**Quantum Effects Through a Fractal Theory of Motion**

M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54172

### **1. Introduction**

16 Quantum Mechanics

722 Advances in Quantum Mechanics

**References**

paper).

[2] E. Witten, Nucl. Phys. B188 513 (1981).

[4] D.L. Pursey, Phys. Rev. D33 2267 (1986).

World Scientific Publishing Co. (2011).

[6] B. Jensen, JHEP 11, 059 (2011).

New York: Wiley, 4th ed. (1986).

[3] G. Darboux, C.R. Acad. Sci. (Paris) 94, 1456 (1882).

[1] E. Schrödinger, Proc. R. Irish Acad. **A46**, 9 (1940); **A46**, 183 (1940); A47, 53 (1941), L.Infeld and T.E. Hull, Rev. Mod. Phys. 23, 21 (1951) (see also the references in this

[5] A. Gangopadhyaya , J.V. Mallow and C. Rasinariu , *Supersymmetric quantum mechanics*

[8] W.E. Boyce, R.C. DiPrima, *Elementary Differential Equations and Boundary Value Problems*

[7] E. L. Ince,*Ordinary differential equations* Dover Publications, Inc. (1956).

Scale Relativity Theory (SRT) affirms that the laws of physics apply in all reference systems, whatever its state of motion and its scale. In consequence, SRT imply [1-3] the followings:

Let *F* (*x*) be a fractal function in the interval *x* ∈ *a*, *b* and let the sequence of values for *x* be:

$$\mathbf{x}\_a = \mathbf{x}\_{0'}\mathbf{x}\_1 = \mathbf{x}\_0 + \boldsymbol{\varepsilon}\_r\mathbf{x}\_k = \mathbf{x}\_0 + k\boldsymbol{\varepsilon}\_r\mathbf{x}\_n = \mathbf{x}\_0 + n\boldsymbol{\varepsilon} = \mathbf{x}\_b \tag{1}$$

We can now say that *F* (*x*, *ε*) is a –scale approximation.

Let us now consider as a *ε*¯-scale approximation of the same function. Since *F* (*x*) is everywhere almost self-similar, if *ε* and *ε*¯ are sufficiently small, both approximations *F* (*x*, *ε*) and must lead to same results. By comparing the two cases, one notices that scale expansion is related to the increase *dε* of *ε*, according to an increase *dε*¯ of *ε*¯. But, in this case we have:

$$
\frac{d\varepsilon}{d\varepsilon} = \frac{d\overline{\varepsilon}}{\overline{\varepsilon}} = d\,\rho\tag{2}
$$

© 2013 Agop et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Agop et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

situation in which we can consider the infinitesimal scale transformation as being

$$
\mathfrak{a}' = \mathfrak{z} + d\mathfrak{z} = \mathfrak{z} + \mathfrak{e}d\mathfrak{p} \tag{3}
$$

where *P*(*F* ) is now an unknown function. The simplest explicit suggested form for *P*(*F* ) is

*P F A BF A B const* ( ) =+ = ,, . (10)

= + (11)

Quantum Effects Through a Fractal Theory of Motion

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725

, (12)

(14)

(15)

*B* - = (13)

linear dependence [2]

we obtain:

in which case the differential equation (9) takes the form:

Hence by integration and substituting:

position and, divergently, on *ε* / *ε*0)

( ) 0 ln

*B* = t

0

é ù æ ö æ ö = + ê ú ç ÷ ç ÷ ç ÷ ê ú è ø è ø ë û

0 *F F* 1

e

0 0 (, ) 1

e e

The following particular cases are to be considered:

power-law dependence on resolution is obtained:

*Fp F p p*

where *ξ*(*p*) is a function depending on parameterization of the fractal curve.

e

0 *A F*

0

e

 e

We can now generalize the previous result by considering that *F* is dependent on parameter‐ ization of the fractal curve. If *p* characterizes the position on the fractal curve then, following the same algorithm as above, the solution will be as a sum of two terms i.e. both classical and differentiable (depending only on position) and fractal, non-differentiable (depending on

( ) ( )

x

**1.** in asymptotic small scale regime *ε ε*0, *τ* is constant (with no scale dependence) and

= + ê ú ç ÷ è ø ë û

t

( ) 0

t e

e

é ù æ ö

*p*

*d* e e

*dF A BF*

Such transformation in the case of function *F* (*x*, *ε*), leads to:

$$F\left(\mathbf{x}, \boldsymbol{\varepsilon}^{\prime}\right) = F\left(\mathbf{x}, \boldsymbol{\varepsilon} + \varepsilon d\,\boldsymbol{\rho}\right) \tag{4}$$

respectively, if we limit ourselves to a first order approximation:

$$F(\mathbf{x}, \boldsymbol{\varepsilon}') = F(\mathbf{x}, \boldsymbol{\varepsilon}) + \frac{\partial F(\mathbf{x}, \boldsymbol{\varepsilon})}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}' - \boldsymbol{\varepsilon}) = F(\mathbf{x}, \boldsymbol{\varepsilon}) + \frac{\partial F(\mathbf{x}, \boldsymbol{\varepsilon})}{\partial \boldsymbol{\varepsilon}} \boldsymbol{\varepsilon} d\boldsymbol{\rho} \tag{5}$$

Moreover, let us notice that for an arbitrary but fixed *ε*0, we obtain:

$$\frac{\partial \ln \left( \varepsilon / \varepsilon\_0 \right)}{\partial \varepsilon} = \frac{\partial \left( \ln \varepsilon - \ln \varepsilon\_0 \right)}{\partial \varepsilon} = \frac{1}{\varepsilon} \tag{6}$$

situation in which (5) can be written as:

$$F(\mathbf{x}, \boldsymbol{\varepsilon}') = F(\mathbf{x}, \boldsymbol{\varepsilon}) + \frac{\partial F(\mathbf{x}, \boldsymbol{\varepsilon})}{\partial \ln(\varepsilon / \varepsilon\_0)} d\boldsymbol{\rho} = \left[ 1 + \frac{\partial}{\partial \ln(\varepsilon / \varepsilon\_0)} d\boldsymbol{\rho} \right] F(\mathbf{x}, \boldsymbol{\varepsilon}) \tag{7}$$

Therefore, we can introduce the dilatation operator:

$$
\stackrel{\frown}{D} = \frac{\stackrel{\frown}{\partial}}{\overline{\partial} \ln \left( \varepsilon / \varepsilon\_0 \right)} \tag{8}
$$

At the same time, relation (8) shows that the intrinsic variable of resolution is not *ε*, but ln(*ε* / *ε*0).

The fractal function is explicitly dependent on the resolution (*ε* / *ε*0), therefore we have to solve the differential equation:

$$\frac{dF}{d\ln\left(\varepsilon/\varepsilon\_0\right)} = P\left(F\right) \tag{9}$$

where *P*(*F* ) is now an unknown function. The simplest explicit suggested form for *P*(*F* ) is linear dependence [2]

$$P\left(F\right) = A + BF,\\ A\_\prime B = const.\tag{10}$$

in which case the differential equation (9) takes the form:

$$\frac{dF}{d\ln\left(\varepsilon/\varepsilon\_0\right)} = A + BF\tag{11}$$

Hence by integration and substituting:

$$B = -\tau\_{\prime} \tag{12}$$

$$-\frac{A}{B} = F\_0 \tag{13}$$

we obtain:

situation in which we can consider the infinitesimal scale transformation as being

'

*Fx Fx d* ( ,' , e

( ) ( ) , , ( ) ,' , ' ,

*Fx Fx F x d* e

= + -= +

 e e

( ) ( ) 0 0 ln ln ln

 e

, ,' , 1 , ln ln

 r

( ) 0 ln

At the same time, relation (8) shows that the intrinsic variable of resolution is not *ε*, but

The fractal function is explicitly dependent on the resolution (*ε* / *ε*0), therefore we have to solve

( ) ( )

*dF P F*

0 ln

*d* e e e e

= + =+ ê ú ¶ ¶ ë û

*Fx Fx d d Fx* e

¶ é ù ¶

e

 e er

*F x F x*

 e1

ee

( ) ( ) ( ) 0 0

e e

¶ ¶

 e

=+ =+ *d d* (3)

) = + ( ) (4)

 e

¶ ¶ (5)

 e

¶ ¶- = = ¶ ¶ (6)

r

Ù ¶ = ¶ (8)

= (9)

 e (7)

 er

e e e e er

Such transformation in the case of function *F* (*x*, *ε*), leads to:

respectively, if we limit ourselves to a first order approximation:

Moreover, let us notice that for an arbitrary but fixed *ε*0, we obtain:

ee

*F x*

e e

*D*

e

( ) ( ) ( )

 e

( ) ( ) ( )

ee

Therefore, we can introduce the dilatation operator:

e

724 Advances in Quantum Mechanics

situation in which (5) can be written as:

ln(*ε* / *ε*0).

the differential equation:

$$F\left(\frac{\mathcal{E}}{\mathcal{E}\_0}\right) = F\_0 \left[1 + \left(\frac{\mathcal{E}\_0}{\mathcal{E}}\right)^r\right] \tag{14}$$

We can now generalize the previous result by considering that *F* is dependent on parameter‐ ization of the fractal curve. If *p* characterizes the position on the fractal curve then, following the same algorithm as above, the solution will be as a sum of two terms i.e. both classical and differentiable (depending only on position) and fractal, non-differentiable (depending on position and, divergently, on *ε* / *ε*0)

$$F(p, \varepsilon/\varepsilon\_0) = F\_0(p) \left[ 1 + \xi(p) \left( \frac{\varepsilon\_0}{\varepsilon} \right)^{\varepsilon(p)} \right] \tag{15}$$

where *ξ*(*p*) is a function depending on parameterization of the fractal curve.

The following particular cases are to be considered:

**1.** in asymptotic small scale regime *ε ε*0, *τ* is constant (with no scale dependence) and power-law dependence on resolution is obtained:

( ) ( ) ( ) ( ) 0 0 0 (, ) a b *Fp T p T p F pQp* t e e e e æ ö = ç ÷ è ø = (16)

**iv.** The local differential time invariance is broken, so the time-derivative of the fractal

lim a

lim b

Both definitions are equivalent in the differentiable case *dt* → −*dt*. In the non-differentiable situation, these definitions are no longer valid, since limits are not defined anymore. Fractal theory defines physics in relationship with the function behavior during the "zoom" operation on the time resolution *δt*, here identified with the differential element *dt* (substitution princi‐ ple), which is considered an independent variable. The standard field *Q*(*t*) is therefore replaced by fractal field *Q*(*t,dt*), explicitly dependent on time resolution interval, whose derivative is not defined at the unnoticeable limit *dt* →0. As a consequence, this leads to the two derivatives

(20)

727

Quantum Effects Through a Fractal Theory of Motion

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(21)

( ) ( )

( ) ( )

field *Q* can be written two-fold:

+

0

+

*d Q Qt t Qt dt t d Q Qt Qt t dt t*

+D - = D

of the fractal field *Q* as explicit functions of the two variables *t* and *dt*,

*d Q Qt t t Qt t*

+D D - D = D

D - -D D = D

*d Q Qt t Qt t t*

( ) ( )

, , lim a

, , lim b

*i* , i.e.

1,2, *i i dX dx i* ± ± == (22)

± ±± = + (23)

± = (24)

( ) ( )

Notation "+" corresponds to the forward process, while "-" to the backward one.

*iii dX dx d*

0 *i d*x

**vi.** The differential fractal part satisfies the fractal equation:

x

*t*

D ®

*t*

0

+

*dt t*

*dt t*

*t*

D ®

+

Since, according to (19), we can write:

and it results:

*t*

D ®

0

**v.** We denote the average of these vectors by *d x*±

D ®

0

**2.** in the asymptotic big scale regime *ε ε*0, *τ* is constant (with no scale dependence) and, in terms of resolution, one obtains an independent law:

$$F(p, \varepsilon/\varepsilon\_0) \to F\_0(p) \tag{17}$$

Particularly, if *F* (*p*, *ε* / *ε*0) are the coordinates in given space, we can write

$$X(p, \varepsilon/\varepsilon\_0) = \mathbf{x}\left(p\right)\left[1 + \xi\left(p\right)\left(\frac{\varepsilon\_0}{\varepsilon}\right)^r\right] \tag{18}$$

In this situation, *ξ*(*p*) becomes a highly fluctuating function which can be described by stochastic process while *τ* represents (according to previous description) the difference between fractal and topological dimensions. The result is a sum of two terms, a classical, differentiable one (dependent only on the position) and a fractal, non-differentiable one (dependent both on the position and, divergently, on *ε* / *ε*0). This represents the importance of the above analysis.

By differentiating these two parts we obtain:

$$dX = d\mathfrak{x} + d\mathfrak{y} \tag{19}$$

where *dx* is the classical differential element and *dξ* is a differential fractal one.

**iii.** There is infinity of fractal curves (geodesics) relating to any couple of points (or starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvature. As a consequence, we have replaced velocity on a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics [4] where the motion of the fluid is described in terms of its velocity field *v* =(*x*(*t*), *t*), density *ρ* =(*x*(*t*), *t*) and, possibly, its pressure. We shall, indeed, recover the fundamental equations of fluid mechanics (Euler and continuity equations), but we shall write them in terms of a density of probability (as defined by the set of geodesics) instead of a density of matter and adding an addi‐ tional term of quantum pressure (the expression of fractal geometry).

**iv.** The local differential time invariance is broken, so the time-derivative of the fractal field *Q* can be written two-fold:

$$\begin{aligned} \frac{d\_+\mathcal{Q}}{dt} &= \lim\_{\Delta t \to 0\_+} \frac{\mathcal{Q}\{t + \Delta t\} - \mathcal{Q}\{t\}}{\Delta t} & \mathbf{a} \\ \frac{d\_-\mathcal{Q}}{dt} &= \lim\_{\Delta t \to 0\_-} \frac{\mathcal{Q}\{t\} - \mathcal{Q}\{t - \Delta t\}}{\Delta t} & \mathbf{b} \end{aligned} \tag{20}$$

Both definitions are equivalent in the differentiable case *dt* → −*dt*. In the non-differentiable situation, these definitions are no longer valid, since limits are not defined anymore. Fractal theory defines physics in relationship with the function behavior during the "zoom" operation on the time resolution *δt*, here identified with the differential element *dt* (substitution princi‐ ple), which is considered an independent variable. The standard field *Q*(*t*) is therefore replaced by fractal field *Q*(*t,dt*), explicitly dependent on time resolution interval, whose derivative is not defined at the unnoticeable limit *dt* →0. As a consequence, this leads to the two derivatives of the fractal field *Q* as explicit functions of the two variables *t* and *dt*,

$$\begin{aligned} \frac{d\_+\mathcal{Q}}{dt} &= \lim\_{\Delta t \to 0\_+} \frac{\mathcal{Q}\{t + \Delta t, \Delta t\} - \mathcal{Q}\{t, \Delta t\}}{\Delta t} & \mathbf{a} \\ \frac{d\_-\mathcal{Q}}{dt} &= \lim\_{\Delta t \to 0\_-} \frac{\mathcal{Q}\{t, \Delta t\} - \mathcal{Q}\{t - \Delta t, \Delta t\}}{\Delta t} & \mathbf{b} \end{aligned} \tag{21}$$

Notation "+" corresponds to the forward process, while "-" to the backward one.

**v.** We denote the average of these vectors by *d x*± *i* , i.e.

$$
\left\langle dX\_{\pm}^{i} \right\rangle = d\alpha\_{\pm \prime}^{i} i = 1,2 \tag{22}
$$

Since, according to (19), we can write:

$$dX^i\_{\pm} = d\mathfrak{x}^i\_{\pm} + d\mathfrak{z}^i\_{\pm} \tag{23}$$

and it results:

( )

æ ö = ç ÷ è ø

0

t e

**2.** in the asymptotic big scale regime *ε ε*0, *τ* is constant (with no scale dependence) and, in

( ) ( ) 0

x

= + ê ú ç ÷ ê ú è ø ë û

In this situation, *ξ*(*p*) becomes a highly fluctuating function which can be described by stochastic process while *τ* represents (according to previous description) the difference between fractal and topological dimensions. The result is a sum of two terms, a classical, differentiable one (dependent only on the position) and a fractal, non-differentiable one (dependent both on the position and, divergently, on *ε* / *ε*0). This represents the importance of

*dX dx d* = +

where *dx* is the classical differential element and *dξ* is a differential fractal one.

x

**iii.** There is infinity of fractal curves (geodesics) relating to any couple of points (or

tional term of quantum pressure (the expression of fractal geometry).

starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvature. As a consequence, we have replaced velocity on a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics [4] where the motion of the fluid is described in terms of its velocity field *v* =(*x*(*t*), *t*), density *ρ* =(*x*(*t*), *t*) and, possibly, its pressure. We shall, indeed, recover the fundamental equations of fluid mechanics (Euler and continuity equations), but we shall write them in terms of a density of probability (as defined by the set of geodesics) instead of a density of matter and adding an addi‐

é ù æ ö

b

t e

e

® (17)

(19)

(16)

(18)

(, ) a

e

( ) ( ) 0 0 *Fp F p* ,e e

Particularly, if *F* (*p*, *ε* / *ε*0) are the coordinates in given space, we can write

0 *Xp xp p* (, ) 1

( ) ( ) ( )

0

*T p F pQp*

0

*Fp T p*

e e

=

terms of resolution, one obtains an independent law:

e e

the above analysis.

