Probability Structure of Time Fractional Schrödinger Equation

Vol. 116 (2009) ACTA PHYSICA POLONICA A No. 2

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 α = 1 mechanics is given by 2 and we express the weighted probability density in terms ∂ρ + ∇ · 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 ρ = ψ∗ψ 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 fractional 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 ∝ ωα where 0 < α < 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 ∂αψ ~2 ∂2ψ i α~ = − + V (x)ψ, (3) α ∂tα 2m ∂x2 2 α where ~α = 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 χn(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 ψ(x, t) = ϕ(t)χ(x) and an energy relationship of side of the probability current equation vanishes as well. the form E = ~ ωα. The unit of E is as usual the unit of Therefore the probability equation in this case is α · ¸ energy. The unit of ~α is the unit of energy multiplied by ∂ρn Cn α −1 = 2Re , (13) (s) and the unit of ω is (s) . From Eq. (3) we obtain ∂t t1−αΓ (α) dαϕ(t) ωα − ϕ(t) = 0, (4) where ·µ ¶ ¸ dtα i α ∂αψ C = ψ∗ n . (14) and n n ∂tα ~2 d2χ(x) t=0 − + [V (x) − E]χ(x) = 0. (5) Now knowing the solution of the ϕ functions are Mittag– 2m dx2 Leffler with a complex argument we find There exists a multitude of definitions [9] like Riemann– · ¸ ∂ρ D E ((iω t)α) Liouville, Weyl, Reisz, and Caputo for the fractional n n α n = 2Re 1−α , (15) derivative. However, in this paper we only use the Ca- ∂t t Γ (α) 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- ωαχ2 (x) D = n n . (16) tional derivative for α > 0 is defined by n i α Z t (n) α C α 1 f (τ) D f(t) =0 Dt f(t) = α+1−n dτ, Γ (n − α) 0 (t − τ) 3. Limiting case of α = 1 − ε n − 1 < α < n, and t > 0, (6) When the order of fractional derivative α is close to where n is an integer number and f (n)(τ) denotes the unity, namely where α = 1 − ε, with small ε, then one n-th derivative of the function f(τ). It is possible to use can use an expansion. The method has been introduced an identity for the Caputo derivative of order 0 < α < 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 α ∂ ψ realm of fractional calculus. 2 ˜ ∂tα ∂ψ η ∂ ψ ξ ˜ t=0 = − α 2 + α ψ + 1−α , (7) In this method for the Caputo fractional derivative of ∂t i ∂x i t Γ (α) order (1 − ε) one defines where 1−ε (1) 1 ~2 V (x) ∂1−αψ D f(t) = f (t) + εD1f(t) + ..., (17) ˜ 1 η = , ξ = and ψ = 1−α . (8) where D f(t) is 2m~α ~α ∂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 (τ) ln(t − τ)dτ, (18) ∂ρ ∂j ηA + ξB C 0 + η = 2Re + , (9) where γ is the Euler constant. In previous works we have ∂t ∂x i α t1−αΓ (α) applied this method to the case of fractional oscillation where [10] and fractional relaxation phenomena [12]. ∂ψ˜ ∂ψ∗ A = ,B = ψ∗ψ˜ 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 α ¶ ¸ ∗ ∂ ψ ϕ(t) = ϕ0(t) + εϕ1(t) + ... (19) C = ψ α , (10) ∂t t=0 Upon insertion of Eqs. (17)–(19) in Eq. (4) we find the fractional probability current density is given by dϕ ω 0 − ϕ = 0, (20) ψ∗ ∂ψ˜ ψ ∂ψ˜∗ dt i 0 j = + , (11) i α ∂x (−i)α ∂x with solution ϕ0(t) = exp(−iωt). and the symbol “Re” in Eq. (9) stands for the real part. For the correction term we get dϕ ω ³ ω ω ´ The general quantum state for this theory is Ψ = 1 − ϕ = − D1 + ln ϕ , (21) P dt i 1 1 i i 0 cnψn(x, t) and the basis state ψn(x, t) is given by with the initial condition ϕ1(0) = 0. ψn(x, t) = χn(x)ϕn(t). (12) 1 Now we consider the quantity D1ϕ0. From expression However, it is hard to make any quantitative predictions (18) we have from the probability current equation for a general quan- D1ϕ = D1 exp(−iωt) = −iω ln(t) − iω exp(−iωt) tum state. 1 0 1 Z t So we only consider the basis states. One can show that −ω2 exp(−iωt) ln(t − τ)dτ. (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 £ ¤ iω ln s LT D1ϕ = . (23) 1 0 s + iω Taking the Laplace transform of Eq. (21) and with the help of expression (23) we ﬁnd iω ln s iω ln(−iω) Φ (s) = − + . (24) 1 (s + iω)2 (s + iω)2 Therefore in the time domain [13] we will have

