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Comment on ``Fractional Quantum Mechanics Comment on “Fractional quantum mechanics” and “Fractional Schrödinger equation” Yuchuan Wei To cite this version: Yuchuan Wei. Comment on “Fractional quantum mechanics” and “Fractional Schrödinger equation”. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2016, 93, 10.1103/physreve.93.066103. hal-01588657 HAL Id: hal-01588657 https://hal.archives-ouvertes.fr/hal-01588657 Submitted on 16 Sep 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 PHYSICAL REVIEW E 00, 006100 (2016) 2 Comment on “Fractional quantum mechanics” and “Fractional Schrodinger¨ equation” * 3 Yuchuan Wei () 4 Nanmengqian, Dafeng, Wuzhi, Henan, 454981, China 5 (Received 19 July 2015; revised manuscript received 14 January 2016; published xxxxxx) 6 In this Comment we point out some shortcomings in two papers [N. Laskin, Phys.Rev.E62, 3135 (2000); 66, 7 056108 (2002)]. We prove that the fractional uncertainty relation does not hold generally. The probability 8 continuity equation in fractional quantum mechanics has a missing source term, which leads to particle 9 teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be 10 viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an 11 observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be 12 viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be 13 viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how 14 to teleport a particle to an arbitrary destination. 15 DOI: 10.1103/PhysRevE.00.006100 16 I. INTRODUCTION (ii) The fractional probability continuity equation obtained 41 by Laskin [3]was 42 17 Historically quantum mechanics based on a non-New- 18 tonian kinetic energy has been studied widely [1]. In ∂ ρ + ∇ · jα = 0, (4) 19 Refs. [2,3], standard quantum mechanics [4] was generalized ∂t 20 to fractional quantum mechanics. The Schrodinger¨ equation where the probability density and current density were defined 43 21 was rewritten as as 44 ∂ ρ = ψ∗ψ, i ψ(r,t) = Hαψ(r,t), ∂t − ∗ − − ∗ j =−iD α 1[ψ (−∇2)α/2 1∇ψ − ψ(−∇2)α/2 1∇ψ ]. α α α Hα = Tα + V = Dα|p| + V (r). (1) (5) 22 As usual, ψ(r,t) is a wave function defined in the three- In fact, a source term was missing, which indicates another 45 23 dimensional space and dependent on time t, Dα is a constant way of probability transportation, probability teleportation. 46 24 dependent on the fractional parameter 1 <α 2, r and p (iii) The relationship between fractional quantum mechan- 47 25 are the position and momentum operators, respectively, ics and the real world was not given and it was almost 48 26 is the Plank constant, and m is the mass of a particle. The impossible to find the applications of this theory. Here we will 49 27 fractional Hamiltonian operator Hα is the sum of the fractional point out that the relativistic kinetic energy can be viewed as an 50 28 kinetic energy Tα and the potential energy V (r). When α = 2, approximate realization of the fractional kinetic energy, which 51 29 taking D2 = 1/(2m), the fractional kinetic energy becomes makes the probability teleportation a practical phenomenon. 52 30 the classical kinetic energy Now we will discuss these shortcomings in order. For the 53 + convenience, please be reminded that the symbol H in [2,3] 54 p2 T = = T (2) should be H. 55 2 2m 31 and the fractional Schrodinger¨ equation becomes the standard II. FRACTIONAL UNCERTAINTY RELATION 56 32 Schrodinger¨ equation. When 1 <α<2, the fractional kinetic A. The uncertainty relation is independent of wave equations 57 33 energy operator is defined by the momentum representation 34 [2]. However, there exist three shortcomings in this recent For simplicity, we do not consider wave functions that 58 35 quantum theory. are not square integrable. Suppose that ψ(x) is a normal- 59 36 (i) The Heisenberg uncertainty relation was generalized to ized square-integrable wave function defined on the x axis. 60 37 the fractional uncertainty relation [2] Heisenberg’s uncertainty relation says [4] 61 2 2 1/μ 1/μ (x) (p) , (6) |x|μ |p|μ > , μ<α, 1 <α 2. (3) 2 (2α)1/μ where 62 38 It seems unsuitable to call this inequality fractional uncer- x = x − x,p= p − p. (7) 39 tainty relation and this inequality does not hold mathemati- 40 cally. As usual, x and p stand for the one-dimensional position 63 and momentum operators and x and p