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International Journal of Theoretical and Mathematical 2015, 5(4): 58-65 DOI: 10.5923/j.ijtmp.20150504.02

The Infinite Square Well Problem in the Standard, Fractional, and Relativistic

Yuchuan Wei1,2

1International Center of , Three Gorges University, China 2Department of Oncology, Wake Forest University, NC

Abstract There has been a continuous argument on the correctness of the Laskin‟s solution for the infinite square well problem in the fractional quantum mechanics. In this paper, we prove that the Laskin‟s functions are not amathematical solution to the fractional Schrödinger and the equation does not have any nonzero solutions at all in the of . As in the standard quantum mechanics, we view the infinite square well problem as the limit of the finite well problem, and define the solution for the infinite square well problem as the limit of the solution for the finite square well problem. Using the simple property of the infinite operators, we show that Laskin‟s function can be the limit solution for the infinite square well problem. The single-sided well problem and the 3 dimensional well problem are included. This method works for the same problems in the relativistic quantum mechanicsas well. Keywords Fractional quantum mechanics, Relativistic quantum mechanics, Fractional Schrödinger equation, Relativistic Schrödinger equation

standard Schrödinger equation either, and it is nothing but 1. Introduction the limit of the solution to the finite square problem. Without a mathematical definition of the eigenvalues and In 2000, Laskin introduced the fractional quantum of a Hamiltonian operator with local infinity, mechanics [1-3]. As an example he solved the infinite square the argument will be endless. Therefore, in this paper we well problem in a piecewise fashion [3]. However, in 2010 mathematically define the infinite square well problem as a Jeng, et al [4] criticized that it was meaningless to solve a limit of the finite well. This viewpoint is also useful for other nonlocal equation in a piecewise fashion and they potentials with infinity, such as the coulomb potential in the demonstrated that it was impossible for the , the single-sided , etc. function to satisfy the fractional Schrödinger equation near Since it is difficult to solve the fractional Schrödinger‟s the boundary inside the well. In a series of papers [5-8], equation with a finite square well potential, we wish a direct Bayin insisted that he explicitly completed the calculation in way to find the solution of the infinite square well problem. Jeng‟s paper and the functions did satisfy the fractional Fortunately, we can express the fractional Hamiltonian in Schrödinger equation inside the well. Hawkins and Schwarz terms of the standard Hamiltonian. In the same way, we can [9] claimed that the Bayin‟s calculation contained serious mistakes. Luchko [10] provided some evidence that the also solve the infinite square well problem in the relativistic solution did not satisfy the equation outside the well. On the quantum mechanics, since the fractional and relativistic other hand, Dong [11] re-obtained the Laskin‟s solution by quantum mechanics are closely related. We acknowledge solving the fractional Schrödinger equation with the path that these solutions need to be verified when the solutions to method. It is not easy for readers to judge their the finite square well problem are reported later. mathematical argument [12, 13], but weagree with Jeng‟s We first recall the infinite square well problem in standard opinion, including that the piecewise way to solve the quantum mechanics, and then solve the problem in the equation is wrong, and that the solution does not satisfy the fractional quantum mechanics and in the relativistic quantum fractional Schrödinger equation, since we will inarguably mechanics. show that the Laskin‟s functions do not satisfy the fractional Schrödinger equation ( 1) anywhere on the x-axis. However, in fact, this solution does not satisfy the 2. Definition of the Infinite Well Problem * Corresponding author: In this section we will recall the relation between the finite [email protected] (Yuchuan Wei) Published online at http://journal.sapub.org/ijtmp and infinite square well problems in the standard quantum Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved mechanics, and accordingly define the infinite square well

