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Probability Structure of Time Fractional Schrödinger Equation

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Probability Structure of Time Fractional Schrödinger Equation Vol. 116 (2009) ACTA PHYSICA POLONICA A No. 2 Probability Structure of Time Fractional Schrödinger Equation A. Tofighi¤ Department of Physics, Faculty of Basic Science, University of Mazandaran P.O. Box 47416-1467, Babolsar, Iran (Received July 19, 2007; in final form July 1, 2009) We consider the motion of a particle under the influence of a real potential within the framework of time fractional Schrödinger equation in 1 + 1 dimensions. For the basis states we obtain a simple expression for the probability current equation. In the limit where the order of fractional derivative is close to unity we find a fluctuating probability density for the basis states. We also provide a compact expression for the time rate of change of the probability density in this case. For the special case where the order of fractional derivative is equal 1 to 2 we compute the probability density. PACS numbers: 03.65.¡w 1. Introduction Utilizing a powerful expansion method [9, 10] we find the wave function and the probability density. We prove that Fractional quantum mechanics is a theory to discuss for the basis state the probability density increases ini- quantum phenomena in fractal environments. There has tially, then fluctuates around a constant value in later been many investigations on the subject. It is formulated times. We express the time rate of change of this prob- as a path integral over Lévy flight [1, 2], or from Nottale’s ability density in terms of simple function. We see that scale relativity theory [3, 4]. An algebraic treatment of in the steady state it fluctuates around zero. Moreover, the subject is given in [5, 6]. Finally, a deformed momen- we prove that the result of the expansion method is iden- tum operator is introduced in [7]. tical to the result that can be extracted from the exact The probability structure of space-fractional quantum solution. In Sect. 5 we discuss the special case of ® = 1 mechanics is given by 2 and we express the weighted probability density in terms @½ + r ¢ j = 0; (1) squares of absolute value of a complementary error func- @t tion. And finally in Sect. 6 we draw our conclusions. where j is a fractional probability current density and ½ = äà is the usual probability density [2]. In Ref. [8] the Schrödinger equation is considered with 2. Time fractional Schrödinger equation the first order time derivative changed to a Caputo frac- tional derivative. The resulting Hamiltonian is found to In this modern era and with the advent of many be non-Hermitian and time dependent. The structure new materials such as polymers, glasses, liquid crystals, of probability current equation for the free particle case biopolymers one would expect that the law and char- also has been obtained in [8]. In comparison to the space- acteristics of motion of a particle in such environments -fractional quantum mechanics the structure of probabil- (complex systems) is different from that of a simple sys- ity equation is more complicated. tem. In fact, such deviations have been observed for vari- The aim of the present paper is to study this probabil- ety of classical phenomena [11]. In this work we therefore ity structure in detail. In Sect. 2 first we find a general seek an anomalous relation between energy and the an- probability current equation for a particle moving under gular frequency of the form the influence of a real potential. Then by considering the E / !® where 0 < ® < 1; (2) basis states, we find the time rate change of the proba- this is called fractional Planck quantum energy relation. bility density. In Sections 3 and 4 we examine the case This assumption will imply a time fractional Schrödinger where the order of fractional derivative is close to unity. equation of the form @®Ã ~2 @2à i ®~ = ¡ + V (x)Ã; (3) ® @t® 2m @x2 2 ® where ~® = MPc T is a scaled Planck constant. The pa- ¤ e-mail: A.Tofi[email protected] P (114) Probability Structure of Time Fractional Schrödinger Equation 115 rameters MP and TP are Planck mass and Planck time, that Ân(x) is a real function we find that j = 0, moreover respectively, and V (x) is a real potential. Next we as- the contribution of the first two terms of the right hand sume Ã(x; t) = '(t)Â(x) and an energy relationship of side of the probability current equation vanishes as well. the form E = ~ !