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
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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
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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. The related quantities can be calculated as α−1 ∗ 2 α/2−1 Iα =−iDα [∇ψ (−∇ ) ∇ψ = = x 0, p p0, −∇ψ(−∇2)α/2−1∇ψ∗]. (16) ∞ | | = | | = | | ∗ = l x x x ψL(x,0)ψL(x,0)dx , Specifically, if Iα(r,t) > 0, there is a source at position r 135 −∞ π and time t, which generates the probability; when Iα(r,t) < 0, 136 | | = | − | p p p0 there is a sink at position r and time t, which destroys the 137 ∞ | − | probability. 138 = l | − | − p p0 l = p p0 exp dp . It is easy to find cases where the source term is not zero. 139 2 −∞ l For example, take the wave function 140 (11) ψ = ψ1 + ψ2, 100 Therefore, we have the inequality ψ (x,t) = exp(ik x)exp(−iE t), (17) l 1 1 1 |x| |p| = = < . (12) = − π l π 2 ψ2(x,t) exp(ik2x)exp( iE2t), + α α 101 (ii) At t = 0, keep μ = 1 as a constant and let α → 1 . with k1 >k2 > 0, E1 = Dα(k1) , and E2 = Dα(k2) , 141 102 Since the parameter of the Levy wave packet ν = α,thetwo which is a superposition of two solutions to the fractional 142
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143 Schrodinger¨ equation for a free particle. We have α−1 ∗ 2 α/2−1 2 α/2−1 ∗ Iα =−iDα [∇ψ (−∇ ) ∇ψ − ∇ψ(−∇ ) ∇ψ ] =− α−1 ∗ + ∗ −∇2 α/2−1 + − + −∇2 α/2−1 ∗ + ∗ iDα [(ψ1 ψ2 ) ( ) (ψ1 ψ2) (ψ1 ψ2) ( ) (ψ1 ψ2 ) ] α−1 ∗ 2 α/2−1 ∗ 2 α/2−1 2 α/2−1 ∗ 2 α/2−1 ∗ =−iDα [ψ (−∇ ) ψ + ψ (−∇ ) ψ − ψ (−∇ ) ψ − ψ (−∇ ) ψ ] 1 2 2 1 1 2 2 1 α−1 α−1 ∗ α−1 ∗ α−1 ∗ α−1 ∗ =−iDα k1k ψ ψ + k k ψ ψ − k1k ψ ψ − k k ψ ψ 2 1 2 1 2 2 1 2 1 2 1 2 2 1 α−1 α−1 α−1 ∗ ∗ =−iDα k1k − k k (ψ ψ − ψ ψ ) 2 1 2 1 2 1 2 = α−1 α−1 − α−1 − − − 2Dα k1k2 k1 k2 sin[(k2 k1)x (E2 E1)t/ ], (18)
144 which is not zero unless α = 2. with 171 145 The source term indicates that the probability is no longer ρ = ψ∗ψ, 146 locally conserved. As Laskin proved in [9], the total proba- =− α−1 α−2 ∗∇ − ∇ ∗ 147 bility in the whole space is conserved. Here the probability jα iDα k (ψ ψ ψ ψ ). (26)
148 transportation in the fractional quantum mechanics becomes This completes the proof. 172 149 unusual: Some probabilities can disappear at a region and We emphasize that in scatter problems the source term 173 150 simultaneously appear at other regions, but the total probability I1<α<2(r,t) = 0 at the detector’s location, though the potential 174 151 does not change. In other words, some probabilities can at the detector may be zero. There are two reasons for this: 175 152 teleport from one place to another. Furthermore, if the particle (i) The particle is not free so the relation (22) does not hold and 176 153 has mass and charge, probability teleportation will imply mass (ii) the kinetic energy of particles from the scattering source 177 154 teleportation and charge teleportation. We need to pay close may not be exactly the same, i.e., they may not be strictly 178 155 attention to this phenomenon as mass teleportation contradicts monoenergetic. How to develop a scattering model based on 179 156 our life experience and charge teleportation contradicts the the correct continuity equation (15) is an important problem 180 157 classical electrodynamics. in quantum mechanics. 181