International Journal of Theoretical and 2015, 5(4): 58-65 59

problem in the fractional quantum mechanics. 0 xa , . Vxi     (8) 2.1. The Finite and Infinite Potential Wells in the  xa  Standard Quantum Mechanics At xa , the potential has two infinite jumps. The In the standard quantum mechanics [14], the one subscript imeans the infinite square well. dimensional -independent Schrödinger equationis Obviously the potential Vi is not real in physics and not H x  E x , (1) well-defined in mathematics, since we do not know how to determine the value of the multiplication V x() x where  x is a defined on the x-axis, and i outside the well, even if the wave function outside the well is E is an . The Hamiltonian operator zero. (Does 0 equal 0,1, or  ?) In physics this H T V x (2) potential is used as a simplified model for „a very deep finite square well‟. is the summation of the and the Conventionally the Schrödinger equation potentialenergy of a . 22d The standard kinetic is  V x  x  E   x (9) 2m dx2 i pd2 2 2 , (3) T    2 is solved in a piecewise way again [15]. The wave function 22mmdx outside the well is zero because the potential is infinite and where p is the one dimensional operator the wavefunction inside the well is a sine or cosine function. The revised continuity condition for this case is that the wave d functions must be continuous but their should not  . (4) dx be. The infinite square well problem has the simple and well-known solution As usual, m is the of the particle and ℏ is the reduced Plank constant.  1 n A finite square well [15] is defined as  sin x a x a (),  n x   a 2a (10)  0 xa  0,xa , , Vx    (5) f V x a  0 n2 2 2 (11) En  2 where V0  0 is the depth of the well and 2a is the width of 8ma the well. The subscript f means the finite square well. for n 1,2,3, . The eigenequation of the finite square well problem is However, we emphasize that this solution does not satisfy the Schrödinger (9). Take the ground state as an 22d  V x  x  E   x . (6) example, we have 2m dx2 f 22d This equation can be solved separately in three regions V x  x 2m dx2 11i first and then the piecewise solutions are connected by the . (12) continuity condition that the wave function and its derivative  2 must be continuous. This process generates the eigenvalues E11 x   ()() x  a   x  a  4ma a Exfn ()and eigenstates  fn ()x , with n 1,2,3 . The The wave function satisfies the Schrödinger equation state with n=1 is the ground state [15]. everywhere except at x=a and x=-a. This is not a small flaw One can easily verify that the explicit solutions satisfy the for a , and we have to say that the wave eigenequation at every point on the whole x-axis functions in (10) are not the solution of the Schrödinger 22d equation at all. The extra terms in the above equation comes , (7) from the discontinuity of the derivative of the wave function, 2 fn V f x  fn x  E  fn  x 2m dx but if we keep the derivative of the wavefunction continuous, which convinces us that the continuity condition we use is we can only get a trivial solution  x  0 , which is suitable. In fact this continuity condition comes from the meaningless in physics. We have to say that the Schrödinger Schrödinger equation. equation with the infinite square well potential does not have The infinite square well potential is defined as solutions in mathematics. (See the appendix if one feels

60 Yuchuan Wei: The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics surprised.) where the coefficient D  mc2 /() mc  with χ a Since the infinite square well is a limit of the finite square well, conventionally we call the limit of the solutions of the positive , and c is the of the . When finite square well problem   2 , taking   1/ 2 and hence Dm 1/ (2 ) , the fractional kinetic energy recovers the standard kinetic energy, limfn (x ) n ( x ), lim E fn ( x ) E n , (13) VV  2 00 i.e. T2  p/ (2 m ) T . Originally Laskin [1-3] required the solution of the infinite square well, and symbolically the fractional parameter 12 , but in this paper we write allow 02 . The fractional Schrödinger equation is 22d n V i x  n x  E n  n  x . (14) 2m dx2 HxE   x (17)