®. The unit of E is as usual the unit of Therefore the probability equation in this case is ® · ¸ energy. The unit of ~® is the unit of energy multiplied by @½n Cn ® ¡1 = 2Re ; (13) (s) and the unit of ! is (s) . From Eq. (3) we obtain @t t1¡®¡ (®) d®'(t) !® ¡ '(t) = 0; (4) where ·µ ¶ ¸ dt® i ® @®Ã C = ä n : (14) and n n @t® ~2 d2Â(x) t=0 ¡ + [V (x) ¡ E]Â(x) = 0: (5) Now knowing the solution of the ' functions are Mittag– 2m dx2 Leffler with a complex argument we find There exists a multitude of definitions [9] like Riemann– · ¸ @½ D E ((i! t)®) Liouville, Weyl, Reisz, and Caputo for the fractional n n ® n = 2Re 1¡® ; (15) derivative. However, in this paper we only use the Ca- @t t ¡ (®) puto fractional derivative as it is easier to apply the initial where the coefficient Dn is given by conditions in this type. The left (forward) Caputo frac- !®Â2 (x) D = n n : (16) tional derivative for ® > 0 is defined by n i ® Z t (n) ® C ® 1 f (¿) D f(t) =0 Dt f(t) = ®+1¡n d¿; ¡ (n ¡ ®) 0 (t ¡ ¿) 3. Limiting case of ® = 1 ¡ " n ¡ 1 < ® < n; and t > 0; (6) When the order of fractional derivative ® is close to where n is an integer number and f (n)(¿) denotes the unity, namely where ® = 1 ¡ ", with small ", then one n-th derivative of the function f(¿). It is possible to use can use an expansion. The method has been introduced an identity for the Caputo derivative of order 0 < ® < 1 in [9]. The main advantage of this expansion method is and find [8]: h i that it opens the way for analytical calculations in the ® @ à realm of fractional calculus. 2 ~ @t® @à ´ @ à » ~ t=0 = ¡ ® 2 + ® à + 1¡® ; (7) In this method for the Caputo fractional derivative of @t i @x i t ¡ (®) order (1 ¡ ") one defines where 1¡" (1) 1 ~2 V (x) @1¡®Ã D f(t) = f (t) + "D1f(t) + :::; (17) ~ 1 ´ = ;» = and à = 1¡® : (8) where D f(t) is 2m~® ~® @t 1 1 (1) (1) The discussion of [8] for the probability current equation D1f(t) = f (0) ln t + γf (t) is only for a free particle. In the case of a particle moving Z t under the influence of a real potential we find (2) · ¸ + f (¿) ln(t ¡ ¿)d¿; (18) @½ @j ´A + »B C 0 + ´ = 2Re + ; (9) where γ is the Euler constant. In previous works we have @t @x i ® t1¡®¡ (®) applied this method to the case of fractional oscillation where [10] and fractional relaxation phenomena [12]. @Ã~ @ä A = ;B = äÃ~ Essentially one treats the case of low-level fractionality @x @x as a perturbation to the integer case. One defines an and expansion of the physical variable of interest, namely ·µ ® ¶ ¸ ¤ @ à '(t) = '0(t) + "'1(t) + ::: (19) C = à ® ; (10) @t t=0 Upon insertion of Eqs. (17)–(19) in Eq. (4) we find the fractional probability current density is given by d' ! 0 ¡ ' = 0; (20) ä @Ã~ à @Ã~¤ dt i 0 j = + ; (11) i ® @x (¡i)® @x with solution '0(t) = exp(¡i!t). and the symbol “Re” in Eq. (9) stands for the real part. For the correction term we get d' ! ³ ! ! ´ The general quantum state for this theory is ª = 1 ¡ ' = ¡ D1 + ln ' ; (21) P dt i 1 1 i i 0 cnÃn(x; t) and the basis state Ãn(x; t) is given by with the initial condition '1(0) = 0. Ãn(x; t) = Ân(x)'n(t): (12) 1 Now we consider the quantity D1'0. From expression However, it is hard to make any quantitative predictions (18) we have from the probability current equation for a general quan- D1' = D1 exp(¡i!t) = ¡i! ln(t) ¡ i! exp(¡i!t) tum state. 1 0 1 Z t So we only consider the basis states. One can show that ¡!2 exp(¡i!t) ln(t ¡ ¿)d¿: (22) it is always possible to find real functions for the bound 0 state solutions of Eq. (5). Now from Eq. (4) and the fact If we compute the Laplace transform (LT) of expression 116 A. Tofighi (22) we obtain £ ¤ i! ln s LT D1' = : (23) 1 0 s + i! Taking the Laplace transform of Eq. (21) and with the help of expression (23) we find i! ln s i! ln(¡i!) © (s) = ¡ + : (24) 1 (s + i!)2 (s + i!)2 Therefore in the time domain [13] we will have '1(t) = i!t exp(¡i!t)[Ei(i!t) ¡ ln(i!)] ¡ 1 + exp(¡i!t) + i!t exp(¡i!t) ln(¡i!); (25) where E i (t) is the exponential integral defined by Z t exp(y) 13 Ei(t) = dy: (26) Fig. 2. z versus t for " = 0:01 and ! = 10 . ¡1 y The solution will take a simpler form in terms of an- other exponential integral defined by ¡E1(z) = Ei(¡z)¡ In Fig. 2 we show the variation of the parameter z = i¼ [14], so '(t) is d½n(x;t) =Â2 (x) versus time. £ dt n '(t) = exp(¡i!t) + " ¡ 1 + exp(¡i!t) Here we restrict ourselves to the case of basis states.
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