726 Advances in Quantum Mechanics

By differentiating these two parts we obtain:

$$\left\langle d\_{\mathbb{P}\_{\pm}}^{\mathbb{E}^i} \right\rangle = 0 \tag{24}$$

**vi.** The differential fractal part satisfies the fractal equation:

$$d\_{\pm} \xi^i = \lambda\_{\pm}^i \left(dt\right)^{1/D\_F} \tag{25}$$

where only the first three terms were used in Nottale's theory (*i.e*. second order terms in the motion equation). Relations (29) are valid in any point both for the spatial manifold and for the points *X* on the fractal curve (selected in relations 29). Hence, the forward and backward

2 3 1 1

The following aspects should be mentioned: the mean value of function *f* and its derivatives

average of their products coincides with the product of averages. Consequently, the equations

2 3 1 1 2 6

*QQ Q d Q dt Q d d Xd X d Xd Xd X t X X XXX* ± ± ± ± ±±±

( )

±±± ±± ±

+

*i ki k j j*

2

*i j*

coordinate average, the situation can still be different. Firstly, let us focus on the averages

( )

*dt dt* -

( )

*dt dt* -

(2 1 ) *i i j j DF d d*

*d*±*ξ j*

(3 1 ) *i k ik j j DF ddd*

 lll

 ll

¶ ¶ = +Ñ × + + +

1 2

*Q Q d Q dt Q d d xd x d d*

± ± ±± ±±

*i j j i k*

xxx

*d*−*ξ j* . If *i* ≠ *j*, these averages are zero due to the independence of *d*±*ξ i*

*i j j i k*

*i j j i k*

and *dt* are independent; therefore, the

Quantum Effects Through a Fractal Theory of Motion

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729

is null (see 24), for higher order of fractal

(32)

and

*X* (30)

*i j j i k*

*X* (31)

( )

*i i j j*

x x

± ± ±± = (33)

. Now, using equations (25), we can write:

± ± ± ±+± = (34)

*d*±*ξ k* . If *i* ≠ *j* ≠*k*, these averages are zero due to

2 6

*Q QQ d Q dt Q d d Xd X d Xd Xd X t X X XXX* ± ± ± ± ±±±

¶ ¶¶ = +Ñ × + + ¶ ¶ ¶ ¶¶¶

¶¶ ¶ = +Ñ + + ¶ ¶ ¶ ¶¶¶

*t X X*

*X*

¶ ¶ ¶

x x

and *d*±*ξ k*

*Q d xd xd x d d d*

average values of these relations take the form:

coincide with themselves and the differentials *d*±*X i*

or more, using equations (23) with characteristics (24),

3

¶

¶¶¶

. So, using (25), we can write:

Then, let us consider the averages *d*±*ξ i*

on *d*±*ξ j*

Then, equations (32) may be written as follows:

x x x

*XXX*

*i k j*

Even if the average value of the fractal coordinate *d*±*ξ i*

1 6

*d*+*ξ j* and *d*−*ξ i*

independence of *d*±*ξ i*

(30) become:

*d*+*ξ i*

*d*±*ξ j*

where *λ*± *i* are some constant coefficients and *DF* is a constant fractal dimension. We note that the use of any Kolmogorov or Hausdorff [1, 5, 6-8] definitions can be accepted for fractal dimension, but once a certain definition is admitted, it should be used until the end of analyzed dynamics.

**vii.** The local differential time reflection invariance is recovered by combining the two derivatives, *d*+ / *dt* and *d*− / *dt*, in the complex operator:

$$\frac{d\ddot{d}}{dt} = \frac{1}{2} \left( \frac{d\_+ + d\_-}{dt} \right) - \frac{\mathrm{i}}{2} \left( \frac{d\_+ - d\_-}{dt} \right) \tag{26}$$

Applying this operator to the "position vector", a complex velocity yields

$$\hat{\mathbf{V}} = \frac{\hat{d}\mathbf{X}}{dt} = \frac{1}{2} \left( \frac{d\_+\mathbf{X} + d\_-\mathbf{X}}{dt} \right) - \frac{i}{2} \left( \frac{d\_+\mathbf{X} - d\_-\mathbf{X}}{dt} \right) = \frac{\mathbf{V}\_+ + \mathbf{V}\_-}{2} - i\frac{\mathbf{V}\_+ - \mathbf{V}\_-}{2} = \mathbf{V} - i\mathbf{U} \tag{27}$$

with:

$$\begin{aligned} V &= \frac{V\_+ + V\_-}{2} & \mathbf{a} & \mathbf{a} \\ \mathbf{U} &= \frac{V\_+ - V\_-}{2} & \mathbf{b} \end{aligned} \tag{28}$$

The real part, *V*, of the complex velocity *V* **^** , represents the standard classical velocity, which does not depend on resolution, while the imaginary part, *U*, is a new quantity coming from resolution dependant fractal.

#### **2. Covariant total derivative**

Let us now assume that curves describing particle movement (continuous but non-differen‐ tiable) are immersed in a 3-dimensional space, and that *X* of components *X i* (*i* =1 ¯ , 3) is the position vector of a point on the curve. Let us also consider a fractal field *Q*( *X*, *t* ) and expand its total differential up to the third order:

$$d\_{\pm}Q = \frac{\partial Q}{\partial t}dt + \nabla Q \cdot d\_{\pm} \mathbf{X} + + \frac{1}{2} \frac{\partial^2 Q}{\partial \mathbf{X}^i \partial \mathbf{X}^j} d\_{\pm} \mathbf{X}^i d\_{\pm} \mathbf{X}^j + \frac{1}{6} \frac{\partial^3 Q}{\partial \mathbf{X}^i \partial \mathbf{X}^j \partial \mathbf{X}^k} d\_{\pm} \mathbf{X}^i d\_{\pm} \mathbf{X}^j d\_{\pm} \mathbf{X}^k \tag{29}$$

where only the first three terms were used in Nottale's theory (*i.e*. second order terms in the motion equation). Relations (29) are valid in any point both for the spatial manifold and for the points *X* on the fractal curve (selected in relations 29). Hence, the forward and backward average values of these relations take the form:

$$
\left\langle d\_{\pm}Q \right\rangle = \left\langle \frac{\partial Q}{\partial t}dt \right\rangle + \left\langle \nabla Q \cdot d\_{\pm}X \right\rangle + \frac{1}{2} \left\langle \frac{\partial^{2}Q}{\partial X^{i}\partial X^{j}}d\_{\pm}X^{i}d\_{\pm}X^{j} \right\rangle + \frac{1}{6} \left\langle \frac{\partial^{3}Q}{\partial X^{i}\partial X^{j}\partial X^{k}}d\_{\pm}X^{i}d\_{\pm}X^{j}d\_{\pm}X^{k} \right\rangle \tag{30}
$$

The following aspects should be mentioned: the mean value of function *f* and its derivatives coincide with themselves and the differentials *d*±*X i* and *dt* are independent; therefore, the average of their products coincides with the product of averages. Consequently, the equations (30) become:

$$d\_{\pm}Q = \frac{\partial Q}{\partial t}dt + \nabla Q \left< d\_{\pm}X \right> + \frac{1}{2} \frac{\partial^{2}Q}{\partial X^{i}\partial X^{j}} \left< d\_{\pm}X^{i}d\_{\pm}X^{j} \right> + \frac{1}{6} \frac{\partial^{3}Q}{\partial X^{i}\partial X^{j}\partial X^{k}} \left< d\_{\pm}X^{i}d\_{\pm}X^{j}d\_{\pm}X^{k} \right> \tag{31}$$

or more, using equations (23) with characteristics (24),

( ) 1/ *i i DF d dt*

the use of any Kolmogorov or Hausdorff [1, 5, 6-8] definitions can be accepted for fractal dimension, but once a certain definition is admitted, it should be used until the end of analyzed

**vii.** The local differential time reflection invariance is recovered by combining the two

*i* are some constant coefficients and *DF* is a constant fractal dimension. We note that

± ± = (25)

è øè ø (26)

**^** , represents the standard classical velocity, which

*i j j i k*

**X** (29)

(28)

(*i* =1 ¯

, 3) is the

x l

derivatives, *d*+ / *dt* and *d*− / *dt*, in the complex operator:

Applying this operator to the "position vector", a complex velocity yields

è øè ø

2 2 *d i dd dd dt dt dt* +- +- æ öæ ö + - = ç ÷ç ÷ -

2 2 22 *d i dd dd i i*

2

+ =

*V V V*

*V V U*

+ -

2

+ -

tiable) are immersed in a 3-dimensional space, and that *X* of components *X i*

does not depend on resolution, while the imaginary part, *U*, is a new quantity coming from

Let us now assume that curves describing particle movement (continuous but non-differen‐

position vector of a point on the curve. Let us also consider a fractal field *Q*( *X*, *t* ) and expand

2 3 1 1 2 6

*Q QQ d Q dt Q d d Xd X d Xd Xd X t X X XXX* ± ± ± ± ±±±

¶ ¶¶ = +Ñ × ++ + ¶ ¶ ¶ ¶¶¶

*i j j i k*

+ - + - +- +- æ öæ ö + - +- = = - = - =- ç ÷ç ÷

*X X X X X VV VV V V U* (27)

a

b

ˆ 1

*dt dt dt*

The real part, *V*, of the complex velocity *V*

resolution dependant fractal.

**2. Covariant total derivative**

its total differential up to the third order:

where *λ*±

728 Advances in Quantum Mechanics

dynamics.

with:

ˆ 1 ˆ

$$\begin{split} d\_{\pm} \mathbf{Q} &= \frac{\partial \mathbf{Q}}{\partial t} dt + \nabla \mathbf{Q} \cdot d\_{\pm} \mathbf{X} + \frac{1}{2} \frac{\partial^{2} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j}} \Big( d\_{\pm} \mathbf{x}^{i} d\_{\pm} \mathbf{x}^{j} + \left< d\_{\pm} \boldsymbol{\xi}^{i} d\_{\pm} \boldsymbol{\xi}^{j} \right> \Big) + \\ \frac{1}{6} \frac{\partial^{3} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j} \partial \mathbf{X}^{k}} \Big( d\_{\pm} \mathbf{x}^{i} d\_{\pm} \mathbf{x}^{j} d\_{\pm} \mathbf{x}^{k} + \left< d\_{\pm} \boldsymbol{\xi}^{i} d\_{\pm} \boldsymbol{\xi}^{j} d\_{\pm} \boldsymbol{\xi}^{k} \right> \Big) \end{split} \tag{32}$$

Even if the average value of the fractal coordinate *d*±*ξ i* is null (see 24), for higher order of fractal coordinate average, the situation can still be different. Firstly, let us focus on the averages *d*+*ξ i d*+*ξ j* and *d*−*ξ i d*−*ξ j* . If *i* ≠ *j*, these averages are zero due to the independence of *d*±*ξ i* and *d*±*ξ j* . So, using (25), we can write:

$$\left\langle d\_{\pm} \xi^{i} d\_{\pm} \xi^{j} \right\rangle = \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \left( dt \right)^{\left( 2f D\_{\mathbb{F}} \right) - 1} dt \tag{33}$$

Then, let us consider the averages *d*±*ξ i d*±*ξ j d*±*ξ k* . If *i* ≠ *j* ≠*k*, these averages are zero due to independence of *d*±*ξ i* on *d*±*ξ j* and *d*±*ξ k* . Now, using equations (25), we can write:

$$\left\langle d\_{\pm} \xi^{i} d\_{\pm} \xi^{j} d\_{\pm} \xi^{k} \right\rangle = \lambda\_{\pm}^{i} \lambda\_{+}^{j} \lambda\_{\pm}^{k} \left(dt\right)^{\left(3/D\_{\mp}\right) - 1} dt \tag{34}$$

Then, equations (32) may be written as follows:

$$\begin{split} d\_{\pm} \mathbf{Q} &= \frac{\partial \mathbf{Q}}{\partial t} dt + d\_{\pm} \mathbf{x} \cdot \nabla \mathbf{Q} + \frac{1}{2} \frac{\partial^{2} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j}} d\_{\pm} \mathbf{x}^{i} d\_{\pm} \mathbf{x}^{j} + \frac{1}{2} \frac{\partial^{2} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \left( dt \right)^{\{2l\mathcal{D}\_{\mathbb{r}}\}-1} dt + \\ \frac{1}{6} \frac{\partial^{3} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j} \partial \mathbf{X}^{k}} d\_{\pm} \mathbf{x}^{i} d\_{\pm} \mathbf{x}^{k} + \frac{1}{6} \frac{\partial^{3} \mathbf{Q}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j} \partial \mathbf{X}^{k}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \lambda\_{\pm}^{k} \left( dt \right)^{\{3l\mathcal{D}\_{\mathbb{r}}\}-1} dt \end{split} \tag{35}$$

This relation also allows us to define the fractal operator:

( ) ( )

*dt*

lll

4

*F*

 lll

ll

lll

*D*

( ) ( )

*dt*

Particularly, by choosing:

ˆ ˆ

12

*F*

*V*

the fractal operator (39) takes the usual form:

derivative operator, *d* / *dt*, with the complex operator ∂

*t t*

diffusion equation in its covariant form:

**dissipative approximation**

tive), (*V* ∧

¶ ¶

*D*

( ) ( )

*ii ii jj jj*

++ -- ++ --

*i*

 ll

 lll  ll

*i j*

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http://dx.doi.org/10.5772/54172

^ / ∂*t* (this results in a generalization of

(39)

731

*i k j*

*XXX*

++ - =- = ij D (40)

+++ - + =- = ijk D (41)

*V* **D D** (42)

( ) ( ) 2 1 2 3 2 (3 1 ) 3 0

*V D D dt* (43)

3

2 1 2

 d

3 *D D F F i dt dt*

We now apply the principle of scale covariance and postulate that the passage from classical (differentiable) to "fractal" mechanics can be implemented by replacing the standard time

Nottale's [1, 2] principle of scale covariance). Consequently, we are now able to write the

This means that at any point on a fractal path, the local temporal ∂*t Q*, the non-linear (convec‐

⋅∇)*Q*, the dissipative, *ΔQ*, and the dispersive, ∇3*Q*, terms keep their balance.