The piecewise way to solve the Schrödinger equation for HTV. (18) the infinite square well, together with the aforementioned revised continuity condition, is nothing but an easy and For the infinite square well problem, we have direct way to get the limit of the finite square well solution d 2 2 . (19) without solving the finite square well problem. D()()2  x  Vi  x  x  E   x This definition for the infinity potential is applicable for dx several important cases in quantum mechanics. It helps us to Laskin [3] solved the equation above in a piecewise answer the following questions: fashion, and got the same wave functions and a new energy (1) in what sense the wave functions of the hydrogen level atom [14]can satisfy the Schrödinger equation at  1 n sin x a x a (), the origin r  0 where the coulomb potential  (20)  n x   a 2a V( r ) 1/ r is undefined. (r is the of a  0,xa point to the origin.)   (2) why the wave functions for the potential n 2 V( r )  1/ r  / r [14], where   0 is EDn  . (21) 2a much smaller than 1, can be divergent at the origin while we claim that the wave functions should be However, in 2010 Jeng, et al [4] criticized that it was be continuous and differentiable everyday, meaningless to solve the nonlocal equation (19) in a (3) why only the odd (rather than even) piecewise fashion and by contradiction they demonstrated eigen-functions of the simple harmonic oscillator that the ground state function did not satisfy the fractional [15] are the solutions of the single-sided oscillator Schrödinger equation. We believe that Jeng et al is correct and Basin is wrongin 1 22  m x x  0 this argument. There are two reasons. Vx()  2 . (15)  x 0 (1) The inarguable evidence is that the solution does not satisfy the fractional Schrödinger equation (4) why half-sine functions are the solution of the (  1 ) anywhere on the x-axis. Taking the single-sided infinite square well ( or infinite wall) ground state as an example, we have the fact but half-cosine functions are not. We will explain 2 this case further in the section III. 2dd 1/2  D1( ) 1 () x D 1   ()*(1/) x  x In one word, the different continuity conditions of wave dx2 dx functions in quantum mechanics are used in order that one D1 x   x  x  can directly find the limit solution for a potential with Si(  )Si(   )cos (22) infinity, instead of solving the Schrödinger equation with a 2aa2 2a 2 2 a a corresponding finite potential first and then calculating the D1 x   x  x  limit of the solutions. Ci(  )  Ci(  ) sin . 2aa2 2a 2 2 a a 2.2. The Laskin’s Solution of the Infinite Square Well The above function is not proportional to 1()x inside In 2000, Laskin generalized the classical kinetic and the well, it is not zero outside the well, and it is divergent at momentum relation to the boundaries xa . Here Si and Ci are sine and cosine   /2 integral functions, respectively. ||pd2  1 22, (16) T D || p 2 mc  D    2 (2) In fact, the fractional Schrödinger equation with mc dx the infinite square well (19) does not have any

International Journal of Theoretical and Mathematical Physics 2015, 5(4): 58-65 61

nonzero solutions. Since the potential Vx() is square well problems. infinity, to avoid V()() x x be infinity, we have 3.1. The 1D Infinite Square Well Problem to let ()x be zero outside the well, that is, For the 1D infinite square well problem, the fractional ()x is a compactly supported function. Hamiltonian

Thuswe have V( x ) ( x ) 0 on the x-axis (if H T Vi () x (24) we take 00  ). For 0 2, the kinetic can be expressed in terms of the standard Hamiltonian energy operator is nonlocal, so the resultant  H T Vi () x . (25) function D | p | ( x ) will be extended outside the well. Therefore Equation (19) requires as a compacted function to equal a non-compacted H (2 m )/2 D H /2 . (26) function. This is impossible. Therefore there are  no nonzero solutions to the fractional Schrödinger Here is the deduction of the above operator relationship equation (0 2) with an infinite square /2  /2  /2  /2 well potential. By the way, Luchko [10] ever had a (2m ) D H (2 m ) D T V x conjecture that the solution for the infinite square  (2m )/2 D T /2 | x | a well problem in fractional quantum mechanics     should be treated as a new special function.  /2  /2 (2m ) D  V x | x | a Now let us define the infinite square well problem and its D( p2 ) /2 | x | a solutionin the fractional quantum mechanics    mathematically.  ||xa  Definition 1. The solution of the infinite square well T || x a (27) problem.   Suppose the fractional Schrödinger equation for the finite ||xa  square well potential  0 |xa | 2 T  d   T|| x  a 2   D()()2 f x  V f x  x  E f  f  x dx  0 |xa | T  has the solutions  fn ()x and E fn with n 1,2,3, . ||xa  If their limits exist TVH   .

limfn (x )  n ( x ), lim E  fn E  n , (23) Therefore we have VV00  H( x ) (2 m )/2 D H /2 ( x ) we say that they are the solutions for the infinite square well nn  /2  /2 . (28) problem, and symbolically write  (2m ) D  Enn  ( x ) 2  2 d n D()()n x  V i x   n x  E  n   n  x .  Dx  n () dx2 2a Obviously it will be difficult and complicated to solve the We see that the wave functions of the fractional infinite square well problem according to the above Hamiltonian are the same functions of the original definition. Before any strict solutions have been reported in Hamiltonian, and the new energy levels are the publications, let us develop a tentative way to solve the  infinite square well problem based on some intuitive n EDn   . (29) property of the infinity operator. 2a That is to say, the Laskin‟s solution is re-obtained in a non-piecewise fashion. 3. The Infinite Square Well Problems in In the above deduction, we use some formal operations of Fractional Quantum Mechanics the infinity operator  , such as Based on the relation between the standard and fractional T     Hamiltonians, one can construct a non-piecewise method for the infinite square well. There are three cases, (1) one  T   (30) dimensional, (2) three dimensional and (3) one-sided infinite  /2  .