ϕ1(t) = iωt exp(−iωt)[Ei(iωt) − ln(iω)] − 1

+ exp(−iωt) + iωt exp(−iωt) ln(−iω), (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 ε = 0.01 and ω = 10 . −∞ 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 = iπ [14], so ϕ(t) is dρn(x,t) /χ2 (x) versus time. £ dt n ϕ(t) = exp(−iωt) + ε − 1 + exp(−iωt) Here we restrict ourselves to the case of basis states. ¤ Moreover, in this section we consider the media with low- −iωt exp(−iωt)E1(−iωt) . (27) -level fractionality, where the order of fractional deriva- Therefore the probability distribution of a stationary tive is close to unity. Then it is possible to make quanti- state is tative statements about the probability structure of time ρ (x, t) = ψ ψ∗ = χ2 (x) {1 + 2ε [−1 + exp(−iωt) fractional Schrödinger equation. n n n n We find that in the beginning the source term of the ³π ío +ωt − Si(ωt) , (28) probability current equation creates some particles. But 2 in the fluctuating region whenever the slope of the curve where Si(t) is the sine-integral function defined by in Fig. 1 is negative it absorbs some particles, and when Z t sin(x) the slope of the curve is positive it creates particles. Si(t) = dx. (29) This process continues until we reach the steady state 0 x In Fig. 1 we draw the parameter y = ρ (x, t)/χ2 (x) ver- when the value of the probability remains approximately n n constant. sus time. As shown in this figure first we have an increase 13 in the value of this parameter followed by fluctuations For a value of ω = 10 we see from Fig. 1 that it takes around the value 1 + 2ε. a few picoseconds for the system to attain its equilibrium value. For higher values of frequencies this time will be much shorter. We see that the probability of the basis states are time dependent. But considering the range of energies in ordinary quantum mechanics which is usually of order of a few electron-volts, which corresponds to frequencies of order 1016, then the main fluctuations which is due to the imposition of fractional derivative occurs within a few millipicoseconds. Therefore the basis states of the theory in the case of a media with low-level fractionality are quasi-stationary. As far as the time rate of change of the probability is concerned, Fig. 2 shows a decreasing function followed by fluctuations around zero. This trend is what is given by Eq. (15) as well. Fig. 1. y versus t for ε = 0.01 and ω = 1013. The system attains its equilibrium value in few picoseconds, so the basis states are quasi-stationary. 4. Comparison with the exact result It is useful to evaluate the time rate of change of this Here we compare the result of our expansion method probability density. The result is with the exact calculation. The Mittag–Leffler function with a complex argument can be expressed by [8]: dρn(x, t) 2 = −2χn(x)ε si(ωt), (30) exp(iωt) (iω)α sin(απ) dt E ((iωt)α) = − where si(t) is the complimentary sine-integral defined by α Z α π π ∞ exp(−rt)rα−1 dr si(t) = Si(t) − . (31) × . (32) 2 2α α α 2α 0 r − 2(iω) cos(απ)r + (iω) Probability Structure of Time Fractional Schrödinger Equation 117 µ ¶ r ∗ This actually corresponds to ϕ (t). Now with α = 1 − ε, 1 1 −iπ √ ωt ζ 2 = (−iωt) 2 = exp ωt = (1 − i) . (41) and a small ε we obtain 4 2 ∗ exp(iωt) From Eqs. (38), (40), (41) we obtain the time depen- ϕ (t) = E1−ε = Z 1 − ε dent part of the wave function ∞ exp(−rt)dr −iωε . (33) ϕ(t) = exp(−iωt) erfc[(1 − i)G] (42) (r + iω)2 q 0 where G = ωt . To evaluate the integral of Eq. (33) which we denote by 2 λ we make the following change of variable r0 = r + iω: The parameter y which is measure of probability den- Z ∞ exp(−r0t)dr0 sity in this case is λ = exp(iωt) . (34) 02 ρn(x, t) ¯ ¯2 i ω r ¯ ¯ y = 2 = erfc[(1 − i)G] . (43) Using integration by parts and the definition of the ex- χn(x) ponential integral we find In Fig. 3 the parameter y is drawn versus G. The prob- · ¸ ability density starts from unity and then it fluctuates i 1 λ = − exp(iωt) exp(−iωt) + tE1(iωt) . (35) around the value which is equal to 4 in the present ω α2 case. Therefore from Eq. (33) we get exp(iωt) ϕ∗(t) = 1 − ε · ¸ i +iεω exp(iωt) exp(−iωt) + tE (iωt) , (36) ω 1 which is equal to £ ϕ∗(t) = exp(iωt) + ε − 1 + exp(iωt) ¤ +iωt exp(iωt)E1(iωt) . (37) But this is the complex conjugate of expression (27). So the expansion method produces the exact result up to order ε.