3 2

 d

( ) ( )

\_ 2 *i i j j*

 ll

\_ 2 2 *ik ik j j*

( ) ( ) 2 1 3 2 (3 1 ) 3 ˆ 2 ˆ

¶ ¶ - - = + ×Ñ- D + Ñ

 lll

( )

( )

*Q Q D D F F Q i Q dt Q t t* Ù Ù ¶ ¶ æ ö - - = + ×Ñ - ç ÷ D + Ñ = ç ÷ ¶ ¶ è ø

**3. Fractal space-time and the motion equation of free particles in the**

Newton's fundamental equation of dynamics in the dissipative approximation is:

+++ --- +++ ---

3 1 3

*t t X X*

¶ ¶ ¶ é ù = + ×Ñ+ +- - + ¶ ¶ ê ú ë û ¶ ¶

 ll

ll

¶ é ù + +- - ê ú ë û ¶¶¶

*ikik ikik jj jj*

*i*

 lll

If we divide by *dt* and neglect the terms containing differential factors (for details on the method see [9, 10]), equations (38a) and (38b) are reduced to:

$$\frac{d\_{\pm}\mathbb{Q}}{dt} = \frac{\partial\mathbb{Q}}{\partial t} + V\_{\pm} \cdot \nabla\mathbb{Q} + \frac{1}{2} \frac{\partial^{2}\mathbb{Q}}{\partial\mathbf{X}^{i}\partial\mathbf{X}^{\top}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{\{\prime\prime\prime\}-1} + \frac{1}{6} \frac{\partial^{3}\mathbb{Q}}{\partial\mathbf{X}^{i}\partial\mathbf{X}^{\top}\partial\mathbf{X}^{k}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \lambda\_{\pm}^{k} \left(dt\right)^{\{\prime\prime\prime\}-1} \tag{36}$$

These relations also allow us to define the operator:

$$\frac{d\_{\pm}}{dt} = \frac{\partial}{\partial t} + V\_{\pm} \cdot \nabla + \frac{1}{2} \frac{\partial^{2}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \{dt\}^{(2\text{\textquotedblleft D}\_{\mathbb{T}}) - 1} + \frac{1}{6} \frac{\partial^{3}}{\partial \mathbf{X}^{i} \partial \mathbf{X}^{j} \partial \mathbf{X}^{k}} \lambda\_{\pm}^{i} \lambda\_{\pm}^{j} \lambda\_{\pm}^{k} \{dt\}^{(3\text{\textquotedblleft D}\_{\mathbb{T}}) - 1} \tag{37}$$

Under these circumstances, let us calculate (∂ ∧ *Q* / ∂*t* ). Taking into account equations (26), (27) and (37), we shall obtain:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 1 3 1 2 3 2 1 3 1 1 2 11 1 1 2 2 4 12 11 1 1 2 2 4 12 2 *F F F F i j j D D i k i j j i k i j j D D i k i j j i k Q dQ dQ dQ dQ i t dt dt dt dt Q QQ Q dt dt t X X XXX Q QQ Q dt dt t X X XXX i Q* l l lll l l lll Ù +- +- - - + ++ +++ - - - - - --- ¶ é ù æ ö = +- - = ê ú ç ÷ ¶ ë û è ø ¶ ¶¶ = + ×Ñ + + + ¶ ¶ ¶ ¶¶¶ ¶ ¶¶ + + ×Ñ ++ + - ¶ ¶ ¶ ¶¶¶ ¶ - ¶ *V V* ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 1 3 1 2 3 2 1 3 1 2 1 2 2 12 2 2 2 12 22 4 *F F F F F i j j D D i k i j j i k i j j D D i k i j j i k D ii i jj j i i Qi Q Q dt dt t X X XXX iQ i i Q i Q Q dt dt t X X XXX Q dt i Q i t* l l lll l l lll ll ll ll - - + ++ +++ - - - -- --- - +- +- ++ -- ++ ¶ ¶ - ×Ñ - - + ¶ ¶ ¶¶¶ ¶ ¶¶ + + ×Ñ + + = ¶ ¶ ¶ ¶¶¶ ¶ æ ö + - = + - ×Ñ + ç ÷ +- - ¶ è ø *V V VV VV* ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 1 3 2 1 2 3 1 12 4 12 *F F F i j i j D ikik ikik jj jj i k j D ii ii jj jj i j D ikik iki jj j Q X X dt Q i XXX Q Q dt Q i t X X dt i* l l lll lll lll lll ll ll ll ll lll lll lll ll - - - +++ --- +++ --- - Ù ++ -- ++ -- - +++ --- +++ -- ¶ é ù + ê ú ë û ¶ ¶ ¶ é ù + +- - = ê ú ë û ¶¶¶ ¶ ¶ é ù = + ×Ñ + +- - + ¶ ê ú ë û ¶ ¶ +- - *V* ( ) 3 *j k i k j Q XXX* l- ¶ é ù ê ú ë û ¶¶¶ (38)

This relation also allows us to define the fractal operator:

$$\begin{split} \frac{\hat{\mathcal{O}}}{\partial t} &= \frac{\hat{\mathcal{O}}}{\partial t} + \hat{\mathcal{V}} \cdot \nabla + \frac{\left(dt\right)^{(2/D\_{r})-1}}{4} \Bigg[ \Big(\lambda\_{+}^{i}\lambda\_{+}^{j} + \lambda\_{-}^{i}\lambda\_{-}^{j}\Big) - i\Big(\lambda\_{+}^{i}\lambda\_{+}^{j} - \lambda\_{-}^{i}\lambda\_{-}^{j}\Big) \Bigg] \frac{\hat{\mathcal{O}}^{2}}{\partial \mathbf{X}^{i}\partial \mathbf{X}^{j}} + \\ &+ \frac{\left(dt\right)^{(3/D\_{r})-1}}{12} \Bigg[ \Big(\lambda\_{+}^{i}\lambda\_{+}^{j}\lambda\_{+}^{k} + \lambda\_{-}^{i}\lambda\_{-}^{j}\lambda\_{-}^{k}\Big) - i\Big(\lambda\_{+}^{i}\lambda\_{+}^{j}\lambda\_{+}^{k} - \lambda\_{-}^{i}\lambda\_{-}^{j}\lambda\_{-}^{k}\Big) \Bigg] \frac{\hat{\mathcal{O}}^{3}}{\partial \mathbf{X}^{i}\partial \mathbf{X}^{j}\partial \mathbf{X}^{k}} \end{split} \tag{39}$$

Particularly, by choosing:

( ) ( )

lll

lll

( )

*i j j i k*

*i j j i k*

*i j j i k*

¶ ¶ ¶¶¶

*i j j i k*

*ii i jj j*

++ -- ++

*i k j*

*XXX*

*i k j Q XXX*

( )

( )

( )

 ll

¶ é ù + ê ú ë û ¶ ¶

*i j*

). Taking into account equations (26), (27)

2

*i j*

*Q X X* (38)

*i j*

l l

*F*

(35)

( ) ( )

( ) ( )

*i j j D D F F i k*

( ) ( )

*i j j D D F F i k*

*V* (37)

( ) 2 3 1 1 2 1 3 1

( ) 2 3 1 1 2 1 3 1

2 3 2 1 3 1

*F F*

lll

*F F*

*F F*

lll

*F F*

ll ll

lll

lll

2 3 2 1 3 1

2 3 2 1 3 1

( )

2 3 2 1 3 1

*V* (36)

*F*

l l

2 2 2 1

*i i j j D*

3 3 3 1

*Q Q d xd xd x dt dt*

( )

*i j j i k*

∧ *Q* / ∂*t*

( ) ( )

*i j j D D i k*

*Q QQ Q dt dt t X X XXX Q QQ Q dt dt t X X XXX*

¶ ¶¶ = + ×Ñ + + + ¶ ¶ ¶ ¶¶¶

¶ ¶¶ + + ×Ñ ++ + - ¶ ¶ ¶ ¶¶¶

*i j j D D i k*

¶ ¶ - ×Ñ - - +

*i i Qi Q Q dt dt t X X XXX*

*i j j D D i k*

2 1

*D*

¶ ¶¶ + + ×Ñ + + = ¶ ¶ ¶ ¶¶¶

*i Q i*

*iQ i i Q i Q Q dt dt t X X XXX*

*F*

*VV VV* ( )

( ) ( )

*ii ii jj jj*

++ -- ++ --

 ll

 lll

 ll ll

( ) 3 *j k*

l-

2 1 2

*i j j D D i k*

( ) ( )

( ) ( )

( ) ( )

2 6

( )

*d dt dt*

2 6

*dt t X X XXX* l l

¶ ¶ ¶ = + ×Ñ+ + ¶ ¶ ¶ ¶¶¶

( )

11 1 1 2 2 4 12 11 1 1 2 2 4 12

+ ++ +++

2 2 12

( ) ( )

+++ --- +++ ---

¶ é ù + +- - = ê ú ë û ¶¶¶

*ikik iki jj j*

+++ --- +++ --

+- -

ll ll

*i*

 lll

¶ é ù ê ú ë û ¶¶¶

*ikik ikik jj jj*

 lll

*Q Q dt Q i t X X*

¶ ¶ é ù = + ×Ñ + +- - + ¶ ê ú ë û ¶ ¶

3 1 3

*dt Q i*

+ ++ +++

( ) ( )

= + - ×Ñ + ç ÷ +- -

( )

( )

( )

22 4

2 2 2 12

*dt t X X XXX* l l

¶¶ ¶ = + ×Ñ + + ¶ ¶ ¶ ¶¶¶

*d Q QQ Q Q dt dt*

*t X X X X*

±±± ±±±

± ± ± ± ± ±

¶ ¶ ¶ ¶ ¶

+

1 1 6 6

730 Advances in Quantum Mechanics

*x*

*i k j j i k*

method see [9, 10]), equations (38a) and (38b) are reduced to:

*XXX XXX*

¶ ¶

¶¶¶ ¶¶¶

These relations also allow us to define the operator:

Under these circumstances, let us calculate (∂

l l

+- +-

l l

l l

l l

*Q dt*

+- +-

lll

( ) ( )

lll

( )

 lll

4

*F*

 lll

*D*

¶ æ ö + -

¶ è ø

*Q dQ dQ dQ dQ i t dt dt dt dt*

¶ é ù æ ö = +- - = ê ú ç ÷ ¶ ë û è ø

and (37), we shall obtain:

*V*

*V*

*V*

*V*

1 2

Ù

2

¶ - ¶

*i Q*

*t*

( ) ( )

( ) ( )

*dt*

3 1

*V*

*D*

*F*

12

12

*F*

*D*

1 1 2 2

*Q QQ d Q dt d Q d xd x dt dt*

*i k j j i k D*

*i i j j*

If we divide by *dt* and neglect the terms containing differential factors (for details on the

*i j j i k*

¶ ¶¶ = + ×Ñ + + +

lll

$$
\lambda\_+^i \lambda\_+^j = -\lambda\_-^i \lambda\_-^j = \mathbf{2} \mathbf{D} \mathbf{S}^\dagger \tag{40}
$$

$$
\lambda\_+^i \lambda\_+^j \lambda\_+^k = -\lambda\_-^i \lambda\_-^j \lambda\_+^k = 2\sqrt{2} \mathbf{D}^{\dagger 2} \delta^{\dagger \mathbf{k}} \tag{41}
$$

the fractal operator (39) takes the usual form:

$$\frac{\hat{\mathcal{C}}}{\partial t} = \frac{\hat{\mathcal{C}}}{\hat{\mathcal{C}}t} + \hat{\mathbf{V}} \cdot \nabla - i\mathbf{D} \left(dt\right)^{\left(2\not{/D}\_{\mathcal{V}}\right)-1} \Delta + \frac{\sqrt{2}}{3} \mathbf{D}^{3\not{/2}} \left(dt\right)^{\left(3\not{/D}\_{\mathcal{V}}\right)-1} \nabla^{3} \tag{42}$$

We now apply the principle of scale covariance and postulate that the passage from classical (differentiable) to "fractal" mechanics can be implemented by replacing the standard time derivative operator, *d* / *dt*, with the complex operator ∂ ^ / ∂*t* (this results in a generalization of Nottale's [1, 2] principle of scale covariance). Consequently, we are now able to write the diffusion equation in its covariant form:

$$\frac{\stackrel{\frown}{\partial}\mathcal{Q}}{\partial t} = \frac{\partial \mathcal{Q}}{\partial t} + \left(\stackrel{\frown}{\mathbf{V}} \cdot \nabla\right) \mathcal{Q} - \mathrm{i}\mathcal{D}\left(dt\right)^{\left(2\circ\mathcal{D}\_{\mathbb{F}}\right)-1} \Delta \mathcal{Q} + \frac{\sqrt{2}}{3} \mathcal{D}^{\dagger 2} \left(dt\right)^{\left(3\circ\mathcal{D}\_{\mathbb{F}}\right)-1} \nabla^{\mathrm{3}} \mathcal{Q} = 0 \tag{43}$$

This means that at any point on a fractal path, the local temporal ∂*t Q*, the non-linear (convec‐ tive), (*V* ∧ ⋅∇)*Q*, the dissipative, *ΔQ*, and the dispersive, ∇3*Q*, terms keep their balance.

### **3. Fractal space-time and the motion equation of free particles in the dissipative approximation**

Newton's fundamental equation of dynamics in the dissipative approximation is:

$$m\frac{\stackrel{\frown}{\partial}\stackrel{\frown}{V}}{\partial t} = -\nabla \Phi \tag{44}$$

If we consider the flow of *V* induced by a uniform translational motion of a plane spaced a distance *Y* above a stationary parallel plane (Fig. 1), and if the 'fluid' velocity increases from zero (at the stationary plane) to *U* (at the moving plane) like in the case of simple Couette flow,

rate of shear deformation *d U*

**Figure 1.** Uniform translational motion of a plane spaced a distance Y above a stationary parallel plane.

*d T*

We study an important case, of the one-dimensional flow along the Ox axis :

*V k* = z

To resume, the model considered here consists in analyzing the transport of *V*, along a small elementary distance *Λ*, by the 'particles' of a *Newtonian fluid* moving with velocity *U*, where

( ) *x* (50)

*dy Y* = = h

 h

Fluids which obey (49) in the above situation are known as *Newtonian fluids*, which have a very small coefficient of viscosity. When such 'fluids' flow at reasonable velocities it is found that viscous effects appear only in thin layers on the surface of objects or surfaces over which the 'fluid' flows. That is, if one continues the analogy, and questions how is *V* transported by the motion of 'fluid' particles with the velocity *U*, in second equation (46), one can assume that the mechanism of transfer of *V* from one particle of 'fluid' to another is achieved over small

*V U*

For many fluids it is found that the magnitude of the shearing stress is related to the rate of

*dy Y* = = *V*

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(48)

733

(49)

or simple shear flow, then

shear proportionally:

distances (in thin layers, as stated above).

the stress tensor obeys (49), i.e.

where *m* is the mass, *V* ∧ the instantaneous velocity of the particle, *Φ*the scalar potential and

$$\frac{\stackrel{\frown}{\mathcal{O}}}{\frac{\partial}{\partial t}} = \frac{\stackrel{\frown}{\mathcal{O}}}{\frac{\partial}{\partial t}} + \left(\stackrel{\frown}{\mathcal{V}} \cdot \nabla\right) - i\mathcal{D}(dt)^{(2fD\_{\mathbb{F}})-1} \,\,\Delta\tag{45}$$

is the fractal operator in the dissipative approximation.

In what follows, we study what happens with equation (44), in the free particle case (*Φ* = 0), if one considers the space-time where particles move changes from classical to nondifferentiable.

According to Nottale [11], the transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic). Separating the real and imaginary parts, (44) becomes:

$$\begin{aligned} \frac{\partial \mathbf{V}}{\partial t} + \mathbf{V} \cdot \nabla \mathbf{V} &= 0 \\ \mathbf{U} \cdot \nabla \mathbf{V} &= -\mathbf{D} \Delta \mathbf{V} \end{aligned} \tag{46}$$

where *V* **^** =*V* −*i U* is the complex velocity defined through (27) and *D* defines the amplitude of the fractal fluctuations (D=*D*(*dt*) (2/*DF* )−1 ).

Let us analyze in what follows, the second equation (46) which, one can see, may contain some interesting physics. If we compare it with Navier-Stokes equation, from fluid mechanics [12]

$$\frac{D\mathbf{v}}{Dt} = \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = \nu \nabla^2 \mathbf{v} \tag{47}$$

we can see the left side of (46) gives the rate at which *V* is transported through a 'fluid' by means of the motion of 'fluid' particles with the velocity *U*; the right hand side gives the diffusion of *V*, (*D* which is the amplitude of the fractal fluctuations, plays here the role of the 'cinematic viscosity' of the 'fluid'). One can notice, in those regions in which the right hand side

of (47) is negligible, *Dv/Dt =* 0. This means that in inviscid flows, for instance, *V* **∧** is frozen into the 'particles of the fluid'. Physically this is due to the fact that in an inviscid 'fluid' shear stresses

are zero, so that there is no mechanism by which *V* can be transferred from one 'fluid' particle to another. This may be the case for the transport of *V* by *U* in the second equation (II.3).

**∧**

If we consider the flow of *V* induced by a uniform translational motion of a plane spaced a distance *Y* above a stationary parallel plane (Fig. 1), and if the 'fluid' velocity increases from zero (at the stationary plane) to *U* (at the moving plane) like in the case of simple Couette flow, or simple shear flow, then

*m t*

*t t*

is the fractal operator in the dissipative approximation.

where *m* is the mass, *V*

732 Advances in Quantum Mechanics

where *V*

the fractal fluctuations (D=*D*(*dt*)

∧

Ù Ù ¶ = -ÑF ¶

*V* (44)

*V D dt* (45)

*v v v* (47)

**∧**

can be transferred from one 'fluid' particle

is frozen into

(46)

the instantaneous velocity of the particle, *Φ*the scalar potential and

( ) (2 1 ) *DF i*

In what follows, we study what happens with equation (44), in the free particle case (*Φ* = 0), if one considers the space-time where particles move changes from classical to nondifferentiable. According to Nottale [11], the transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic). Separating the real and imaginary parts, (44) becomes:

D

Let us analyze in what follows, the second equation (46) which, one can see, may contain some interesting physics. If we compare it with Navier-Stokes equation, from fluid mechanics [12]

we can see the left side of (46) gives the rate at which *V* is transported through a 'fluid' by means of the motion of 'fluid' particles with the velocity *U*; the right hand side gives the diffusion of *V*, (*D* which is the amplitude of the fractal fluctuations, plays here the role of the 'cinematic viscosity' of the 'fluid'). One can notice, in those regions in which the right hand side

the 'particles of the fluid'. Physically this is due to the fact that in an inviscid 'fluid' shear stresses

to another. This may be the case for the transport of *V* by *U* in the second equation (II.3).

**∧**

¶ + ×Ñ =

**V V V**

×Ñ = - D

**UV V**

*D* 2

¶ = + ×Ñ = Ñ

*t*

¶

(2/*DF* )−1 ).