62 Yuchuan Wei: The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics

3.2. The 3D Infinite Square Well Problem With n, m , l  1,2,3 . In quantum mechanics, the 3D infinite square well is defined 3.3. One-sided Infinite Square Well The 1D one-sided infinite square well is defined as  0x a ,yz  a , and  a VV()r x, y, z  (31)  otherwise.  x  0,  Ux    (42) 0 x > 0. Here r  (,,)x y z is a point in the 3D Euclidean . Please notice that the 3D infinite square well potential is The standard Hamilton is the summation of three one dimensional infinite square p2 wells, H U x . (43) 2m VVVVx,, y z  x  y  z (32) i i i The Schrödinger equation since 0 0  0,0  , and  . p2 By the way, a 3D finite square well potential does not have H()()() x   x  U x  x (44) such a simple relation, so it is very difficult and complicated 2m to solve. Here we see that the use of the infinity potential  has the solution can greatly simplify the problem with a very big finite potential.  00x  ()x  (45) The standard Hamiltonian is sin(kx ) x  0 p2 2 2 2 2 H  V()(,,)r      V x y z (33) 22k 22mmxxx222 E  (46)  2m where p is the 3D . with k  0 . The Schrödinger‟s equation Here we repeat that the wave function in (45) does not H(,,)(,,) x y z  E  x y z (34) satisfy the Schrödinger equation (44) at x=0. Then one natural question is why we do not call has the solution  00x  (,,)()()()x y z  n x  m y  l z (35) ()x  (47) cos(kx ) x  0  22 2 2 2 , (36) a solution. Elmn 2 () n  m  l 8ma In fact, for a finite single–sided square well with .  ()x is defined in (10) n, m , l  1,2,3 n V0 x 0, Uxf     (48) The fractional Hamilton is 0 x>0, 2  /2 HDV pr () the solution to the Schrödinger equation is  /2 . (37)  Aexp( x ) x  0 222 ()x (49) DV     ()r f   222 sin(kx0 ) x 0 xxx 22 Again, we have the same relation between the standard Here the parameters k  0 and  2/mV0 k . and fractional Hamiltonian The other parameters A and  are chosen so that this 0 /2 /2 solution is continuous and differentiable at x=0. It is easy to H (2 m ) D H (38) verify that Therefore, the fractional Schrödinger equation  0x  0, H(,,)(,,) x y z  E  x y z (39) limf (x )  (50) V0 sin(kx ) x  0. has the solution This is the reason why the half-sine function in (45) is a (,,)()()()x y z   x  y  z (40) n m l solution, but the half-cosine function ()x in (47) is not.  The fractional Hamiltonian is  2 2 2 /2 Enml  D() n  m  l , (41)  2a H D T U x . (51)

International Journal of Theoretical and Mathematical Physics 2015, 5(4): 58-65 63

Based on (26), the fractional Schrödinger equation  H()()() x  D T  x  U x  x (52) has the solution  00x  ()x  (53) sin(kx ) x  0  E D k  (54) with k  0 .

4. Infinite Square Well Problems in Relativistic Quantum Mechanics

According to [14], the relation between the kinetic energy Tr and the 3D momentum p is 2 2 2 4 Tr p c m c . (55) The subscript r means special relativity. The relativistic Hamiltonian [14] is

2 2 2 4 Hr p c  m c  V r . (56) In Summerfield‟s [14], this Hamiltonian leaded to an energy formula for the coulomb potential, which accurately matches the hydrogen with the . Accordingly, the relativistic Schrödinger equation is  i(,)(,)(,)r t p2 c 2  m 2 c 4  r t  Vr  r t . (57) t The square root operator above is defined by the Fourier Transformation of the wave function, 1 p2cmct 2 2 4( r , )  di 3 p exp( p  r / ) p 2 cmc 2  2 4 exp(  i p  r '/ ) ( r ', td ) 3 r '. (58) (2 )3 RR33