5. The special case of α = 1 2 p Fig. 3. y versus G = ωt/2 for α = 1 and ω = 1013. For a general quantum state and a general value of the 2 parameter α it is hard to describe the behavior of the probability density in time fractional Schrödinger equation. It is found [8] that the value of total probability 6. Concluding remarks 1 for a free particle increases to the value α2 in the limit t → ∞ (assuming the total probability equals unity at We have discussed the probability structure of time t = 0). fractional Schrödinger equation. For the case when the 1 order of fractional derivative is close to unity we find a In this section we discuss the case of α = 2 . The Mittag–Leffler function in this case is [15]: fluctuating probability density. However, considering the 1 1 1 range of energies in ordinary quantum mechanics which is E 1 (ζ 2 ) = exp(ζ)[1 + erf(ζ 2 )] = exp(ζ)erfc(−ζ 2 ), 2 usually of order of a few electron-volts, which corresponds to frequencies of order 1016, then the main fluctuation ζ ∈ C, (38) which is due to imposition of fractional derivative occurs where C denotes the set of complex numbers and erf (erfc) within a few-millipicoseconds. So the basis states of the denotes the (complementary) error function defined as theory are quasi-stationary. Furthermore, we see that in Z ζ 2 2 the steady state the probability is practically conserved. erf(ζ) = √ exp(−u )du, For a general value of α like α = 1 we find that the ap- π 0 2 proach to the equilibrium value is much slower than the erfc(ζ) = 1 − erf(ζ), ζ ∈ C. (39) 1 case of low-level fractionality. Furthermore, the proba- In Eq. (38) by ζ 2 we mean the principal value of the bility is not conserved in this case. square root of ζ in the complex plane cut along the neg- As it has been noted in Ref. [8] there are two ways ative real axis. to fractionalize the time derivative. The first option has For our case µ ¶ been expressed by Eq. (3). The second option is to con- iπ ζ = −iωt = exp − ωt, (40) sider 2 ∂αψ ~2 ∂2ψ i~ = − + V (x)ψ. (44) as for any complex number ζ the principal value of argζ α ∂tα 2m ∂x2 is the unique value of argζ such that −π < argζ ≤ π. The wave function and the probability density for a free Therefore particle within this option have been studied in [16, 17]. 118 A. Tofighi

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