*Dt t*

¶ *v v*

of (47) is negligible, *Dv/Dt =* 0. This means that in inviscid flows, for instance, *V*

are zero, so that there is no mechanism by which *V*

0

**^** =*V* −*i U* is the complex velocity defined through (27) and *D* defines the amplitude of

n

Ù Ù ¶ ¶ æ ö - = + ×Ñ - ç ÷ D ç ÷ ¶ ¶ è ø

**Figure 1.** Uniform translational motion of a plane spaced a distance Y above a stationary parallel plane.

For many fluids it is found that the magnitude of the shearing stress is related to the rate of shear proportionally:

$$T = \eta \frac{d\mathbf{V}}{dy} = \eta \frac{\mathbf{U}}{Y} \tag{49}$$

Fluids which obey (49) in the above situation are known as *Newtonian fluids*, which have a very small coefficient of viscosity. When such 'fluids' flow at reasonable velocities it is found that viscous effects appear only in thin layers on the surface of objects or surfaces over which the 'fluid' flows. That is, if one continues the analogy, and questions how is *V* transported by the motion of 'fluid' particles with the velocity *U*, in second equation (46), one can assume that the mechanism of transfer of *V* from one particle of 'fluid' to another is achieved over small distances (in thin layers, as stated above).

We study an important case, of the one-dimensional flow along the Ox axis :

$$\mathcal{V} = \mathcal{L}\left(\mathbf{x}\right) \; k \tag{50}$$

To resume, the model considered here consists in analyzing the transport of *V*, along a small elementary distance *Λ*, by the 'particles' of a *Newtonian fluid* moving with velocity *U*, where the stress tensor obeys (49), i.e.

$$\frac{d\mathcal{L}\left(\mathbf{x}\right)}{d\mathbf{x}} = \frac{\mathcal{L}\left(\mathbf{x}\right)}{\Lambda} \tag{51}$$

Let us try a solution of the form *ζ(x)=A exp ((i/ħ)S(x))*. Substituting this solution into the time-

2 2 2 2

2

hh h h (58)

( )

( )

1

± ¢ ¢ - ò = (62)

counts the oscillations of the velocity wave

è ø (61)

*x*

2

h hh h (57)

r

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735

c g

2 2 2 2

æ ö æ ö

*dx dx dx dx*

2 2 0 or 0 *d S dS d S dS i ki*

0 0 1 1 2 2 ... ... 0 , ( ( )) *d dS dS dS dS i k x*

é ù æ ö ê ú + +- + + + = > ç ÷ ë û è ø

0 0

1 2 0 ln

dk 2 , i 2 , dx 2 *d dS dS dS dS i dk i k dS dx dx dx dx dx k* æ ö ç ÷ = = =

)*d x* ′

*d S dS dS i S i kx*

*dS k S k x dx*

Assume that *ħ*can, in some sense, be regarded as a small quantity and that *S(x)* can be expanded

independent Schrödinger equation (52) we get:

in powers of *ħ*, *S(x) = S0(x)+ ħ S1(x)* +....

*dx*

*dx dx dx dx dx*

2 0 2 2

0 0 1

z

*dx dx dx*

2 1

0 0 1 1

( ) ( ) 1 2 *x i k x dx*

*x Ak e*

*x* ℏ*k*(*x* ′

function. An increase of *2π ħ* corresponds to an additional phase of *2π*.

*dS*1 *dx*

*dx* é ù

2

Then,

We assume that | *dS*0

We have used:

Therefore, for *χ>γ(x)*

In the classically allowed region *S*0 = ± *∫*

like in the case of simple Couette flow, or simple shear flow.

Consequently, the second eq. (46) reduces to the scalar equation

$$
\zeta''(x) + K^2(x)\zeta(x) = 0\tag{52}
$$

which is the time independent Schrödinger equation, and

$$K^2\left(x\right) = \frac{1}{\Lambda \mathcal{D}} \mathcal{U}\left(x\right) \tag{53}$$

with *Λ* and *D* having the significance of a small elementary distance and of the 'cinematic viscosity' (or amplitude of the fractal fluctuations), respectively, and *U(x)* is the velocity of the 'Newtonian fluid', which is nothing but the imaginary part of the complex velocity [13]. In what follows, we solve this equation accurately by means of the WKBJ approximation method with connection formulas.

#### **3.1. Solving the Schrödinger type equation by means of the WKBJ approximation method**

Let us re-write (53) in the form

$$\mathcal{K}^2\left(\mathbf{x}\right) = \frac{1}{\Lambda \mathbf{D}} \mathcal{U}\left(\mathbf{x}\right) = \frac{2m^2 c}{\hbar^2} \left(\boldsymbol{\chi} - \boldsymbol{\mathcal{V}}\left(\mathbf{x}\right)\right) \tag{54}$$

where we take *D* =ℏ / 2*m* and consider the small elementary distance the Compton length *Λ* =ℏ / *mc* [14]. Therefore, the Schrödinger equation (52) splits into:

$$\frac{d^2}{dx^2}\zeta\left(\mathbf{x}\right) + k^2\left(\mathbf{x}\right)\zeta\left(\mathbf{x}\right) = 0, \quad \mathbf{x} > \mathbf{y} \text{ or } \frac{d^2}{dx^2}\zeta\left(\mathbf{x}\right) - \rho^2\left(\mathbf{x}\right)\zeta\left(\mathbf{x}\right) = 0, \quad \mathbf{x} < \mathbf{y} \tag{55}$$

where

$$k(\mathbf{x}) = \sqrt{\frac{2\mu\left(\mathbf{x} - \boldsymbol{\gamma}\left(\mathbf{x}\right)\right)}{\hbar^2}}, \rho\left(\mathbf{x}\right) = \sqrt{\frac{2\mu\left(\boldsymbol{\gamma}\left(\mathbf{x}\right) - \boldsymbol{\chi}\right)}{\hbar^2}} \text{ with } \mu = m^2 c \tag{56}$$

*χ* is a limit velocity and *γ(x)* a 'velocity potential'.

Let us try a solution of the form *ζ(x)=A exp ((i/ħ)S(x))*. Substituting this solution into the timeindependent Schrödinger equation (52) we get:

$$i\hbar \frac{d^2S}{d\chi^2} - \left(\frac{dS}{d\chi}\right)^2 + \hbar^2 k^2 = 0 \text{ or } i\hbar \frac{d^2S}{d\chi^2} - \left(\frac{dS}{d\chi}\right)^2 - \hbar^2 \rho^2 = 0\tag{57}$$

Assume that *ħ*can, in some sense, be regarded as a small quantity and that *S(x)* can be expanded in powers of *ħ*, *S(x) = S0(x)+ ħ S1(x)* +....

Then,

*dx x* ( ) ( )

() () () 2

 z

with *Λ* and *D* having the significance of a small elementary distance and of the 'cinematic viscosity' (or amplitude of the fractal fluctuations), respectively, and *U(x)* is the velocity of the 'Newtonian fluid', which is nothing but the imaginary part of the complex velocity [13]. In what follows, we solve this equation accurately by means of the WKBJ approximation method

**3.1. Solving the Schrödinger type equation by means of the WKBJ approximation method**

( ) ( ) ( ) ( )

1 2*m c K x Ux* = = -

L*D* h

() () () () () ()

( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 , with *x x k x x m c*

2 2 2 2 0, or 0, *d d x kx x x xx*

 cg

r

h h

2

c g*x*

2

 z

mg

 r

> c

+ => - =< (55)

 z

> m

 cg

where we take *D* =ℏ / 2*m* and consider the small elementary distance the Compton length

 z

= L (51)

¢¢ *x Kx x* + = 0 (52)

( ) ( ) 2 1 *K x Ux* = L*D* (53)

(54)

(56)

*dx* z

like in the case of simple Couette flow, or simple shear flow.

Consequently, the second eq. (46) reduces to the scalar equation

z

which is the time independent Schrödinger equation, and

2

2 2

*dx dx*

 z

mc g

*χ* is a limit velocity and *γ(x)* a 'velocity potential'.

*Λ* =ℏ / *mc* [14]. Therefore, the Schrödinger equation (52) splits into:

with connection formulas.

734 Advances in Quantum Mechanics

Let us re-write (53) in the form

z

where

$$i\hbar \frac{d}{dx} \left[ \frac{d\mathbf{S}\_0}{dx} + \hbar \frac{d\mathbf{S}\_1}{dx} + \dots \right] - \left( \frac{d\mathbf{S}\_0}{dx} + \hbar \frac{d\mathbf{S}\_1}{dx} + \dots \right)^2 + \hbar^2 k^2 = 0 \,, \quad \{\mathbf{x} > \boldsymbol{\gamma}(\mathbf{x})\} \tag{58}$$

We assume that | *dS*0 *dx* | > > | ℏ *dS*1 *dx* | and collect terms with equal powers of *ħ*.

$$-\left[\frac{d\mathbf{S}\_0}{d\mathbf{x}}\right]^2 + \hbar^2 k^2 = 0 \quad \Rightarrow \quad \mathbf{S}\_0 = \pm \int\_{\mathbf{x}} \hbar k \left(\mathbf{x}'\right) d\mathbf{x}' \tag{59}$$

$$i\frac{d^2S\_0}{dx^2} - 2\frac{dS\_0}{dx}\frac{dS\_1}{dx} = 0 \quad \Rightarrow \quad S\_1 = \frac{1}{2}i\ln k\left(x\right) \tag{60}$$

We have used:

$$\mathrm{i}\frac{d}{d\mathbf{x}}\left(\frac{dS\_0}{d\mathbf{x}}\right) = 2\frac{dS\_0}{d\mathbf{x}}\frac{dS\_1}{d\mathbf{x}}\quad,\quad\mathrm{i}\frac{d\mathbf{k}}{d\mathbf{x}} = 2\mathrm{k}\frac{dS\_1}{d\mathbf{x}}\quad,\quad dS\_1 = \frac{\mathrm{i}}{2}\frac{dk}{k}\tag{61}$$

Therefore, for *χ>γ(x)*

$$
\zeta'(x) = Ak^{-\frac{1}{2}}e^{\frac{\pm i \int k(x')dx'}{x}} \tag{62}
$$

In the classically allowed region *S*0 = ± *∫ x* ℏ*k*(*x* ′ )*d x* ′ counts the oscillations of the velocity wave function. An increase of *2π ħ* corresponds to an additional phase of *2π*.

Similarly, in regions where *χ<γ(x)* we have:

$$
\zeta'(\mathbf{x}) = A \rho^{-\frac{1}{2}} e^{\frac{\pm \int \rho(\mathbf{x}') d\mathbf{x}'}{\mathbf{x}}} \tag{63}
$$

In the region between *x1* and *x2* it is of the form:

z

*γ*(*x*)=*χ* + *K*2(*x* − *x*2).

Let us define *z* = −( 2*μK*1

defined through:

and

( ) ( ) ( ) 1 1 2 2 2 2 2 *x x*

In the neighborhood of *x1* the time-independent Schrödinger equation then becomes:

1 2 2 1 2 0 *d K x x*

and in the neighborhood of *x2* the time-independent Schrödinger equation becomes:

2 2 2 2 2 0 *d K x x*

(*x* − *x*1). Then we obtain *d* 2

0 1 cos 3

p

1 4

*Ai z z z*

*Ai z z z*

*s Ai z sz ds*

m

h

m

h

2

*dx* z

2

*dx* z

( )

ℏ2 ) 1 3

which for large *|z|* has the asymptotic form

( )

( )

2

( )

p

*z*

1 4

p*z*

*x Ak e Ak e*

*i k x dx i k x dx*

= + ¢ (66)

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(67)

737

(68)

(70)

(71)

+ - ¢ ¢ ¢ ¢ - - ò ò

At *x = x1* and *x = x2* the velocity wave function *ζ* and its derivatives have to be continuous. Near *x1* and *x2* we expand the velocity potential well *γ(x)* in a Taylor series expansion in *x* and neglect all terms of order higher than *1*. Near *x1* we have *γ*(*x*)=*χ* − *K*1(*x* − *x*1), and near *x2* we have

( )

( )

equation which vanish asymptotically as *z* → ∞ or *x* → -∞ are the Airy functions. They are

3

3 2

æ ö ç ÷ - > ç ÷ è ø

p

æ ö ç ÷ -+ <

1 2 ~ exp , ( 0) 3

( ) 3 2

1 2 ~ sin , ( 0) 3 4

è ø -

¥ æ ö

z

> z

> > *ζ*

*d z* 2 − *zζ* =0 near *x1*. The solutions of this

= + ç ÷ ç ÷ è ø ò (69)

+ -=

For our first order expansion to be accurate we need that the magnitude of higher order terms decreases rapidly. We need | *dS*0 *dx* | > > | ℏ *dS*1 *dx* | or |*k* | > > | 1 2*k dk dx* |. The local deBroglie wavelength is *λ =* 2*π/k*. Therefore, | *λ* 4*π dλ dx* | < <*λ* , i.e. the change in *λ* over a distance *λ/*4*π* is small compared to *λ*. This holds when the velocity potential *γ(x)* varies slowly and the momentum is nearly constant over several wavelengths.

Near the classical turning points the WKBJ solutions become invalid, because *k* goes to zero here. We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.

#### **3.2. Velocity potential γ (x) and the bound states**

We want to find the velocity wave function in a given velocity potential well *γ(x)*. Assuming that the limit velocity of the particle is*χ* and that the classical turning points are *x1* and *x2*, *x1<x2*, i.e. we have a velocity potential well with two sloping sides (Fig. 2).

**Figure 2.** Bound state problem.

For *x < x1* the velocity wave function is of the form:

$$
\zeta\_1(\mathbf{x}) = A\_1 \rho^{-\frac{1}{2}} e^{\frac{1}{2} \int \rho(\mathbf{x'}) dx'} \tag{64}
$$

For *x > x2* the velocity wave function is of the form:

$$
\zeta\_3(\mathbf{x}) = A\_3 \rho^{-\frac{1}{2}} e^{-\int \rho(\mathbf{x}') d\mathbf{x}'} \tag{65}
$$

In the region between *x1* and *x2* it is of the form:

Similarly, in regions where *χ<γ(x)* we have:

decreases rapidly. We need | *dS*0

736 Advances in Quantum Mechanics

**Figure 2.** Bound state problem.

wavelength is *λ =* 2*π/k*. Therefore, | *λ*

( ) ( ) 1 2 *x*

*xA e*

*dx* | > > | ℏ

4*π dλ*

*x1<x2*, i.e. we have a velocity potential well with two sloping sides (Fig. 2).

 r

z

momentum is nearly constant over several wavelengths.

**3.2. Velocity potential γ (x) and the bound states**

For *x < x1* the velocity wave function is of the form:

For *x > x2* the velocity wave function is of the form:

*x dx*

*dx* | or |*k* | > > | 1

± ¢ ¢ - ò = (63)

2*k dk*

*dx* | < <*λ* , i.e. the change in *λ* over a distance *λ/*4*π* is

*dx* |. The local deBroglie

r

For our first order expansion to be accurate we need that the magnitude of higher order terms

*dS*1

small compared to *λ*. This holds when the velocity potential *γ(x)* varies slowly and the

Near the classical turning points the WKBJ solutions become invalid, because *k* goes to zero here. We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.

We want to find the velocity wave function in a given velocity potential well *γ(x)*. Assuming that the limit velocity of the particle is*χ* and that the classical turning points are *x1* and *x2*,

> ( ) ( ) 1 2

( ) ( ) 1 2 3 3 *x*

 r

*xA e*

 r *x dx*

*x dx*

r

¢ ¢ - ò = (64)

r

1 1 *x*

z

z

*xA e*

$$
\zeta\_2 \left( \mathbf{x} \right) = A\_2 k^{-\frac{1}{2}} e^{\frac{-1}{2} \int\_x^{\left[ k \left( x' \right) \right] dx'}} + A\_2' k^{-\frac{1}{2}} e^{-\frac{1}{x} \int\_x^{\left[ k \left( x' \right) \right] dx'}} \tag{66}
$$

At *x = x1* and *x = x2* the velocity wave function *ζ* and its derivatives have to be continuous. Near *x1* and *x2* we expand the velocity potential well *γ(x)* in a Taylor series expansion in *x* and neglect all terms of order higher than *1*. Near *x1* we have *γ*(*x*)=*χ* − *K*1(*x* − *x*1), and near *x2* we have *γ*(*x*)=*χ* + *K*2(*x* − *x*2).

