the two strips on either × ∫ ] (0<] <1) . side of the branch cut, which is a standard procedure. Naber 0 𝑑𝜏 (𝑡−𝜏) finally gives the solution as Thisisincorrect,ascanbecheckedeasilybytakingthelimit 1/] ] →1.Theleft-handside → 𝜓(𝑡, 𝑥) as the derivative of zero 𝑒−𝑖𝜔 𝑡 Ψ=Ψ { −𝐹 (𝜔(−𝑖)],𝑡)}. order,buttheright-handside → 0becauseoftheΓ function 0 ] ] (76) in the denominator. The correct form of equation is 𝑡 𝑡→∞ 1−] 1 𝑑𝜓 (𝜏,) 𝑥 𝑑𝜏 He argues that in the limit of the total probability 𝐷 𝜓 (𝑡, 𝑥) = ∫ . 2 𝑡 1−] (71) 1/] Γ (]) 0 𝑑𝜏 (𝑡−𝜏) arises basically from the first term and is equal to , assuming that the wave function was initially normalized. As a consequence, the weight factor in (N18) should be (𝑡 − Since ] <1,thetotalprobabilityis>1, a result difficult to 1−] −] 𝜏) and not (𝑡 − 𝜏) , which Naber uses to give a physical understand physically. However, it is shown later that the significance to the entity he has introduced. The correct solution derived in this paper yields a probability that → 0 weight factor in (N18) would → 1inthelimit] →1. in the limit 𝑡→∞. (b) In (N11), Naber gives an identity satisfied supposedly The solution corresponding to (74)isgivenby(62)in by fractional Caputo derivatives of order less than 1, repro- this paper. The separation into two parts corresponding to duced here for convenience in (72). (76) can be carried out by considering the inverse Laplace transform of (61): ] 1−] ] 𝑑𝑦 [𝐷𝑡 𝑦 (𝑡)] 𝐷 𝐷 𝑦 (𝑡) = − 𝑡=0 . (72) 𝑡 𝑡 𝑑𝑡 𝑡1−]Γ (]) 1 𝛾+𝑖∞ 𝑒𝑠𝑡𝑠𝛽−1𝜓̃ (𝑘,) 0 𝑑𝑠 𝜓̃ (𝑘,) 𝑡 = ∫ . 𝛽 2 (77) 2𝜋𝑖 𝛾−𝑖∞ 𝑠 +𝑖𝐷 𝑘 This identity is not correct and cannot be found anywhere. 𝑓 The identity satisfied by the Caputo derivatives is given in[5] and reproduced later in the current notation for ] <1 As usual, the Bromwich contour is replaced by the Hankel contour. The two contributions arise from (i) the residue at −] ] 𝑠 =(𝐷𝑘2)1/𝛽(−𝑖)1/𝛽 𝐷𝑡 𝐷𝑡 𝑦 (𝑡) =𝑦(𝑡) −𝑦(0) . (73) thepoleat 0 𝑓 and (ii) the integral along the two strips from 0 to −∞ introduced along the branch cut. 1−] ] This yields 𝐷𝑡 𝐷𝑡 𝑦(𝑡) = 𝑑𝑦/𝑑𝑡𝑡 =𝜕 𝑦(𝑡) and has been used The latter yields a contribution which decays in time just as earlier in (52)inthispaper. the second term in Naber’s equation (76). The contribution The incorrect identity has been used by Naber to derive from the residue is given by an equation for the probability current, which is obviously 1/𝛽 2 1/𝛽 incorrect. Unfortunately, the incorrect identity, as given by (−𝑖) (𝐷𝑓𝑘 ) 𝑡 Residue 𝑒 Naber, in (72)hasbeenrepeatedlyusedintheliterature. = (78) 𝜓̃ (𝑘,) 0 𝛽 The correct equation for the probability current has been given in this paper in (59), which reduces to the standard continuity equation for the probability current in regular and corresponds to the first term on the right-hand side in quantum mechanics [49]. (76). However, (c) This point concerns the separation of the Mittag- 1/𝛽 −𝑖𝜋/2 1/𝛽 −𝑖𝜋/2𝛽 𝜋 𝜋 Leffler function with an imaginary argument into an oscilla- (−𝑖) = (𝑒 ) = (𝑒 ) = (cos −𝑖sin ) . tory part and a part which decays exponentially with time. 2𝛽 2𝛽 There is nothing wrong with the derivation itself as given (79) Advances in Mathematical Physics 9