Obviously it is difficult to deal with this equation from low to high, the relativistic Schrödinger equation (57) mathematically [14], so this equation had been abandoned will approximately relate to a fractional Schrödinger until recently when we discovered that its energy formula equation, whose parameters  changes from 2 to 1. 5 Therefore the fractional and the relativistic Schrödinger has an extremely valuable  terms, which is 41% of the experimental [16, 17]. As the only exception in equation should be studied at the same time as two sister this paper, the notation  stands for the fine structure equations. constant rather than the fractional order. 4.1. The 1D Infinite Square Well For a low speed , the relativistic equation (57) recovers the Schrödinger equation For the one-dimensional infinite square well problem, the relativistic Hamiltonian is 2  2 p i(,)(,)(,)r t mc   r t  Vr  r t , 2 2 2 4 tm2 Hri p c  m c  V x 2 (60) whose fractional parameter   2 . d  c2 2  m 2 c 4  V x. For a very high speed motion, if the rest energy can be dx2 i neglected approximately, we have Again, we can express the relativistic Hamiltonian in  i(,)(,)(,)r t c p2  r t Vr  r t , (59) terms of the classical Hamiltonian as t H2 mc2 H m 2 c 4 , (61) which is the fractional Schrödinger equation with  1. r Generally speaking, if the speed of a particle increases since

64 Yuchuan Wei: The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics

Hamiltonians (61), we know that the relativistic Schrödinger 2mc2 H m 2 c 4 equation has the solution 22 2d 2 4 00x  2mc   V ( x )  m c . (62)  2m dx2 ()x  (72)  sin(kx ) x  0 2 2 2d 2 4 2 2 2 2 4  c  m c  V() x  H Er  k c m c . (73) dx2 r See the stepwise deduction in (27) for details. 5. Conclusions Therefore the Laskin‟s wave functions  n ()x satisfy the relativistic Schrödinger equation We agree with Jeng et al that the fractional Schrödinger equation cannot be solved in a piecewise fashion, and the H()() x E x (63) r n rn n Laskin‟s functions are not a mathematical solution of this with a new equation. On the other hand, since these functions are not the mathematical solution of the standard Schrödinger equation 2 2 2 2 nc either and are just the limit of the solutions of the finite E m24 c (64) rn 4a2 square well, to disprove the Laskin‟s solution, one needs to solve the finite square well problem and take the limit of the for n 1,2,3, . solutions. Before the solution of the finite square well For the extreme relativistic case, we have problem is reported, we develop a tentative direct method to nc solve the infinite square well problem in three cases using the En  . (65) straightforward property of the infinity operator. Meanwhile, 2a we point out that the relativistic quantum mechanics is an approximate realization of the fractional quantum mechanics 4.2. The 3D Infinite Square Well Problem and the infinite square well problem can be treated in a same The relativistic Hamiltonian way.

2 2 2 4 Note 1. We just noticed that the paper [18] reported some Hrip c  m c  V r (66) initial research on the one dimensional relativistic infinite can be expressed in terms of the classical Hamiltonian square well, but to utilize the existing mathematical results, they used a completely different definition on the infinite p2 square well, which had no relation to the finite square well. HVr 2m i They defined 00  in the multiplication Vx(), and artificially demanded the resultant function of the as Hamiltonian operator to become zerooutside the well though they are actually not. From the viewpoint of physics, they H2 mc2 H m 2 c 4 . (67) r changed the infinite square well problem to another one. Therefore, the relativistic Schrödinger equation Note 2. In [4], there were two sentences that “By raising that Hamiltonian to the α/2, we get a plausible HE()()rr   (68) rr fractional Laplacian and Eq. (7) is indeed a solution. has the solution However, this is not the Riesz fractional derivative.” These words remind us that Jeng et al knew the method used in this (69) (,,)()()()x y z  n x  m y  l z paper but thought it did not since the operator in this method was not Riesz fractional derivative. In fact, this is the 2  c 2 2 2 2 4 Riesz fractional derivative, see Equation (7) in [1] for the Ernml () n  m  l  m c . (70) 2a definition.