In the neighborhood of *x1* the time-independent Schrödinger equation then becomes:

$$\frac{d^2\zeta}{dx^2} + \frac{2\mu K\_1}{\hbar^2}(x - x\_1)\zeta = 0\tag{67}$$

and in the neighborhood of *x2* the time-independent Schrödinger equation becomes:

$$\frac{d^2\zeta}{dx^2} - \frac{2\mu K\_2}{\hbar^2} \left(\mathbf{x} - \mathbf{x}\_2\right) \zeta = 0\tag{68}$$

Let us define *z* = −( 2*μK*1 ℏ2 ) 1 3 (*x* − *x*1). Then we obtain *d* 2 *ζ d z* 2 − *zζ* =0 near *x1*. The solutions of this equation which vanish asymptotically as *z* → ∞ or *x* → -∞ are the Airy functions. They are defined through:

$$Ai(z) = \frac{1}{\pi} \int\_0^\varepsilon \cos\left(\frac{s^3}{3} + sz\right) ds\tag{69}$$

which for large *|z|* has the asymptotic form

$$A\mathbf{i}(z) \sim \frac{1}{2\sqrt{\pi z^4}} \exp\left(-\frac{2}{3}z^{\frac{3}{2}}\right), \quad \text{( $z > 0$ )}\tag{70}$$

and

$$Ai(z) \sim \frac{1}{\sqrt{\pi} \frac{1}{(-z)^{\frac{1}{4}}}} \sin \left( \frac{2}{3} (-z)^{\frac{3}{2}} + \frac{\pi}{4} \right), \quad (z < 0) \tag{71}$$

If the limit velocity *χ* is high enough, the linear approximation to the velocity potential well remains valid over many wavelengths. The Airy functions can therefore be the connecting velocity wave functions through the turning point at *x1*.

If we define *z* =( 2*μK*2 ℏ2 ) 1 3 (*x* − *x*2) then we find *d* 2 *ζ d z* 2 − *zζ* =0 near *x = x2* and the Airy functions can also be the connecting velocity wave functions through the turning point at *x2*. Here *z* → ∞ or *x* → ∞.

In the neighborhood of *x1* we have

$$k^2 = -\rho^2 = \left(\frac{2\,\mu K\_1}{\hbar^2}\right)^{\frac{1}{3}} \left(\mathbf{x} - \mathbf{x}\_1\right) = -\left(\frac{2\,\mu K\_1}{\hbar^2}\right)^{\frac{1}{3}} \mathbf{z} \tag{72}$$

( )

1

1

, we have

2

*x*

1

*x*

2

*x*

1

*x*

1 3

*A A*

=

*x x*

p

*kdx n n* p

*x x*

z

z

phases must be the same modulo *π* :

Knowing that *∫*

*x*1

This can be re-written as

with

*x*2 = *∫ x*1

*x* + *∫ x*

*x*2

must continue in region 2 as

( )

*x dx*

*x x*

*A e x xx* r

( ) ( ) 1 2 2

*x x A k kd x xx*

*x x*

*x x*

2

*kdx kdx n*

4 4

 p

1 , 1,2,3,... 2

1 1

2 2 2

or

p

¢ ¢ + =- - + ò ò (80)

æ ö ¢ =- = ç ÷ è ø ò (81)

*h dx n dx n h* æö æö P= - P= - ç÷ ç÷ èø èø ò òÑ (82)

Both expressions for *ζ2(x)* are approximations to the same eigenfunction. We therefore need

*x*

2 23 2 sin

è

 - æ ö = + ç ÷ ¢ ç ÷ø

1 1 2 2 1 3 2 sin 2 sin

*A k kdx A k kdx* - p

1 2 3 23

 r ( ) 2

4

4 4

æ ö æ ö ç ÷ ¢ ¢ + = ç ÷ + ç ÷ ç ÷ è ø è ø

For (79) to be satisfied, the amplitudes of each side must have the same magnitude, and the

p

<

2

 p

ò ò (79)

>= (77)

ò (78)

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Therefore

$$\int\_{x\_1}^{x} \rho d\mathbf{x}' = \left(\frac{2\,\mu K\_1}{\hbar^2}\right)^{\frac{1}{3}} \int\_{x\_1}^{x} \sqrt{z} d\mathbf{x}' = -\oint\_0 \sqrt{z'} dz' = -\frac{2}{3}z^{\frac{3}{2}}\tag{73}$$

Similarly

$$\int\_{x\_1}^{x} k dx' = \left(\frac{2\,\mu K\_1}{\hbar^2}\right)^{\frac{1}{3}} \int\_{x\_1}^{x} \sqrt{-z} dz' = -\int\_0^{\underline{x}} \sqrt{-z'} dz' = \frac{2}{3} \left(-z\right)^{\frac{3}{2}}\tag{74}$$

By comparing this with the asymptotic forms of the Airy functions we note that

$$\mathcal{L}\_1(\mathbf{x}) = A\_1 \rho^{-\frac{1}{2}} e^{-\frac{1}{2} \int \rho(\mathbf{x'}) d\mathbf{x'}} \quad \left(\mathbf{x} < \mathbf{x\_1}\right) \tag{75}$$

must continue on the right side as

$$\mathcal{L}\_2\left(\mathbf{x}\right) = 2A\_1k^{-\frac{1}{2}}\sin\left(\int\_{x\_1}^{x} kdx' + \frac{\pi}{4}\right) \left(\mathbf{x} > \mathbf{x}\_1\right) \tag{76}$$

In the neighborhood of *x2*we similarly find that

$$\mathcal{L}\_3\left(\mathbf{x}\right) = A\_3 \rho^{-\frac{1}{2}} e^{-\frac{1}{2} \int \rho(\mathbf{x'}) d\mathbf{x'}} \quad \left(\mathbf{x} > \mathbf{x}\_2\right) \tag{77}$$

must continue in region 2 as

If the limit velocity *χ* is high enough, the linear approximation to the velocity potential well remains valid over many wavelengths. The Airy functions can therefore be the connecting

*ζ*

also be the connecting velocity wave functions through the turning point at *x2*. Here *z* → ∞ or

( )

2 2 1

3 3 1 2

2 2

*K dx zdx z dz z*

æ ö ¢ = =- =- ¢ ¢¢ ç ÷

0

3 3 1 2

2 2

¢ = - =- - = - ¢ ¢¢ ç ÷

*K kdx zdx z dz z* æ ö

By comparing this with the asymptotic forms of the Airy functions we note that

1 2 1 11

 r

1 2 2 sin

 - æ ö = + ç ÷ ¢ ç ÷ø

0

( )

*x dx*

*x x*

+ ¢ ¢ - ò

*A e x xx* r

( ) ( ) 1 2 1 1

*x x A k kd x xx*

è

*x*

( ) 1

4

p

>

èø èø h h

3 3 2 2 1 1

2 2 *K K k x x z*

æö æö =- = - =- ç÷ ç÷

1 1

 m

3

( )

<= (75)

ò (76)

3

*d z* 2 − *zζ* =0 near *x = x2* and the Airy functions can

(72)

(73)

(74)

velocity wave functions through the turning point at *x1*.

r

1 1

*x x*

1 1

*x x*

2

( )

z

z

In the neighborhood of *x2*we similarly find that

must continue on the right side as

m

r

2

m

(*x* − *x*2) then we find *d* 2

m

1

*x xx*

è ø ò òò h

1

*x xx*

è ø ò òò h

ℏ2 ) 1 3

In the neighborhood of *x1* we have

If we define *z* =( 2*μK*2

738 Advances in Quantum Mechanics

*x* → ∞.

Therefore

Similarly

$$\mathcal{L}\_2\left(\mathbf{x}\right) = 2A\_3k^{-\frac{1}{2}}\sin\left(\int\_{\mathbf{x}}^{\mathbf{x}\_2} k d\mathbf{x}' + \frac{\pi}{4}\right) \left(\mathbf{x} < \mathbf{x}\_2\right) \tag{78}$$

Both expressions for *ζ2(x)* are approximations to the same eigenfunction. We therefore need

$$2A\_1k^{-\frac{1}{2}}\sin\left(\int\_{x\_1}^{x}kdx'+\frac{\pi}{4}\right) = 2A\_3k^{-\frac{1}{2}}\sin\left(\int\_{x}^{x\_2}kdx'+\frac{\pi}{4}\right) \tag{79}$$

For (79) to be satisfied, the amplitudes of each side must have the same magnitude, and the phases must be the same modulo *π* :

$$\begin{aligned} \left| A\_1 \right| &= \left| A\_3 \right|\\ \int\_{\infty}^{\infty} k d\mathbf{x}' + \frac{\pi}{4} &= -\int\_{\infty}^{x\_2} k d\mathbf{x}' - \frac{\pi}{4} + n\pi \end{aligned} \tag{80}$$

$$\begin{aligned} \text{Knowing that} & \int\_{x\_1}^{x\_2} \int\_{x\_1}^{x\_2} \text{we have} \\\\ & \int\_{x\_1}^{x\_2} k dx' = \left(n - \frac{1}{2}\right) \pi\_\prime \qquad n = 1, 2, 3, \dots \end{aligned} \tag{81}$$

This can be re-written as

$$\int\_{x\_1}^{x\_2} \Pi d\mathbf{x} = \left(n - \frac{1}{2}\right)\frac{h}{2} \text{ or } \oint \Pi d\mathbf{x} = \left(n - \frac{1}{2}\right)h \tag{82}$$

with

$$\Pi = \left[ 2\,\mu\left(\text{ $\chi$ } - \text{ $\chi$ }\left(\text{ $\chi$ }\right)\right) \right]^{1/2} = \left[ 2\,\mu\text{L}\left(\text{ $\chi$ }\right) \right]^{1/2} = mc \left[ 2\,\frac{\text{L}\left(\text{ $\chi$ }\right)}{c} \right]^{1/2} \tag{83}$$

( ) 1 2 1 2

+

*x dx n*

( ) 1 1 22 2 2 2 *W mx m x c* = = w

3 2 3 2 3 1

+ -- = - ç ÷ è ø

*a*

and we get the expression for the *x* dependence of the velocity term, *γ*(*x*)=*ωx*.

 g

( ) 1 2

m

p

2 2 2 *n*

u

æ ö

*a a*

mc w

where the energy *Wc* of the oscillator *U(x)* with the pulsation *ω*writes

Theleft side term of (86) is an elementary integral and we find:

( ) ( )

where *υ* =*ω a* is the liniar velocity (see the graphic in Fig. 3).

**Figure 3.** Dependence of the limit velocity χ on the linear velocity υ.

 cu

cu

2

p

æ ö é ù ë û - =- ç ÷ è ø ò h (86)

(87)

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h (88)

Here *∮* denote an integral over one complete cycle of the classical motion. The WKBJ method for *γ(x)* velocity potential well with soft walls, therefore, leads to a Wilson-Sommerfeld type quantization rule except that *n* is replaced by *n-*1/2. It leads to a quantization of the complex velocity *U(x)*.

The factor of *π/*2 arises here due to the two phase changes of *π/*4 at *x1* and *x2*. In case where only one of the walls is soft and the other is infinitely steep the factor of 1/2 is replaced by 1/4 in (81). If both walls are infinitely steep, the factor of 1/2 in (81) is replaced by 0.

WKBJ approximation is a semi classical approximation, since it is expected to be most useful in the nearly classical limit of large quantum numbers. The method will not be good for, say, lowest limit velocity states *χ*, so in order to overcome this shortcomings there is a need for a modified semi classical quantization condition. For oscillations between the two classical turning points *x1* and *x2*, we obtain the semi classical quantization condition by requiring that the total phase during one period of oscillation to be an integral multiple of 2*π*; [15] such that

$$2\int\_{x\_1}^{x\_2} k d\alpha' + \phi\_1 + \phi\_2 = 2\pi n$$

where *ϕ1* is the phase loss due to reflection at the classical turning point *x1* and *ϕ2* is the phase loss due to reflection at *x2*. Taking *ϕ1* and *ϕ2* to be equal to *π/2*leads to the modified semiclassical quantization rule, i.e.

$$\int\_{x\_1}^{x\_2} k dx' = \left(n - \frac{m}{4}\right)\pi \tag{85}$$

where *m*is the Maslov index [15], which denotes the total phase loss during one period in units of *π/*2. It contains contributions from the phase losses *ϕ1*and *ϕ2* due to reflections at points *x1* and *x2*, respectively. It is pertinent to note that taking *ϕ1* = *ϕ2* = *π/*2 and an integer Maslov index *m* = 2 in (85), we have the familiar semi classical quantization rule, i.e. (81).

Let us apply the constraint equation (81) to an harmonic oscillator. The condition then is (passing without loss of generality to the limits -*a* to +*a*)

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$$\int\_{-a}^{+a} \left[ 2\mu \left( \chi - \alpha \right) \chi \right]^{\dagger 2} d\chi = \left( n - \frac{1}{2} \right) \pi \hbar \tag{86}$$

where the energy *Wc* of the oscillator *U(x)* with the pulsation *ω*writes

( ) ( ) ( ) ( ) 1 2 1 2 1 2 2 22

Here *∮* denote an integral over one complete cycle of the classical motion. The WKBJ method for *γ(x)* velocity potential well with soft walls, therefore, leads to a Wilson-Sommerfeld type quantization rule except that *n* is replaced by *n-*1/2. It leads to a quantization of the complex

The factor of *π/*2 arises here due to the two phase changes of *π/*4 at *x1* and *x2*. In case where only one of the walls is soft and the other is infinitely steep the factor of 1/2 is replaced by 1/4

WKBJ approximation is a semi classical approximation, since it is expected to be most useful in the nearly classical limit of large quantum numbers. The method will not be good for, say, lowest limit velocity states *χ*, so in order to overcome this shortcomings there is a need for a modified semi classical quantization condition. For oscillations between the two classical turning points *x1* and *x2*, we obtain the semi classical quantization condition by requiring that the total phase during one period of oscillation to be an integral multiple of 2*π*; [15] such that

in (81). If both walls are infinitely steep, the factor of 1/2 in (81) is replaced by 0.

2

*x*

1 2 2 2

*kdx*¢ + += ff

*m kdx n*

æ ö ¢ = - ç ÷

 p

where *ϕ1* is the phase loss due to reflection at the classical turning point *x1* and *ϕ2* is the phase loss due to reflection at *x2*. Taking *ϕ1* and *ϕ2* to be equal to *π/2*leads to the modified semiclassical

4

where *m*is the Maslov index [15], which denotes the total phase loss during one period in units of *π/*2. It contains contributions from the phase losses *ϕ1*and *ϕ2* due to reflections at points *x1* and *x2*, respectively. It is pertinent to note that taking *ϕ1* = *ϕ2* = *π/*2 and an integer Maslov index

Let us apply the constraint equation (81) to an harmonic oscillator. The condition then is

p

*n* ò (84)

è ø ò (85)

1

2

*x*

1

*m* = 2 in (85), we have the familiar semi classical quantization rule, i.e. (81).

(passing without loss of generality to the limits -*a* to +*a*)

*x*

*x*

*x U x mc*

 m

P= - = é ù é ù = ê ú ë û ë û ê ú ë û

mc g

velocity *U(x)*.

740 Advances in Quantum Mechanics

quantization rule, i.e.

*U x*

é ù

*c*

(83)

$$W\_c = \frac{1}{2}m\rho^2 \mathbf{x}^2 = \frac{1}{2}m\gamma^2 \left(\mathbf{x}\right) \tag{87}$$

and we get the expression for the *x* dependence of the velocity term, *γ*(*x*)=*ωx*. Theleft side term of (86) is an elementary integral and we find:

$$\left(\left(\varkappa+\nu\right)^{3/2}-\left(\varkappa-\nu\right)^{3/2}=\frac{3\pi}{2a\left(2\mu\right)^{4l}}\left(n-\frac{1}{2}\right)\hbar\nu\right.\tag{88}$$

where *υ* =*ω a* is the liniar velocity (see the graphic in Fig. 3).

**Figure 3.** Dependence of the limit velocity χ on the linear velocity υ.

We try to estimate a value for the limit velocity *χ*. Let us expand the left side term of (88) in series and keep the first term. If we replace *μ* from (56) and take *a = Λ* (the Compton length), we get:

$$\mathcal{X} \approx \frac{\pi^2 \hbar^2}{8\mu a^2} \left( n - \frac{1}{2} \right)^2 = \frac{\pi^2}{32} (2n - 1)^2 c = \left\{ \frac{\pi^2}{32} c, \ \text{9 } \frac{\pi^2}{32} c, \ \text{25 } \frac{\pi^2}{32} c, \dots \right\} \tag{89}$$

Using trigonometric identities to simplify the right hand side, gives

*A kdx B C* p

1 2 2 2 2 sin ' 4 sin

1

cos

*L*

f-

1

1

*x*

where

equal modulo *π* :

with *n =* 1, 2,....

with *m =* 1, 2,... and *ϕR* given by:

*x*

( )

4 2

é ù æ ö ê ú + = + +- ç ÷ ê ú è ø ë û

( )

The magnitude of the sin function must be equal, and the magnitude of the phases must be

*B C*

*L L*

ê ú + ë û

*B*

é ù

4

2

*kdx kdx n*

Simplifying and combining the integrals gives the quantization condition for region 2:

+ =- - + +

 p

> 1 2

> > 1 2

pf*kdx m*æ ö

( )

*C B*

*R R*

ê ú + ë û

*C*

é ù

4

ê ú = ê ú

1 2 2 2 2

*R*

pf*kdx n*æ ö

' ' 4 4

*x x*

p

2

*x*

1

A similar treatment for the turning point *x3* yields the condition for region 4:

4

*x*

3

1

cos

*x*

34

*R*

f-

q

*x*

12

q

*x x*

ê ú = ê ú

1 2 2 2 2

*L*

*L*

*R*

 pò ò (97)

f

*L*

*L L L*

 p q

ò (94)

 f

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2 22 4 4 *A BC L L* = + (96)

º = -+ ç ÷ è ø ò (98)

º = -+ ç ÷ è ø ò (99)

(95)

743

(100)

It is interesting to note that only the first velocity in (89) is less than the velocity of light, *c*.