Therefore, (78)yields Equation (47)thenbecomes

2 𝜓̃ (𝑘,) 𝑡 Residue 𝑖ℎ𝑓 𝜕 𝜓 (𝑥,) 𝑡 = 𝐶𝐷𝛼/2𝜓 (𝑥,) 𝑡 = II 𝜓̃ (𝑘,) 0 𝜓̃ (𝑘,) 0 0 𝑡 II 2𝑚 𝜕𝑥2 (80) 𝑓 1/𝛽 1/𝛽 (82) 𝑡(𝐷 𝑘2) cos(𝜋/2𝛽) −𝑖𝑡(𝐷 𝑘2) sin(𝜋/2𝛽). 𝑖 =𝑒 𝑓 𝑒 𝑓 − 𝑉 (𝑥,) 𝑡 𝜓 (𝑥,) 𝑡 . ℎ𝑓 The right-hand side of (80) has an amplitude term and an oscillatory factor. Because 𝛽<1,cos(𝜋/2𝛽) is negative, the Multiplyingbothsidesby−ℎ𝑓/𝑖 results in the FTSE for a amplitude factor decays in time exponentially and → 0inthe particle in a potential field as 𝑡→∞ limit . Thus the contribution from the residue due to 2 → → ℎ 𝜕2𝜓 (𝑥,) 𝑡 the pole also 0. Thus the entire wave function 0inthe 𝑖ℎ 𝜕𝛽𝜓 (𝑥,) 𝑡 =− 𝑓 −𝑉(𝑥,) 𝑡 𝜓 (𝑥,) 𝑡 𝑡→∞ 𝑓 𝑡 2 (83) limit of , so does the probability density. Hence the 2𝑚𝑓 𝜕𝑥 total probability also → 0, in contrast to Naber’s result. The question naturally arises why there is this difference which reduces to the standard Schrodinger equation of in the two results, both of which are concerned with the quantum mechanics: Mittag-Leffler functions with complex arguments. The reason 2 2 appears to be that in Naber’s case, the pole occurs at 𝑠0 = 𝜕𝜓 ℎ 𝜕 𝜓 1/] 𝑖ℎ =− +𝑉𝜓. (84) 𝜎 𝑖, whereas in the present paper, the pole occurs (using 𝜕𝑡 2𝑚 𝜕𝑥2 1/] 1/] thesamenotationasNaber)at𝑠0 =𝜎 (−𝑖) .Thesimple Onefinalremarkneedstobemadeconcerningtheuseofthe 𝑖 in Naber’s case leads to the purely oscillatory solution and Planck units for casting the TFSE in terms of dimensionless hence to the result of the probability being greater than unity. quantities. After the derivation, the FTSE in (83)canbecastin In our paper the imaginary number raised to the fractional dimensionless quantities; however, the appropriate fractional power leads to the exponentially decaying solution and hence Planck units must be defined to take care of the fractional leads to the correct limit when 𝑡→∞,namely,zerototal quantities involved in (83). probability. Naber’sresult is a direct consequence of his choice to raise the power of the imaginary number 𝑖 to the fractional power, so as to incorporate a Wick rotation. However, as 9. Discussion and Conclusions had been indicated earlier, this imaginary number cannot be arbitrarily altered as it is connected with the phase factor The TFSE has been derived using the Feynman path integral 𝑖𝑆/ℎ in the Feynman propagator. If it is necessary to include technique for a nonrelativistic particle. As expected the TFSE a Wick rotation, the power of 𝑖 should be changed to ] +1 looks like a time fractional diffusion equation with an imagi- in Naber’s equation (N9) instead of just ].Ifthisisdone, nary fractional diffusion constant but pertains to the realm the total probability would properly → 0inthelimit𝑡→ of subdiffusion only, in contrast to Naber’s generalization ∞.Ifthisisdone,thentheTFSEcanbeinterpretedasthe which includes superdiffusion as well. This is understandable analytic continuation of the fractional diffusion equation, just because the case considered in this paper pertains to the as the regular Schrodinger equation can be considered as the nonrelativistic case. Relativistic considerations would have to analytic continuation of the regular diffusion equation. be included for the superdiffusion case, which would lead to the Klein-Gordon equation in the integer order limit. In adapting the Feynman method, it is shown that it is preferable 8. TFSE for a Particle in a Potential Field to introduce the action 𝑆 as a fractional time integral of the Lagrangian and that it is necessary to introduce a “fractional So far attention has been focused on a free particle. These Planck constant.” In the limit of integer order, the regular considerations can be easily extended to the case of a particle action 𝑆, the regular Schrodinger equation, and the regular moving in a potential field by incorporating a potential term Planck constant are all recovered. It may be of interest to note 𝑉(𝑥, 𝑡) in the Lagrangian 𝐿=𝑇−𝑉.Thiswillnecessitate −𝑉(𝑥, 𝛼/2𝑡)𝜀 /Γ(1 + 𝛼/2) that there is a fractional Planck constant implied in Naber’s incorporating an additional term work also, although it is not explicitly so stated. His equations in the right-hand side of (35) and an additional factor are rendered nondimensional by using the Planck units of −(𝑖/ℎ )(𝑉(𝑥, 𝛼/2𝑡)𝜀 /Γ(1+𝛼/2)) 𝑓 in the exponential in (37). This mass, length, and time and then generalized to fractional results in changing (45)into derivatives after including a Wick rotation. This implies a 𝑡→𝑖𝑡 𝜀𝛽 change of variable of time so that the imaginary 𝜓 (𝑥,) 𝑡 + 𝐶𝐷𝛽𝜓 (𝑥,) 𝑡 number is raised to the same power as the order of the II 0 𝑡 II Γ(𝛽+1) fractional time derivative involved. However, the Planck units may not be the appropriate quantities as the equations involve ∞ 1 𝑎 𝜂2 𝑖 𝑉 (𝑥,) 𝑡 𝜀𝛼/2 = ∫ [− II ]{1− } quantities of fractional dimension and the equations must be 𝐴 exp 𝜀𝛼/2 ℎ Γ (1+𝛼/2) (81) −∞ II 𝑓 made nondimensional after the generalization to fractional derivatives and not before. A fractional Planck constant does 𝜕𝜓 𝜂2 𝜕2𝜓 ×{𝜓 (𝑥,) 𝑡 +𝜂 II + II }𝑑𝜂. show up in Naber’s treatment as well, as the ratio of masses II 2 𝜕𝑥 2 𝜕𝑥 𝑚/𝑀𝑝 [31, 32]. 10 Advances in Mathematical Physics

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