4.3. The One-sided Infinite Square Well Problem ACKNOWLEDGEMENTS The relativistic Hamiltonian for the one-sided square well is This work is supported by the National Natural Foundation. The research on the relativistic Schrödinger d 2 equation was supported by Gansu Industry University during 2 2 2 4 (71) Hr   c2  m c  U x. 1989-1991, with a subject title „On the solvability of an dx equation with a square root operator in relativistic quantum Based on the relation between the relativistic standard mechanics‟. Thanks for the support from my family as well.

International Journal of Theoretical and Mathematical Physics 2015, 5(4): 58-65 65

Appendix 1. ‘’ [3] N. Laskin, “Fractional and quantum mechanics,” Chaos 10, pp780-790 (2000). Usually in quantum mechanics it is also demanded that the [4] M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On derivative of the wavefunction in to the the nonlocality of the fractional Schrödinger equation,” wavefunction itself be continuous; here this demand would Journal of Mathematical Physics 51, 062102 (2010). lead to the only solution the constant zero function, which is not what we desire, so we give up this demand (as [5] S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math.Phys. 53, 042105 this with infinite potential can be regarded as a (2012). nonphysical abstract limiting case, we can treat it as such and "bend the rules"). Note that giving up this demand means [6] S. S. Bayin, “Comment„On the consistency of the solutions of that the wavefunction is not a differentiable function at the the space fractional Schrödinger equation,” Journal of Mathematical Physics 53, 084101 (2012). boundary of the box, and thus it can be said that the wavefunction does not solve the Schrödinger equation at the [7] S. S. Bayin, “Comment„On the consistency of the solutions of boundary points x = 0 and x = L (but does solve everywhere the space fractional Schrödinger equation,” Journal of else). Mathematical Physics 54, 074101 (2013). http://en.wikipedia.org/wiki/Particle_in_a_box [8] S. S. Bayin, “Consistency problem of the solutions of the space fractional Schrödinger equation”, Journal of Mathematical Physics 54, 092101(2013). Appendix 2. An Alternative Definition [9] E. Hawkins and J. M. Schwarz, “Comment„On the on the Infinite Square Well Problem consistency of the solutions of the space fractional Schrödinger equation,” Journal of Mathematical Physics 54, Definition 2. The solution of the infinite square well 014101 (2013). problem. [10] Y. Luchko, “Fractional Schrödinger equation for a particle We define a potential moving in a ,” Journal of Mathematical Physics 2m 54, 012111 (2013). x Vm  x  V0  [11] J. Dong, “Levy path integral approach to the solution of the a fractional Schrödinger equation with infinite square well,” preprint arXiv:1301.3009v1 [math-ph] (2013). with V0 >0 and a>0, m=1,2,3 . [12] J. Tare and J. Esguerra, “Bound states for multiple Dirac-δ Suppose the fractional Schrödinger equation for the finite wells in space-fractional quantum mechanics,” Journal of potential Vxm   Mathematical Physics 55, 012106 (2014). d 2 [13] J. Tare and J. Esguerra, “Transmission through locally D()()2  x  V x  x  E   x periodic potentials in space-fractional quantum mechanics,” dx2 m m m f Physica A: and its Applications 407(2014), pp 43–53. has the solutions mn ()x and Emn with n 1,2,3, . [14] A. Messiah, Quantum Mechanics Vol 1, 2(North Holland If their limits exist Publishing Company 1965). limmn (x )  n ( x ), lim E  mn E  n , mm  [15] M. Bellonia and R.W. Robinettb, “The infinite well and potentials aspedagogical, mathematical and we say that they are the solutions for the infinite square well physical models inquantum mechanics,” Physics Reports problem, and symbolically write 540(2014), pp25-122. 2 d [16] Y. Wei, “The Quantum Mechanics Explanation for the Lamb 2 . D()()2 n x  V i x   n x  E  n   n  x Shift,” SOP Transactions on Theoretical Physics1 (2014), no. dx 4, pp.1-12. [17] Y. Wei, “On the divergence difficulty in perturbation method for relativistic correction of energy levels of H atom,” College Physics14(1995), No. 9, pp25-29. REFERENCES [18] K. Kaleta, M. Kwasnicki, and J. Malecki, “One-dimensional [1] N. Laskin, “Fractional quantum mechanics,” Physical quasi-relativistic particle in a box,” Reviews in Mathematical Review E 62, pp3135-3145 (2000). Physics25, No. 8 (2013) 1350014. [2] N. Laskin, “Fractional Schrödinger equation,” E66, 056108 (2002).