Let us analyze now, one more bound state, the velocity wave function in a given velocity double well potential *γ(x)*.

We begin by deriving a quantization condition for region 2 analogous to (81). Again, applying the boundary condition for region 1 leaves only the exponentially growing solution. Applying the connection formula at *x1* then gives an expression for the velocity wave function in region 2 :

$$\mathcal{L}\_1\left(\mathbf{x}\right) = 2Ak^{-1/2}\sin\left[\int\_{x\_1}^{x} kdx' + \frac{\pi}{4}\right] \tag{90}$$

However, the solution in region 3 must have both growing and decaying solutions present. Considering the region 3 solutions in terms of *x2* and letting BL and CL be the amplitudes of the decaying and growing solutions respectively, the connection formulas give another expression for the velocity wave function in region 2:

$$\mathcal{L}\_2\left(\mathbf{x}\right) = 2B\_L k^{-1/2} \cos \theta + \mathcal{C}\_L k^{-1/2} \sin \theta \tag{91}$$

with

$$
\theta = \int\_{\infty}^{x\_2} k dx' - \frac{\pi}{4} \tag{92}
$$

We equate the two expressions (90), (91) for the velocity function in region 2 and cancel common factors giving

$$2A\sin\left[\int\_{x\_1}^{x} kdx^\cdot + \frac{\pi}{4}\right] = 2B\_L\cos\theta + C\_L\sin\theta\tag{93}$$

Using trigonometric identities to simplify the right hand side, gives

$$2A\sin\left[\int\_{x\_1}^{x} kdx' + \frac{\pi}{4}\right] = \left(4B\_L^2 + C\_L^2\right)^{1/2}\sin\left(\theta + \frac{\pi}{2} - \phi\_L\right)\tag{94}$$

where

We try to estimate a value for the limit velocity *χ*. Let us expand the left side term of (88) in series and keep the first term. If we replace *μ* from (56) and take *a = Λ* (the Compton length),

1 2 1 , 9 , 25 , . . .

 p

(89)

 pp

( ) 2 2 2 2 22 2 2

 p

( )

z

expression for the velocity wave function in region 2:

z

8 2 32 32 32 32 *n nc c c c*

æ ö ì ü ï ï » -= - = ç ÷ í ý è ø ï ï î þ

It is interesting to note that only the first velocity in (89) is less than the velocity of light, *c*.

Let us analyze now, one more bound state, the velocity wave function in a given velocity

We begin by deriving a quantization condition for region 2 analogous to (81). Again, applying the boundary condition for region 1 leaves only the exponentially growing solution. Applying the connection formula at *x1* then gives an expression for the velocity wave function in region

1

ë û

qq

*x Bk Ck* - - = + (91)

= - ò (92)

 q

ò (93)

*x*

However, the solution in region 3 must have both growing and decaying solutions present. Considering the region 3 solutions in terms of *x2* and letting BL and CL be the amplitudes of the decaying and growing solutions respectively, the connection formulas give another

*x*

4

ò (90)

p

1 2 1 2 sin '

*x Ak kdx*

( ) 1 2 1 2 2 2 cos sin *L L*

> 2 ' 4

*x kdx*

2 sin ' 2 cos sin 4

ê ú += + ê ú

*A kdx B C* p

*x*

q

1

*x*

*x*

é ù

ë û

p

We equate the two expressions (90), (91) for the velocity function in region 2 and cancel

*L L*

q

 - é ù = ê ú + ê ú

2

*a* p

h

m

c

742 Advances in Quantum Mechanics

double well potential *γ(x)*.

we get:

2 :

with

common factors giving

$$\phi\_L = \cos^{-1}\left[\frac{2B\_L}{\left(4B\_L^2 + C\_L^2\right)^{1/2}}\right] \tag{95}$$

The magnitude of the sin function must be equal, and the magnitude of the phases must be equal modulo *π* :

$$\mathbf{4}A^2 = \mathbf{4}B\_L^2 + \mathbf{C}\_L^2 \tag{96}$$

$$\int\_{\infty}^{\infty} k d\boldsymbol{x}' + \frac{\pi}{4} = -\int\_{\infty}^{\infty} k d\boldsymbol{x}' - \frac{\pi}{4} + \phi\_L + n\pi \tag{97}$$

Simplifying and combining the integrals gives the quantization condition for region 2:

$$
\theta\_{12} = \bigwedge\_{x\_1}^{x\_2} k dx = \pi \left( n - \frac{1}{2} \right) + \phi\_L \tag{98}
$$

with *n =* 1, 2,....

A similar treatment for the turning point *x3* yields the condition for region 4:

$$
\theta\_{34} \equiv \int\_{x\_3}^{x\_4} k dx = \pi \left( m - \frac{1}{2} \right) + \phi\_{\mathbb{R}} \tag{99}
$$

with *m =* 1, 2,... and *ϕR* given by:

$$\phi\_{\rm R} = \cos^{-1}\left[\frac{2C\_{\rm R}}{\left(4C\_{\rm R}^2 + B\_{\rm R}^2\right)^{1/2}}\right] \tag{100}$$

where BR and CR are the amplitudes of the decaying and growing region 3 solutions in terms of *x3*.

We now have the quantization conditions (98, 99) for regions 2 and 4, but they contain the free parameters *ϕL* and *ϕR*. To eliminate these free parameters, we consider the WKBJ solution in region 3. The coefficients BL, CL, BR, CR define two expressions for solution, which must be equal:

$$\mathcal{L}\_3 = \mathcal{B}\_L \rho^{-1/2} \exp\left[ - \int\_{\infty}^{\infty} \rho d\mathbf{x'} \right] + \mathcal{C}\_L \rho^{-1/2} \exp\left[ \int\_{\infty}^{\infty} \rho d\mathbf{x'} \right] \tag{101}$$

The constraints (98, 99, 105) may be combined to give a single quantization condition for the allowed WKBJ velocity limits *χ* for a double-well velocity potential *γ(x)*. Applying trigono‐

> tan tan exp 2 22 4 *L R*

*C B B C*

*L R*

è øè ø

1

 q

 æ öæ ö = ç ÷ç ÷ = -

Equation (107) may be combined with (98) and (99) to give the WKBJ quantization condition

( ) 12 34 23 1 exp 2

Equation (108) is a nonlinear constraint approximately determining the allowed velocity levels *χ* of a double-well velocity potential *γ(x)* (see Fig. 4) and can be written (taking *ϕR = ϕL = π/*4 in (98) and (99), i.e. the velocity quarter-wave shift in the connection formulas, which is known

1 1 2 1 1 2 () ln 4 ln 2 m,n 1, 2, 3, ... 2 2

 fp

*L R*

ë û é ù - = × - + × - + =- = í ý ê úê ú ç÷ ç ÷ ï ï î þ ë ûë û èø è ø ò h (109)

( ) 23

q

= - (108)

Quantum Effects Through a Fractal Theory of Motion

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1 2

 f (107)

745

metric identities to (95) and (100), and plugging into (105) gives

*L R*

 f

for a double-well potential in terms of the phase integrals *θ12* and *θ34* :

4 *ctg ctg* qq

to optimize the tunneling effect between two oscillating waves [17] ) as :

 *x dx ctg n ctg m* p

f

confirming the results given in [16].

( ) 3

 c

**Figure 4.** Tunneling potential barrier.

In terms of the momentum *Π* we have :

mg

2

*x*

*x*

$$\mathcal{L}\_3 = B\_R \rho^{-1/2} \exp\left[\int\_x^{x\_3} \rho dx'\right] + \mathcal{C}\_R \rho^{-1/2} \exp\left[-\int\_x^{x\_3} \rho dx'\right] \tag{102}$$

Equations (101) and (102) each contain a term that grows exponentially with *x* and a term that decays exponentially with *x*. Equating the growing terms from each equation and the decaying term from each equation gives two constraints:

$$B\_L \exp\left[-\int\_{x\_2}^x \rho d\mathbf{x}^\prime\right] = B\_R \exp\left[\int\_x^{x\_3} \rho d\mathbf{x}^\prime\right] \tag{103}$$

$$\mathbf{C}\_{L}\exp\left[\int\_{x\_{2}}^{x}\rho d\mathbf{x}^{\prime}\right] = \mathbf{C}\_{R}\exp\left[-\int\_{x}^{x\_{3}}\rho d\mathbf{x}^{\prime}\right] \tag{104}$$

Combining the integrals in these constraints gives

$$\frac{B\_L}{B\_R} = \frac{C\_R}{C\_L} = \exp\left(\theta\_{23}\right) \tag{105}$$

with

$$
\theta\_{23} \equiv \int\_{x\_2}^{x\_3} \rho dx'\tag{106}
$$

The constraints (98, 99, 105) may be combined to give a single quantization condition for the allowed WKBJ velocity limits *χ* for a double-well velocity potential *γ(x)*. Applying trigono‐ metric identities to (95) and (100), and plugging into (105) gives

$$
\tan \phi\_L \tan \phi\_R = \left(\frac{\mathcal{C}\_L}{2\mathcal{B}\_L}\right) \left(\frac{\mathcal{B}\_R}{2\mathcal{C}\_R}\right) = \frac{1}{4} \exp\left(-2\theta\_{23}\right) \tag{107}
$$

Equation (107) may be combined with (98) and (99) to give the WKBJ quantization condition for a double-well potential in terms of the phase integrals *θ12* and *θ34* :

$$
\varepsilon t \text{g} \theta\_{12} c \text{tg} \theta\_{34} = \frac{1}{4} \text{exp} \left( -2 \theta\_{23} \right) \tag{108}
$$

confirming the results given in [16].

where BR and CR are the amplitudes of the decaying and growing region 3 solutions in terms

We now have the quantization conditions (98, 99) for regions 2 and 4, but they contain the free parameters *ϕL* and *ϕR*. To eliminate these free parameters, we consider the WKBJ solution in region 3. The coefficients BL, CL, BR, CR define two expressions for solution, which must be

2 2

ë û ëû

 r  r

 r

ò ò (102)

ò ò (103)

ò ò (104)

(105)

º *dx* ò (106)

ò ò (101)

*x x*

3 3

*x x*

Equations (101) and (102) each contain a term that grows exponentially with *x* and a term that decays exponentially with *x*. Equating the growing terms from each equation and the decaying

> exp ' exp ' *x x*

*x x B dx B dx* r

exp ' exp ' *x x*

é ù é ù ê ú = -ê ú ê ú ê ú ë û ë û

*x x C dx C dx*

( ) 23 exp *L R*

3

 r

2 23 ' *x*

*x*

q

é ù é ù ê ú - = ê ú ê ú ê ú ë û ë û

ëû ë û

3

 r

3

 r

 r

*x x*

*x x*

1 2 1 2 3 exp ' exp '

1 2 1 2 3 exp ' exp '

 *B dx C dx* - - é ù éù = -+ ê ú êú

 *B dx C dx* - - éù é ù = +- êú ê ú

 r

*L L*

*R R*

2

2

*L R*

*R L B C B C* = =

q

r

*L R*

 r

zr

zr

term from each equation gives two constraints:

Combining the integrals in these constraints gives

of *x3*.

744 Advances in Quantum Mechanics

equal:

with

Equation (108) is a nonlinear constraint approximately determining the allowed velocity levels *χ* of a double-well velocity potential *γ(x)* (see Fig. 4) and can be written (taking *ϕR = ϕL = π/*4 in (98) and (99), i.e. the velocity quarter-wave shift in the connection formulas, which is known to optimize the tunneling effect between two oscillating waves [17] ) as :

$$\frac{1}{\hbar} \sum\_{\mathbf{r}\_2}^{\mathbf{r}\_2} \left[ \left[ 2\,\mu \left( \gamma(\mathbf{x}) - \mathbf{y} \right) \right]^{\mathbf{J}/2} \,\mathrm{d}\mathbf{x} = \ln \left\{ 4 \cdot \mathrm{c} \mathrm{tg} \left[ \pi \left( n - \frac{1}{2} \right) + \phi\_{\mathbf{L}} \right] \cdot \mathrm{c} \mathrm{tg} \left[ \pi \left( m - \frac{1}{2} \right) + \phi\_{\mathbf{R}} \right] \right\}^{-\mathbf{J}/2} = -\ln 2 \qquad \mathrm{m}, \mathrm{n} = 1, 2, 3, \dots \tag{109}$$

**Figure 4.** Tunneling potential barrier.

In terms of the momentum *Π* we have :

$$\int\_{x\_2}^{x\_3} \Pi d\mathbf{x} = -\hbar \ln 2 \text{ or } \oint \Pi d\mathbf{x} = -2\hbar \ln 2 \tag{110}$$

degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of

Inspired by such studies, a considerable amount of work has been made on the thermody‐ namics of information processing, which include Maxwell's demon problem [20], reversible

Here, considering a double-well velocity potential *γ(x)* and the velocity quarter-wave shift in the connection formulas, a quanta of *ħ* ln2 for the complex velocity*U(x)* of the moving Newtonian 'fluid' occurs. It can be argued that it can be put into a one-to-one correspondence

Furthermore, one gets an interesting result when taking *ϕR = ϕL = π/*2, i.e. the velocity halfwave shift in the connection formulas, when singularities occur in (II.66). We try to solve this case by making use of the vortices theory. Benard in 1908 was the first to investigate the appearance of vortices behind a body moving in a fluid [12]. The body he used was a cylinder. He observed that at a high enough fluid velocity (or Reynolds number based on the cylinder diameter), which depends on the viscosity and width of the body, vortices start to shed behind

> ( ) ( ) 1 2 0 2 () ln 2 ln

> > ( ) ( ) 2

2 *U x c ctg x x* 1

*x dx tg x x l*

where we use *ctg(α + π/*2*) = - tg(α)*, take m *=* n, make the notations *x* =*nl*, *x*0 =*l* / 2 and consider

æ ö é ù p

is the velocity of light. When plotting (113) (see Fig. 6) we see that indeed, singularities are ob‐ tained for *x - x0 = Λ/*2 and for *x - x0 = Λ/*4we get for *U(x)* minima of value the velocity of light, *c*.

Usually, at some distance behind a body placed in a fluid, vortices are arranged at a definite distance *l* apart and with a definite separation *h* between the two rows. The senses of the

p

ì ü ï ï é ù é ù ë û - =- - - í ý ê ú ï ï î þ ë û ò h h (112)

0

=+ - ç ÷ ê ú è ø ë û L (113)

and replace *μ* =*m*2

Quantum Effects Through a Fractal Theory of Motion

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747

*c*, where *c*

to the quanta of information Landauer and other authors discussed about [23, 24].

molecules it results in an increase of entropy of at least *kB*ln2.

computation [21], the proposal of the algorithmic entropy [22] and so on.

the cylinder, alternatively from the top and the bottom of the cylinder.

Consequently, we write (109) in the form

mg

 c

again the one-dimensional case, motion along the Ox axis.

where we assume *l = Λ* (the Compton length), *U(x) = [γ(x) - χ]/*2*π2*

rotation in the two rows are opposite (see Fig. 7).

*x*

Solving (112) one gets

where *∮* denotes an integral over one complete cycle of the classical motion, this time

$$\Pi = \left[ 2\,\mu\left(\gamma\left(\mathbf{x}\right) - \mathbf{x}\right) \right]^{l/2} = \left[ 2\,\mu L\left(\mathbf{x}\right) \right]^{l/2} = mc \left[ 2\,\frac{L\left(\mathbf{x}\right)}{c} \right]^{l/2} \tag{111}$$

since *γ(x)*>*χ*for the integration limits, i.e. region 3 (see Fig. 5). We get again a quantization of the complex velocity*U(x)*, where the levels are equally spaced at a value of *ħ* ln2.

**Figure 5.** Schematic diagram of a double-well potential with three forbidden regions (1, 3, 5) and two allowed re‐ gions (2, 4).

In 1961, Landauer [18] discussed the limitation of the efficiency of a computer imposed by physical laws. In particular he argued that, according to the second law of thermodynamics, the erasure of one bit of information requires a minimal heat generation *kBT* ln2, where *kB*is Boltzmann's constant and *T* is the temperature at which one erases. Its argument runs as follows. Since erasure is a logical function that does not have a single-valued inverse it must be associated with physical irreversibility and therefore requires heat dissipation. A bit has one degree of freedom and so the heat dissipation should be of order *kBT*. Now, since before erasure a bit can be in any of the two possible states and after erasure it can only be in one state, this implies a change in information entropy of an amount −*kB*ln2.

The one-to-one dynamics of Hamiltonian systems [19] implies that when a bit is erased the information which it contains has to go somewhere. If the information goes into observable degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of molecules it results in an increase of entropy of at least *kB*ln2.