stand for their 64 averages on the wave function ψ(x), for example, 65 ∞ = ∗ − ∂ * p ψ (x) i ψ(x)dx. (8) [email protected] −∞ ∂x 2470-0045/2016/00(00)/006100(5) 006100-1 ©2016 American Physical Society COMMENTS PHYSICAL REVIEW E 00, 006100 (2016) 66 This relation holds for all the square-integral functions and sides of the inequality (3) are continuous functions about 103 67 it is a property of the space of square-integrable functions. A α. Taking the limα→1+ of the both sides of the fractional 104 68 complete mathematical proof can be seen in [5]. uncertainty relation (3), we get 105 69 As a kinetic equation, the Schrodinger¨ equation ∂ |x||p| , (13) i ψ(x,t) = Hψ(x,t)(9) 2 ∂t which contradicts inequality (12). Therefore, the fractional 106 70 tells us how to determine the wave-function–time relation uncertainty relation (3) does not hold mathematically. 107 71 ψ x,t H ( ) by the Hamiltonian operator given the initial wave Further, once the fractional uncertainty relation does not 108 72 ψ x, function ( 0). From the viewpoint of geometry, Eq. (9) de- hold for certain α at t = 0, we know that there exists a 109 73 fines a curve in the space of square-integrable functions, which small time neighborhood [0,δ) for which the relation does 110 74 t = passes a given point at time 0. Heisenberg’s uncertainty not hold either since the wave packet has not expanded very 111 75 ¨ relation and the Schrodinger equation are independent. Laskin much. In short, the fractional generalization of the Heisenberg 112 76 ¨ generalized the Schrodinger equation, but wave functions uncertainty relation does not hold generally. 113 77 remains square-integrable functions. In other words, the used We would like to explain why we can take α = 1, which 114 78 function space remains the space of the square-integrable was not included in [2,3]. The case α = 1 is just a step of 115 79 functions, so the Heisenberg uncertainty relation remains true, our proof, like an auxiliary line used in geometry problems. 116 80 regardless of the standard or fractional quantum mechanics. Here we add two points. (i) There exist papers that allow 117 81 In addition, suppose that there is an uncertainty relation 0 <α 2. In [6], Jeng et al. claimed that Laskin’s solutions 118 82 ¨ that holds for all the solutions of the fractional Schrodinger for the infinite square-well problem were wrong by means of 119 83 α α = . equation (1) with certain , e.g., 1 5. Since the initial wave the evidence from the case 0 <α<1. In fact, the evidence 120 84 ψ x, function ( 0) is an arbitrary square-integrable function, from the case α = 1 is more straightforward [7]. (ii) The 121 85 we know that this uncertainty relation holds for the whole fractional Schrodinger¨ equation with α = 1 has many closed- 122 86 space of square-integrable functions. Therefore, there does form solutions [8], which is an easy starting point for the study 123 87 not exist a so-called fractional uncertainty relation. Generally of the fractional Schrodinger¨ equation with 1 <α<2. 124 88 speaking, a generalization [1] of the Schrodinger¨ equation does 89 not generate new uncertainty relations if the wave functions III. PROBABILITY CONTINUITY EQUATION 125 90 remain square-integrable functions. A. Correct probability continuity equation 126 91 B. Fractional uncertainty relation does not hold in mathematics In this section we present the correct probability continuity 127 92 Even with the Levy wave packet [Eq. (35) in [2]], equations in the fractional quantum mechanics and reveal a 128 93 the uncertainty relation (3) does not hold in the sense of different phenomenon of the probability transportation. From 129 94 mathematics. We prove it by contradiction. There are two the fraction Schrodinger¨ equation (1) we can get 130 95 steps. ∂ ∗ ∗ ∗ 96 (i) Let us consider the case μ = 1 and α = 1 first. The i (ψ ψ) = ψ T ψ − ψT ψ . (14) ∂t α α 97 Levy wave packet with ν = 1att = 0is According to Laskin’s definitions of the probability density 131 l ∞ |p − p |l p 1 0 and the current density (5), the correct probability continuity 132 ψL(x,0) = exp − exp i x dp 2 π −∞ 2 equation 133 3 1 l 1 p0 ∂ = exp i x . (10) ρ + ∇ · jα = Iα (15) 2 π x2 + (l/2)2 ∂t 98 The letters L denotes the Levy wave packet and l is a has an extra source term 134 99 reference length.
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