158 B. The case Iα(r,t) = 0 IV. SCHRODINGER¨ EQUATION WITH RELATIVISTIC 182 = = 159 When α 2, it is easy to see that I2(r,t) 0. The fractional KINETIC ENERGY 183 160 continuity equation recovers the standard continuity equation. In Refs. [2,3], the relationship between the fractional 184 161 Proposition. For 1 <α<2, we have Iα(r,t) = 0forafree quantum mechanics and the real world was not given. A natural 185 162 particle with a definite kinetic energy. question is which particle has a fractional kinetic energy. If 186 163 Proof. Since V (r) = 0, the fractional Schrodinger¨ equation there are no fractional particles in our world, why do we 187 164 is ∂ need fractional quantum mechanics? To relate the fractional 188 i ψ(r,t) = T ψ(r,t). (19) quantum mechanics to the real world, we regard relativistic 189 ∂t α quantum mechanics [1,10–12] as an approximate realization 190 165 For a definite energy E, its solution is of fractional quantum mechanics. 191 According to the special relativity, the relativistic kinetic 192 ψ(r,t) = C(θ,φ)exp(ik · r)sinθdθdφexp(−iEt/), energy is 193
2 2 2 4 (20) Tr = p c + m c , (27) α E = Dα(k) , (21) where the subscript r means special relativity. For the case of 194 low speed, the relativistic kinetic energy is approximately the 195 166 where is the unit sphere, (k,θ,φ) is the spherical coordinate summation of the rest energy and the classical kinetic energy 196 167 of the wave vector k, and C(θ,φ) is an arbitrary function. Thus (α = 2) 197 168 we have 2 2 p 2 − − T ≈ mc + = mc + T , (28) (−∇2)α/2 1ψ(r,t) = kα 2ψ(r,t), (22) r 2m 2 − ∗ − ∗ and for the case of extremely high speed, where the rest energy 198 (−∇2)α/2 1ψ (r,t) = kα 2ψ (r,t). (23) can be neglected, the relativistic kinetic energy is the fractional 199 169 In this case the source term vanishes kinetic energy with α = 1, 200 =− α−1 ∇ ∗ −∇2 α/2−1∇ −∇ −∇2 α/2−1∇ ∗ Iα iDα [ ψ ( ) ψ ψ( ) ψ ] Tr ≈|p|c = T1. (29) =− α−1 ∇ ∗∇ −∇2 α/2−1 −∇ ∇ −∇2 α/2−1 ∗ iDα [ ψ ( ) ψ ψ ( ) ψ ] Generally speaking, if the speed of a particle increases 201 α−1 α−2 ∗ ∗ =−iDα k (∇ψ ∇ψ−∇ψ∇ψ ) = 0 (24) from low to high, the relativistic kinetic energy Tr will 202 approximately correspond to a fractional kinetic energy Tα, 203 170 and the continuity equation has a sourceless form whose parameter α changes from 2 to 1. Therefore, the 204 ∂ relativistic kinetic energy is an approximate realization of the 205 ρ + ∇ · j = 0, (25) ∂t α fractional kinetic energy. 206
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207 The Hamiltonian function with the relativistic kinetic scattering theory and experiments. Furthermore, the concept 247 208 energy is [10,11] in this Comment offers a very intuitive explanation for 248
superconductivity and superfluidity. We also point out how 249 2 2 2 4 Hr = p c + m c + V (r). (30) to teleport a particle to an arbitrary destination. 250
209 Historically, using this Hamiltonian function, Sommerfeld 210 calculated the relativistic correction to Bohr’s hydrogen energy ACKNOWLEDGMENTS 251 211 levels, and the fine structure in the hydrogen spectrum was
212 explained exactly [4]. I would like to thank Professor Laskin for alerting me 252 213 The relativistic Schrodinger¨ equation is [1,10,11] to Ref. [17]. I would like to thank the reviewers for their 253 ∂ suggestions on revision. Also I would like to thank Professor 254 i ψ(r,t) = H ψ(r,t). (31) ∂t r Xiaofeng Yang and Wengfeng Chen for the discussion on the 255 relativistic covariance. 256 214 Using the perturbation method, we recently calculated the 215 relativistic correction to the hydrogen energy levels obtained
216 from the Schrodinger¨ equation. The resultant energy levels APPENDIX A: INTUITIVE EXPLANATION FOR THE 257 5 217 contain an α term, which can explain the Lamb shift at an SUPERFLUIDITY AND SUPERCONDUCTIVITY 258 218 accuracy of 41% [13,14]. BASED ON TELEPORTATION 259 219 The continuity equation can be expressed as [1] In [17] Tayurskii and Lysogorskiy applied the fractional 260 ∂ ρ + ∇ · jr = Ir , (32) quantum mechanics to explain some property of the superfluid 261 4 ∂t He. From fractional Schrodinger¨ equation, they correctly 262 220 with the current density and the source term obtained the fractional probability continuity equation 263 1 j =− (ψ∗T ∇−2∇ψ − ψT ∇−2∇ψ∗), ∂ r i r r ρ + ∇ · j = I , (A1) ∂t α α 1 I =− ∇ψ∗T ∇−2∇ψ − ∇ψT ∇−2∇ψ∗ . r ( α α ) (33) i where ρ was viewed as the density of the superfluid, jα as the 264 221 Again, the probability is not locally conserved, but the total mass current density, and Iα as extra sources. In order to keep 265 222 probability in the whole space is conserved [1] consistent with the well-known fluid continuity equation 266 ∂ ∗ 3 = ∗ 3 − ∗ 3 = i ψ ψd r ψ Tr ψd r ψTr ψ d r 0. ∂ + ∇ · = ∂t 3 3 3 ρ j 0, (A2) R R R ∂t (34)