Inspired by such studies, a considerable amount of work has been made on the thermody‐ namics of information processing, which include Maxwell's demon problem [20], reversible computation [21], the proposal of the algorithmic entropy [22] and so on.

Here, considering a double-well velocity potential *γ(x)* and the velocity quarter-wave shift in the connection formulas, a quanta of *ħ* ln2 for the complex velocity*U(x)* of the moving Newtonian 'fluid' occurs. It can be argued that it can be put into a one-to-one correspondence to the quanta of information Landauer and other authors discussed about [23, 24].

Furthermore, one gets an interesting result when taking *ϕR = ϕL = π/*2, i.e. the velocity halfwave shift in the connection formulas, when singularities occur in (II.66). We try to solve this case by making use of the vortices theory. Benard in 1908 was the first to investigate the appearance of vortices behind a body moving in a fluid [12]. The body he used was a cylinder. He observed that at a high enough fluid velocity (or Reynolds number based on the cylinder diameter), which depends on the viscosity and width of the body, vortices start to shed behind the cylinder, alternatively from the top and the bottom of the cylinder.

Consequently, we write (109) in the form

$$\int\_{\infty} \left[ 2\,\mu \left( \gamma(\mathbf{x}) - \mathbf{y} \right) \right]^{1/2} d\mathbf{x} = -\hbar \ln 2 - \hbar \ln \left\{ t g \left[ \frac{\pi}{l} \left( \mathbf{x} - \mathbf{x}\_0 \right) \right] \right\} \tag{112}$$

where we use *ctg(α + π/*2*) = - tg(α)*, take m *=* n, make the notations *x* =*nl*, *x*0 =*l* / 2 and consider again the one-dimensional case, motion along the Ox axis.

Solving (112) one gets

3

*x*

746 Advances in Quantum Mechanics

ln 2 or 2 ln 2

( ) ( ) ( ) ( ) 1 2 1 2 1 2 2 22

since *γ(x)*>*χ*for the integration limits, i.e. region 3 (see Fig. 5). We get again a quantization of

**Figure 5.** Schematic diagram of a double-well potential with three forbidden regions (1, 3, 5) and two allowed re‐

In 1961, Landauer [18] discussed the limitation of the efficiency of a computer imposed by physical laws. In particular he argued that, according to the second law of thermodynamics, the erasure of one bit of information requires a minimal heat generation *kBT* ln2, where *kB*is Boltzmann's constant and *T* is the temperature at which one erases. Its argument runs as follows. Since erasure is a logical function that does not have a single-valued inverse it must be associated with physical irreversibility and therefore requires heat dissipation. A bit has one degree of freedom and so the heat dissipation should be of order *kBT*. Now, since before erasure a bit can be in any of the two possible states and after erasure it can only be in one

The one-to-one dynamics of Hamiltonian systems [19] implies that when a bit is erased the information which it contains has to go somewhere. If the information goes into observable

state, this implies a change in information entropy of an amount −*kB*ln2.

where *∮* denotes an integral over one complete cycle of the classical motion, this time

*x U x mc*

 m

P= - = é ù é ù = ê ú ë û ë û ê ú ë û

the complex velocity*U(x)*, where the levels are equally spaced at a value of *ħ* ln2.

P =- P =- *dx dx* ò ò h h Ñ (110)

*U x*

é ù

*c*

(111)

2

*x*

mg

gions (2, 4).

 c

$$\mathcal{U}I(\mathbf{x}) = c \left( 1 + ctg^2 \left[ \frac{2\pi}{\Lambda} (\mathbf{x} - \mathbf{x}\_0) \right] \right) \tag{113}$$

where we assume *l = Λ* (the Compton length), *U(x) = [γ(x) - χ]/*2*π2* and replace *μ* =*m*2 *c*, where *c* is the velocity of light. When plotting (113) (see Fig. 6) we see that indeed, singularities are ob‐ tained for *x - x0 = Λ/*2 and for *x - x0 = Λ/*4we get for *U(x)* minima of value the velocity of light, *c*.

Usually, at some distance behind a body placed in a fluid, vortices are arranged at a definite distance *l* apart and with a definite separation *h* between the two rows. The senses of the rotation in the two rows are opposite (see Fig. 7).

forbidden region. This uniquely quantum mechanical behavior allows a velocity wave, initially localized in one potential well, to penetrate through the barrier, into the other well (as we will

We already know at the points where *χ* −*γ*(*x*)=0, special treatment is required because *k* is singular. The way of handling the solution near the turning point is a little bit more technical, but the basic idea is that we have a solution to the left and to the right of the turning point, and one needs a formula that interpolates between them. In other words, in the vicinity of the turning point one approximates 2*μ*(*χ* −*γ*(*x*)) / ℏ2 by a straight line over a small interval and solves TISE (time independent Schrodinger equation) exactly. This leads to the following

( ) ( ) 2 1

( ) ( ) 1 1 sin 4

p

é ù ò ê ú - Û-

*k x dx e*

4

*k x dx e*

p

é ù -ò ê ú - Û

*b x dx*

*b x dx*

*x b*

> *x b*

r

r

r

r

ê ú ë û ò (114)

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ê ú ë û ò (115)

see in what follows).

**Figure 8.** Single row of rectilinear vortices.

connection formulas:

Barrier to the right ( x = b turning point )

Barrier to the left ( x = a turning point )

**3.3. Velocity potential γ (x) and the quantum barrier**

cos

*k*

*k*

*x*

*x*

**Figure 6.** The complex velocity *U(x)* singularities' distribution along the Ox axis.

**Figure 7.** Von Karman vortex streets.

In 1912 von Karman expounded a theory of such vortex streets and the drag which a cylinder would experience due to their formation [12]. Since we considered here the one-dimensional case, we get the solution of a single row of rectilinear vortices, which has already been referred to as characterizing a surface of discontinuity (see Fig. 8).

A typical bound state in a double-well velocity potential has two classically allowed regions, where the velocity potential *γ(x)*is less than the limit velocity *χ*. These regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity. As we can see, quantum mechanics predicts that a velocity wave *ζ(x)* travelling in such a potential is most likely to be found in the allowed regions. However, unlike classical mechanics, quantum mechanics predicts that this velocity wave can also be found in the

**Figure 8.** Single row of rectilinear vortices.

forbidden region. This uniquely quantum mechanical behavior allows a velocity wave, initially localized in one potential well, to penetrate through the barrier, into the other well (as we will see in what follows).

#### **3.3. Velocity potential γ (x) and the quantum barrier**

We already know at the points where *χ* −*γ*(*x*)=0, special treatment is required because *k* is singular. The way of handling the solution near the turning point is a little bit more technical, but the basic idea is that we have a solution to the left and to the right of the turning point, and one needs a formula that interpolates between them. In other words, in the vicinity of the

turning point one approximates 2*μ*(*χ* −*γ*(*x*)) / ℏ2 by a straight line over a small interval and solves TISE (time independent Schrodinger equation) exactly. This leads to the following connection formulas:

Barrier to the right ( x = b turning point )

**Figure 7.** Von Karman vortex streets.

748 Advances in Quantum Mechanics

to as characterizing a surface of discontinuity (see Fig. 8).

**Figure 6.** The complex velocity *U(x)* singularities' distribution along the Ox axis.

In 1912 von Karman expounded a theory of such vortex streets and the drag which a cylinder would experience due to their formation [12]. Since we considered here the one-dimensional case, we get the solution of a single row of rectilinear vortices, which has already been referred

2

4

6

8

10

A typical bound state in a double-well velocity potential has two classically allowed regions, where the velocity potential *γ(x)*is less than the limit velocity *χ*. These regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity. As we can see, quantum mechanics predicts that a velocity wave *ζ(x)* travelling in such a potential is most likely to be found in the allowed regions. However, unlike classical mechanics, quantum mechanics predicts that this velocity wave can also be found in the

$$\frac{2}{\sqrt{k}}\cos\left[\stackrel{b}{\int}k(\mathbf{x})d\mathbf{x}-\frac{\pi}{4}\right]\Leftrightarrow\frac{1}{\sqrt{\rho}}e^{-\stackrel{\stackrel{\text{\*}}{\int}\rho(\mathbf{x})d\mathbf{x}}}\tag{114}$$

$$\frac{1}{\sqrt{k}}\sin\left[\bigwedge\_{\mathbf{x}}^{b}\mathbf{k}\left(\mathbf{x}\right)d\mathbf{x}-\frac{\pi}{4}\right]\Leftrightarrow -\frac{1}{\sqrt{\rho}}e^{\frac{\pi}{\rho}\int \rho\left(\mathbf{x}\right)d\mathbf{x}}\tag{115}$$

Barrier to the left ( x = a turning point )

$$\frac{2}{\sqrt{k}}\cos\left[\stackrel{\circ}{\int}k\,(\mathbf{x})d\mathbf{x}-\frac{\pi}{4}\right]\Leftrightarrow\frac{1}{\sqrt{\rho}}e^{-\int\rho\mathbf{(x})dx}\tag{116}$$

Again, using the connection formulas for the case barrier to the right (using (116) and (117) on

*a a*

 a

ëû ëû

 aa

4

p

4

*ref(x)* we are now in position to calculate the

( ) ( )

ò ò h h (125)

 c

mg

p

*a x*

*i kdx i*

*a x*

*i kdx i*

2

a

 p

Quantum Effects Through a Fractal Theory of Motion

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(122)

751

(123)

(124)

(126)

*x x*

ò ò

*ee e e e e*

1 4

 a

 a

2

a

*e*

æ ö ç ÷ + è ø

4

a

a

*inc(x)* and *ζ1*

1 1 2 1

To summarize, for a barrier with large attenuation *e-2α*→0, the tunneling probability equals

2 1 2 2

*e T e dx x dx*

r

æ öæ ö = »= - = - - ç ÷ç é ù ÷ ë û ÷ æ ö è øè ø ç ÷ +

1

æ ö ç ÷ - è ø = @ + =

 a

 a

*e e*

*e e*

*R T R*

a

a

1 4

æ ö ç ÷ + è ø

2 2 exp exp 2

*b b*

*a a*

2

2

4 , 1

1 4

é ù æ öæ ö =- + + - ê ú ç ÷ç ÷ ë û è øè ø

( ) ( )

p

*iu iu*

aa

24 4

éù éù = êú êú - - - =

 a

1 1 2 sin cos

*e ui e u*

*x e kdx i e kdx k k*

1 1 4 4

*i x e ee k*

ì ò ï æ ö ï =- + ç ÷ ï è ø í

> *i x e ee k*

> > ( ) ( )

= =

3

z

z

*inc x e T x*

2

ï ò æ ö ï =- - ç ÷ ïî è ø

= -=

1 1 2 sin cos

*i*

a

*k*

( )

1

z

*inc*

1

z

Having obtained the expression for *ζ1*

2 2

a

a

( ) ( )

*x*

*x*

1

z

z

*ref inc*

1

2

transmission coefficient using:

1 1 4

The reflection coefficient is:

*e*

è ø

a

*ref*

( )

*k k*

(121)), we get for region I:

Hence

( )

2

a

1

z

$$\frac{1}{\sqrt{k}}\sin\left[\int\_{a}^{x}k(x)dx - \frac{\pi}{4}\right] \Leftrightarrow -\frac{1}{\sqrt{\rho}}e^{\int \rho(x)dx} \tag{117}$$

The connection formulas enable us to obtain relationships between the solutions in a region at some distance to the right of the turning point with those in a region at some distance to the left [25-27].

One of the most important problems to which connection formulas apply is that of the penetration of a potential barrier. The barrier is shown in Fig. 4 and the limit velocity *χ* is such that the turning points are at *x = a* and *x = b*.

Suppose that the motion is incident from the left. Some waves will be reflected and some transmitted, so that in region III we will have:

$$\mathcal{L}\_3\left(\mathbf{x}\right) = \frac{1}{\sqrt{k}} e^{\frac{i\int\_0^\pi k d\mathbf{x} - i\frac{\pi}{4}}{k}} = \frac{1}{\sqrt{k}} \cos\left[\frac{\pi}{b}k d\mathbf{x} - \frac{\pi}{4}\right] + \frac{i}{\sqrt{k}} \sin\left[\frac{\pi}{b}k d\mathbf{x} - \frac{\pi}{4}\right] \tag{118}$$

The phase factor is included for convenience of applying the connection formulas.

In region II (using (114) and (115) on (118)) we have:

$$\mathcal{L}\_2\left(x\right) = \frac{1}{2} \frac{1}{\sqrt{\rho}} e^{-\int \rho dx} \stackrel{\stackrel{b}{\longrightarrow}}{\quad} -i \frac{1}{\sqrt{\rho}} e^x \tag{119}$$

Now using

$$\int\_{a}^{b} \rho dx = \int\_{x}^{a} \rho dx + \int\_{a}^{b} \rho dx = -\int\_{a}^{x} \rho dx + a \tag{120}$$

we can write

$$\mathcal{L}\_2\left(\mathbf{x}\right) = \frac{1}{2} \frac{1}{\sqrt{\rho}} e^{\frac{\stackrel{\stackrel{\text{\*}}{}} \rho \text{dx}}{}} e^{-\alpha} - i \frac{1}{\sqrt{\rho}} e^{-\frac{\stackrel{\stackrel{\text{\*}}{}} \rho \text{dx}}{}} e^{\alpha} \tag{121}$$

Again, using the connection formulas for the case barrier to the right (using (116) and (117) on (121)), we get for region I:

$$\begin{split} \mathcal{L}\_1(\mathbf{x}) &= \frac{1}{2} e^{-\alpha} \frac{1}{\sqrt{k}} \sin \left[ \int\_x^a k dx - \frac{\pi}{4} \right] - i \frac{2}{\sqrt{k}} e^{\alpha} \cos \left[ \int\_x^b k dx - \frac{\pi}{4} \right] = \\ &= \frac{1}{2} \frac{1}{\sqrt{k}} e^{-\alpha} \sin(u) - i \frac{2}{\sqrt{k}} e^{\alpha} \cos(u) = \\ &= - \frac{i}{\sqrt{k}} \left[ e^{iu} \left( e^{\alpha} + \frac{1}{4} e^{-\alpha} \right) + e^{-iu} \left( e^{\alpha} - \frac{1}{4} e^{-\alpha} \right) \right] \end{split} \tag{122}$$

Hence

( ) ( ) 2 1

( ) ( ) 1 1 sin 4

p

é ù ò ê ú - Û-

*k x dx e*

ê ú ë û

ê ú ë û

4

*k x dx e*

p

é ù -ò ê ú - Û

*x x dx*

*x x dx*

The connection formulas enable us to obtain relationships between the solutions in a region at some distance to the right of the turning point with those in a region at some distance to the

One of the most important problems to which connection formulas apply is that of the penetration of a potential barrier. The barrier is shown in Fig. 4 and the limit velocity *χ* is such

Suppose that the motion is incident from the left. Some waves will be reflected and some

*i kdx i x x*

*x e kdx kdx kk k*

The phase factor is included for convenience of applying the connection formulas.

11 1

*x e ie* r

> r

1 1 1

*x ee i e e* r

a

*x x a a dx dx*

r

*bab x*

*xxa a*

cos sin

> *b b x x dx dx*

> > r

 ra

 r

p

*b b*

*i*

ëû ëû

 r

4 4

*dx dx dx dx* = + =- + òòò ò (120)

 r

> a

 p (118)

*a x*

> *a x*

r

r

r

r

ò (116)

ò (117)

cos

*k*

*k*

that the turning points are at *x = a* and *x = b*.

transmitted, so that in region III we will have:

( ) 4

*x b*

In region II (using (114) and (115) on (118)) we have:

1 1

( ) 2

z

r

( ) 2

z

2

r

2

 r

p

3

z

left [25-27].

750 Advances in Quantum Mechanics

Now using

we can write

*a*

*a*

$$\begin{cases} \begin{aligned} \zeta\_1^{inc}\left(\mathbf{x}\right) &= -\frac{\dot{i}}{\sqrt{k}} \left(e^{\alpha} + \frac{1}{4}e^{-\alpha}\right) e^{\frac{\dot{i}}{k}\left[kdx - i\frac{\pi}{4}\right]} \\\\ \zeta\_1^{ref}\left(\mathbf{x}\right) &= -\frac{\dot{i}}{\sqrt{k}} \left(e^{\alpha} - \frac{1}{4}e^{-\alpha}\right) e^{-\frac{\dot{i}}{k}\left[kdx + i\frac{\pi}{4}\right]} \end{aligned} \tag{123}$$

Having obtained the expression for *ζ1 inc(x)* and *ζ1 ref(x)* we are now in position to calculate the transmission coefficient using:

$$T = \left| \frac{\zeta\_3(\mathbf{x})}{\zeta\_1^{inc}(\mathbf{x})} \right|^2 = \frac{e^{-2\alpha}}{\left( 1 + \frac{1}{4}e^{-2\alpha} \right)^2} \tag{124}$$

To summarize, for a barrier with large attenuation *e-2α*→0, the tunneling probability equals

$$T = \frac{e^{-2\alpha}}{\left(1 + \frac{1}{4}e^{-2\alpha}\right)^2} \approx e^{-2\alpha} = \exp\left(-\frac{2}{\hbar} \stackrel{b}{\rho} \rho \text{dx}\right) = \exp\left(-\frac{2}{\hbar} \stackrel{b}{\left[\frac{1}{a}\ln\left(\gamma\left(x\right) - \chi\right)\right]} ^{3/2} \text{dx}\right) \tag{125}$$

The reflection coefficient is:

$$R = \left| \frac{\zeta\_1^{ref}\left(\mathbf{x}\right)}{\zeta\_1^{inc}\left(\mathbf{x}\right)} \right|^2 \stackrel{\mathcal{L}}{=} \frac{\left(e^{\alpha} - \frac{1}{4}e^{-\alpha}\right)^2}{\left(e^{\alpha} + \frac{1}{4}e^{-\alpha}\right)^2}, \qquad T + R = 1 \tag{126}$$

and also in the same large attenuation limit, we have:

$$R \approx 1 - e^{-2\alpha} = 1 - \exp\left(-\frac{2}{\hbar} \int\_a \rho dx\right) = 1 - \exp\left(-\frac{2}{\hbar} \prod\_a^b \left[2\,\mu\left(\gamma\left(x\right) - \varkappa\right)\right]^{1/2} dx\right) \tag{127}$$

Since this effect arises from long-range correlations between the dipole moments of the atoms forming the walls of the cavity, that are induced by coupling with the fluctuating electromag‐ netic field, the Casimir energy depends in general on the geometric features of the cavity. For example, we see from (129) that, in the simple case of two parallel slabs, the Casimir energy *EC*is negative and is not proportional to the volume of the cavity, as would be the case for an extensive quantity, but actually depends separately on the area and distance of the slabs. Indeed, the dependence of *EC*on the geometry of the cavity can reach the point where it turns from negative to positive, leading to repulsive forces on the walls. For example [29], in the case of a cavity with the shape of a parallelepiped, the sign of *EC*depends on the ratios among the sides, while in the case of a sphere it has long been thought to be positive. It is difficult to give a simple intuitive explanation of these shape effects, as they hinge on a delicate process of renormalization, in which the finite final value of the Casimir energy is typically expressed as a difference among infinite positive quantities. In fact, there exists a debate, in the current literature, whether some of these results are true or false, being artifacts resulting from an

Quantum Effects Through a Fractal Theory of Motion

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753

There are three well-known technical types of derivation of the Casimir force for different geometries including the simplest geometry of two parallel, uncharged, perfectly conducting plates firstly explored by Casimir. One modern method is the quantum field theoretical approach based on the appropriate Green's function of the geometry of problem [34]. The other technical type is the dimensional regularization method that involves the mathematical complications of the Riemann zeta function and the analytical continuation [34]. The last (the most elementary/the simplest) method is based on modes summation by using the Euler-

The problem of finding the Casimir force, not only for the simplest geometry of two plates or rectangular prism, that we want to study here, but also for other more complicated geometries, indispensably/automatically involves some infinities/irregularities; thus, one should regula‐ rize the calculation for arriving at the desired finite physical result(s). In the Green' function method, one uses the subtraction of two terms (two Green's functions) to do the required regularization. In the dimensional regularization method, although there isn't an explicit subtraction for the regularization of the problem, as is clear from its name, the calculation is regularized dimensionally by going to a complex plane with a mathematically complicated/ ambiguous approach. In the simplest method in which the Euler-Maclurian formula is used, the regularization is performed by the subtraction of the zero-point energy of the free space (no plates) from the energy expression under consideration/calculation (e.g. summation of the

Navier-Stokes equations in scale relativity theory predict that the (vector) velocity field *V* and/ or the (scalar) density field *ρ*, on small distances (the same magnitude as the Compton length) behave like a wave function and are transported by the motion of the Newtonian fluid with

Furthermore, when considering vacuum from the Casimir cavity, a non-differentiable, Newtonian, *2D* non-coherent quantum fluid whose entities (cvasi-particles) assimilated to vortex-type objects, initially non-coherent, become coherent (the coherence of the quantum

interior and exterior zero-point energies of the two parallel plates).

oversimplification in the treatment of the walls [33].

Maclurian integral formula [35-37].

velocity *U*.

One can see from (125) and (127) thatthe velocity wave *ζ(x)* on small distances, with the same order of magnitude as *Λ*, may be influenced by *U(x)*,i.e. it can be transmitted, attenuated or reflected at this scale length. In other words, we get from the calculus, that the velocity field *V* is indeed transported by the motion of the 'Newtonian fluid' particles with the velocity *U(x)* (the imaginary part of the complex velocity [13]).

#### **4. Casimir type effect in scale relativity theory**

In recent years, new and exciting advances in experimental techniques [28] prompted a great revival of interest in the Casimir effect, over fifty years after its theoretical discovery (for a recent review on both theoretical and experimental aspects of the Casimir effect, see Refs. [29-31]). As is well known, this phenomenon is a manifestation of the zero-point fluctuations of the electromagnetic field: it is a purely quantum effect and it constitutes one of the rare instances of quantum phenomena on a macroscopic scale.

In his famous paper, Casimir evaluated the force between two parallel, electrically neutral, perfectly reflecting plane mirrors, placed a distance *L* apart, and found it to be attractive and of a magnitude equal to:

$$F\_{\mathbb{C}} = \frac{\hbar c \pi^2 A}{240L^4} \tag{128}$$

Here, *A* is the area of the mirrors, which is supposed to be much larger than *L*2 , so that edge effects become negligible. The associated energy *EC*

$$E\_{\mathbb{C}} = -\frac{\hbar c \pi^2 A}{720L^3} \tag{129}$$

can be interpreted as representing the shift in the zero-point energy of the electromagnetic field, between the mirrors, when they are adiabatically moved towards each other starting from an infinite distance. The Casimir force is indeed the dominant interaction between neutral bodies at the micrometer or submicrometer scales, and by modern experimental techniques it has now been measured with an accuracy of a few percent (see [28] and references therein).

Since this effect arises from long-range correlations between the dipole moments of the atoms forming the walls of the cavity, that are induced by coupling with the fluctuating electromag‐ netic field, the Casimir energy depends in general on the geometric features of the cavity. For example, we see from (129) that, in the simple case of two parallel slabs, the Casimir energy *EC*is negative and is not proportional to the volume of the cavity, as would be the case for an extensive quantity, but actually depends separately on the area and distance of the slabs. Indeed, the dependence of *EC*on the geometry of the cavity can reach the point where it turns from negative to positive, leading to repulsive forces on the walls. For example [29], in the case of a cavity with the shape of a parallelepiped, the sign of *EC*depends on the ratios among the sides, while in the case of a sphere it has long been thought to be positive. It is difficult to give a simple intuitive explanation of these shape effects, as they hinge on a delicate process of renormalization, in which the finite final value of the Casimir energy is typically expressed as a difference among infinite positive quantities. In fact, there exists a debate, in the current literature, whether some of these results are true or false, being artifacts resulting from an oversimplification in the treatment of the walls [33].

and also in the same large attenuation limit, we have:

(the imaginary part of the complex velocity [13]).

**4. Casimir type effect in scale relativity theory**

instances of quantum phenomena on a macroscopic scale.

effects become negligible. The associated energy *EC*

of a magnitude equal to:

a

752 Advances in Quantum Mechanics

( ) ( ) 1 2 2 2 2 1 1 exp 1 exp 2 *b b*

r

*a a R e dx x dx*

One can see from (125) and (127) thatthe velocity wave *ζ(x)* on small distances, with the same order of magnitude as *Λ*, may be influenced by *U(x)*,i.e. it can be transmitted, attenuated or reflected at this scale length. In other words, we get from the calculus, that the velocity field *V* is indeed transported by the motion of the 'Newtonian fluid' particles with the velocity *U(x)*

In recent years, new and exciting advances in experimental techniques [28] prompted a great revival of interest in the Casimir effect, over fifty years after its theoretical discovery (for a recent review on both theoretical and experimental aspects of the Casimir effect, see Refs. [29-31]). As is well known, this phenomenon is a manifestation of the zero-point fluctuations of the electromagnetic field: it is a purely quantum effect and it constitutes one of the rare

In his famous paper, Casimir evaluated the force between two parallel, electrically neutral, perfectly reflecting plane mirrors, placed a distance *L* apart, and found it to be attractive and

2

2

*L* p

can be interpreted as representing the shift in the zero-point energy of the electromagnetic field, between the mirrors, when they are adiabatically moved towards each other starting from an infinite distance. The Casimir force is indeed the dominant interaction between neutral bodies at the micrometer or submicrometer scales, and by modern experimental techniques it has now been measured with an accuracy of a few percent (see [28] and references therein).

4 240 *C c A F L* p

3 720 *C c A E*

Here, *A* is the area of the mirrors, which is supposed to be much larger than *L*2

mg

 c

= h (128)

= - h (129)

, so that edge

There are three well-known technical types of derivation of the Casimir force for different geometries including the simplest geometry of two parallel, uncharged, perfectly conducting plates firstly explored by Casimir. One modern method is the quantum field theoretical approach based on the appropriate Green's function of the geometry of problem [34]. The other technical type is the dimensional regularization method that involves the mathematical complications of the Riemann zeta function and the analytical continuation [34]. The last (the most elementary/the simplest) method is based on modes summation by using the Euler-Maclurian integral formula [35-37].

The problem of finding the Casimir force, not only for the simplest geometry of two plates or rectangular prism, that we want to study here, but also for other more complicated geometries, indispensably/automatically involves some infinities/irregularities; thus, one should regula‐ rize the calculation for arriving at the desired finite physical result(s). In the Green' function method, one uses the subtraction of two terms (two Green's functions) to do the required regularization. In the dimensional regularization method, although there isn't an explicit subtraction for the regularization of the problem, as is clear from its name, the calculation is regularized dimensionally by going to a complex plane with a mathematically complicated/ ambiguous approach. In the simplest method in which the Euler-Maclurian formula is used, the regularization is performed by the subtraction of the zero-point energy of the free space (no plates) from the energy expression under consideration/calculation (e.g. summation of the interior and exterior zero-point energies of the two parallel plates).

Navier-Stokes equations in scale relativity theory predict that the (vector) velocity field *V* and/ or the (scalar) density field *ρ*, on small distances (the same magnitude as the Compton length) behave like a wave function and are transported by the motion of the Newtonian fluid with velocity *U*.

Furthermore, when considering vacuum from the Casimir cavity, a non-differentiable, Newtonian, *2D* non-coherent quantum fluid whose entities (cvasi-particles) assimilated to vortex-type objects, initially non-coherent, become coherent (the coherence of the quantum fluid reduces to its ordering in vortex streets) due to the constraints induced by the presence of slabs. Casimir type forces are derived which are in good agreement with other theoretical results and experimental data, for both cases: two metallic slabs, parallel to each other, placed at a distance *d* apart, that constitute the plates of the cavity and a rectangle of sides *d1*, *d*.

In other words, non-differentiability and coherence of the quantum fluid due to constraints generate pressure along the *Ox* and *Oy* axis.

For viscous compressible fluids, Navier-Stokes equations

$$
\rho \frac{D\mathbf{v}}{Dt} = \rho X - \nabla p + \mu \nabla^2 \mathbf{v} + \frac{\mu}{3} \nabla \left( \nabla \cdot \mathbf{v} \right) \tag{130}
$$

together with the equation of continuity

$$\frac{D\rho}{Dt} + \rho \nabla \cdot \mathbf{v} = 0,\tag{131}$$

2

 r

r r

×Ñ + Ñ× = ×Ñ + Ñ = *V V*

r

0 a 0 b

where *V* represents the standard classical velocity, which does not depend on resolution, while the imaginary part, *U*, is a new quantity coming from resolution dependant fractal, *υ = μ/ρ* the kinematic viscosity and *D* =*ħ/2m* defines the amplitude of the fractal fluctuations.

The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ballspring combination to be quantized, that is, that the strength of the field to be quantized at each point in space. Canonically, the field at each point in space is a simple harmonic oscilla‐ tor, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vac‐ uum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of vacuum. The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, all of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy.

Let us consider here, vacuum, as a non-differentiable, Newtonian, *2D* non-coherent quan‐ tum fluid whose entities (cvasi-particles) assimilate to vortex-type objects [38] (see Fig.9)

, a

, b

1 sin , d

 f

> f

f

f

1 sin , e

1 f

( )

*Kk d*

*k d K*

0

ò

p

= -

*K u z a z x iy K K a b*

= + ¢ =

=

0 2 2

ò

+ = ¢

p

*k k*

2 1 2 2 2

( )

1 2 2 2 2

and are described by the wave function *Ψ* [39, 40]

with

*U D*(133)

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Y = *cn u k* ( ) ; (134)

c

(135)

where *ρ* is the density, *ν* the velocity of the fluid, *X* the body force, *p* the pressure, *μ* the shear viscosity and *D* / *Dt* ≡*d* / *dt* +*ν* ⋅∇ the Eulerian derivative, apply to *Newtonian* (or near) *fluids*, that is, to fluids in which the stress is linearly related to the rate of strain (as will be assumed further in this section) [12].

Let us see first, what happens with the set of equations (130) and (131), if one considers that the space-time, where particles move, changes from classical to non-differentiable.

We already know, according to Nottale [11], that a transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable *dt* (the non-differentiability of space).

Consequently, replacing *d/dt* with the fractal operator (42) and solving for both real and imaginary parts, (130) and (131) become, in a stationary isotropic case, taking the body force *X* = 0 (constant gravitational field) and ∇*U* = 0 (assuming a constant density of states for the "fluid particles" moving with the velocity *U* – see further in this section):

$$\begin{aligned} V \cdot \nabla V &= -\frac{\nabla p}{\rho} + \upsilon \,\nabla^2 V & \quad \text{a} \\ U \cdot \nabla V + \mathcal{D} \nabla^2 V &= 0 \qquad \text{b} \end{aligned} \tag{132}$$

and

$$\begin{aligned} \mathbf{V} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{V} &= 0 & \mathbf{a} \\ \mathbf{U} \cdot \nabla \rho + \mathcal{D} \nabla^2 \rho &= 0 & \mathbf{b} \end{aligned} \tag{133}$$

where *V* represents the standard classical velocity, which does not depend on resolution, while the imaginary part, *U*, is a new quantity coming from resolution dependant fractal, *υ = μ/ρ* the kinematic viscosity and *D* =*ħ/2m* defines the amplitude of the fractal fluctuations.

The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ballspring combination to be quantized, that is, that the strength of the field to be quantized at each point in space. Canonically, the field at each point in space is a simple harmonic oscilla‐ tor, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vac‐ uum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of vacuum. The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, all of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy.

Let us consider here, vacuum, as a non-differentiable, Newtonian, *2D* non-coherent quan‐ tum fluid whose entities (cvasi-particles) assimilate to vortex-type objects [38] (see Fig.9) and are described by the wave function *Ψ* [39, 40]

1 f

2 2

+ = ¢

*k k*

$$
\Psi = c n \begin{pmatrix} \underline{n} \; k \end{pmatrix} \tag{134}
$$

with

fluid reduces to its ordering in vortex streets) due to the constraints induced by the presence of slabs. Casimir type forces are derived which are in good agreement with other theoretical results and experimental data, for both cases: two metallic slabs, parallel to each other, placed at a distance *d* apart, that constitute the plates of the cavity and a rectangle of sides *d1*, *d*.

In other words, non-differentiability and coherence of the quantum fluid due to constraints

 m

0, *D*

where *ρ* is the density, *ν* the velocity of the fluid, *X* the body force, *p* the pressure, *μ* the shear viscosity and *D* / *Dt* ≡*d* / *dt* +*ν* ⋅∇ the Eulerian derivative, apply to *Newtonian* (or near) *fluids*, that is, to fluids in which the stress is linearly related to the rate of strain (as will be assumed

Let us see first, what happens with the set of equations (130) and (131), if one considers that

We already know, according to Nottale [11], that a transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable *dt* (the non-differentiability of

Consequently, replacing *d/dt* with the fractal operator (42) and solving for both real and imaginary parts, (130) and (131) become, in a stationary isotropic case, taking the body force *X* = 0 (constant gravitational field) and ∇*U* = 0 (assuming a constant density of states for the

2

a

0 b

the space-time, where particles move, changes from classical to non-differentiable.

"fluid particles" moving with the velocity *U* – see further in this section):

2

r

Ñ ×Ñ = - + Ñ

*VV V*

×Ñ + Ñ =

*UV V D*

*p* u

 + Ñ× = r

( ) 2 3

= -Ñ + Ñ + Ñ Ñ× *v X vv* (130)

*v* (131)

(132)

m

generate pressure along the *Ox* and *Oy* axis.

754 Advances in Quantum Mechanics

together with the equation of continuity

further in this section) [12].

space).

and