Tayurskii and Lysogorskiy claimed that in the superfluid 267 223 Similarly, for a free particle with a definite kinetic energy, Iα ≈ 0, since the wave function describing the He atoms could 268 224 we have Ir = 0. be assumed to be 269 225 Since the relativistic kinetic energy is true and the classical
226 kinetic energy is approximate, the probability continuity ψ(r) = C exp(ip · r/), (A3) 227 equation with the source term (32) can be true and the p p 228 popular probability continuity equation in standard quantum 229 mechanics [4] is approximate. Therefore, we need to base our 270 230 scattering model on the continuity equation with the source where the summation goes over the momentums with approx- |p| 271 231 term, i.e., Eq. (32), calculate the variation between the present imately equal . 272 232 model and the traditional model, and design experiments to This is very difficult to understand. First, we do not know 273 233 observe the phenomenon of the probability teleportation. why the atoms are on such a special state. Second, since the V r 274 234 Since the relativistic Schrodinger¨ equation (31) is not potential ( ) is not zero or a constant [see Eq. (13) in [17]], 275 235 relativistically covariant [10,11], violates the causality [15], the wave function (A3) is not an eigenfunction of the fractional H t = 276 236 and is nonlocal [16] and complicated [10], the research on this Hamiltonian operator α,soevenifat 0 the momentum |p| 277 237 equation has been criticized since the early days of quantum magnitudes are the same, they will become different soon. I 278 238 mechanics. A positive experimental result on the probability On the contrary, we should suppose that the source term α or I 279 239 teleportation will end this situation ultimately. r is not zero generally. Here is our superfluid model based on Eq. (A1)or(32). 280 4 When the superfluid He is still, moving atoms sometimes can 281 240 V. CONCLUSION disappear at one place and appear at another distant place. This 282 241 We proved that the fractional uncertainty relation does explains why heat can be conducted easily by the superfluid. 283 242 not hold generally. The missing source term in the fractional When the superfluid flows, some superfluid moves forward 284 243 probability continuity equation leads to particle teleportation. normally, which has friction, and some teleports from one 285 244 According to the special relativity, the classical kinetic energy place to another, which has no friction. Thus we do not need 286 245 is just an approximation in the low-speed case, so it would to artificially divide the superfluid into a normal component 287 246 be a hopeful direction to study particle teleportation in and a superfluid component. 288
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289 Now it becomes an urgent task to observe whether the mass Suppose that a particle is located in an interval on the 313 290 teleportation really exists in the superfluid experiment. An easy negative half of x axis, described by a wave function 314
291 way is to measure the velocity of the superfluid in the capillary nonzero x ∈ [−b,−a] 292 tube and the mass coming out from the tube to check whether ψ(x) = 293 they are consistent. 0 x/∈ [−b,−a]. 294 Similarly, based on the concept of teleportation, the super- 295 conductivity can be viewed as such a phenomenon in which Here a
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