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ВІСНИК ЛЬВІВ. УН-ТУ

Серія фізична. 2002. Вип.35. С.200-207

VISNYK LVIVUNIV. Ser.Physic. 2002. № 35. Р.200-207

PACSnumber(s):47.15.Ki, 47.15.Hg, 61.30.Jf, 67.20fk, 02.00.00

 

QUANTUM VORTEX SHEETS

 

Z. Bak

Institute ofphysics, Pedagogical university of Czenstochowa, Al. Armii Krajowej 13/15, 42-200 Czestochowa, Poland e-mail: z. bak(d).wsp. czest.pi

Topological defects are present in many quantum as well as classical systems. We discuss the nature of stationary, half-infinite vortex sheet in a quantum system. With help of the fractional calculus we construct new representation of spherical harmonics with fractional quantum numbers and show, that solutions of the quantum vortex sheet can be given in terms of spherical harmonics with half-integer quantum numbers.

Key words: quantum vortex sheet, fractional calculus, fractional spherical harmonics,

 

 

In Friedel's (1922) classical polarized light observations, in thin layers of nematic crystals, a few types of singularities were observed (see [1] and references thetrein). Each type of singularity represents itself a discontinuity in nematic ordering at the molecular level.

It is well-known fact, that chaotic quantum dynamics of a single particle can be pictured as a turbulent classical fluid [2]. Such description allows better understanding the important dynamical features, such as topological singularities, by its analogy to the better understood classical fluids. However, there are many other physical systems in which topological singularities occur. Under some critical conditions ordered systems exhibit existence of regions, within which their parameters of order (displacement, velocity or polarization vector) show singular behaviour. Such topological defects called vortices arise when the physical system is unstable with respect to various processes of relaxation, leading to a highly nonuniform state with lower total free energy. Generally, the concept of vortices appears in description of solids, fluid formulation of quantum dynamics and cosmology (see [3] and references therein). Topological defects in solids comprise dislocations, disclinations, stacking faults or textures. Typical defects form a vortex lines (e.g. dislocations in solids, vortices in type-II superconductors) or vortex sheets (e.g. textures in liquid crystals). Line defects in 3D are most commonly studied in microscopic scales, while in meso- and macroscales the vortex sheets received much attention. The latter case is the main subject of this paper.

Usually it is assumed, that both line and sheet vortices have no structure. This allows to reduce dimensionality of the problem: vortex line in 3D reduces to a point defect in 2D, while the vortex sheet in 3D reduces to a vortex line in 2D. We will consider more general case of half-infinite vortex sheet /HIVS/ with an internal structure. Except for the quantum vortex lines there have been only several attempts in

©BakZ.,2002

the past to study vortices in 3D systems and all focused on the stationary systems. This is justified in view of the conclusion that topological defects preserve their integrity during time evolution [2]. In our considerations we will adopt this assumption negleeting the effects of vortex dynamics. In the following, within fractional calculus approach we will give mathematical formulation of a stationary, internally structured HIVS, which is nonuniform and infinitely extended in third dimension.

 

Suppose, that the physical system is described by the Schroedinger equation (for simplicity we assume the constants like h, 2m to be equal unity)

(і)

Let us limit our considerations to the search for zero-energy (E=0) HIVS solutions [2]. Moreover since we are interested in the study of HIVS systems, we assume that the potential V(r) =0 everywhere except of the half-plane, which is to be identified as the HIVS. Let us consider solutions of the equation

V2¥ =-К(г)У = -К(г,Є)5(ф)Ч/(Є,ф),


(2)

where 8(<p) is the Dirac delta function of the toroidal angle ф and V(r,6) is a regular function of spherical coordinates, which determines the internal structure of HIVS. From Eq. (1) follows that outside HIVS it reduces to the Laplace equation V2 Ч^гДф) = 0 and V24>+K(r,6)4/=0 within the HIVS region.

In their approach Chang et al. [3] have shown, that existence of quantum vortex sheets is closely linked to the existence of an angular-momentum-like quantum numbers. We will show, that the HIVS can be described with help of spherical harmonics with fractional indices. The advent of the fractional quantum Hall effect showed, that quantum number fractionalization is a robust possibility in condensed-matter physics. Thus, first of all let us discuss the method, which allows us to catch the fractal spherical harmonics as the analytical continuation of the conventional spherical harmonics, that have fractional quantum numbers.

It is well known that Laplace equation (2) along with the regular solutions given by spherical harmonics labeled by integer indices, has multivalued solutions labeled by the fractional quantum numbers. It is commonly believed that such solution have no physical meaning, however, as wc show below they can be used in construction of the solution of the HIVS problem (1). The singular structure of HIVS cannot be described by regular functions so the search for solutions of the problem must rely on the irregular ones [1]. These multivalued solutions when expressed conventionally by hypergeometric function are not useful in practical calculations. Fortunately they can be expressed (or at least some of them) by fractional continuation of the famous Rodrigues formula [3]. Thus, before we will apply this new representation to the description of HIVS system let us remind the new approach to the generation of fractional harmonics [4]. First of all, for the use of further consideration we will give a shorthand definition of derivative of arbitrary order (generally noninteger) a. Recently, the fractional calculus become a powerful tool in theoretical studies of systems which show fractional spectral dimensionality [5-7].

There are a few definitions of operators of noninteger order differentiation (for extensive review see [8], [9] and references therein), which for absolutely continuous

functions give the same results. For example, the Riemann-Louville (Louville 1832, [8]) derivative (left side) of fractional order а є *R of absolutely continuous function fix) defined on [c, d] с *R is defined as [4], [8].

 

 

Г(а-1) Лс J (дс-/)аЛ

с

where  Г(л:) - is the Euler gamma function. Generally, the Da is a fractional

counterpart of derivative of fractional order Da =d^/ o (for a<0 ) or fractional

integral (for а > 0). In the case of integer а > 0 Eq.(3) gives the same result as the conventional differentiation. For the use of our consideration it is enough to limit a to real numbers, however, in general a can be imaginary or even complex.

 

The derivation of the conventional (single-valued) spherical harmonics is obvious, however , since this procedure is very illustrative to the method presented below let us recall its main points. Suppose that a physical system is described by the Laplace equation (2). The well-known classical solutions of this equation are given by [4]

ччгдф) = лг)у;(е,ф), W

where the Уят (9, (p) are the spherical functions labeled by integer quantum numbers m and n. To construct solutions of Schroedinger equation, that exhibit HIVS symmetry let us first study the structure of the conventional solutions of (4). The spherical functions are given in the form

Уи"(Є,ф)=Ри"(Є,ф)Є'яф, (5)

where the p™ (0, (p) are the associate Legendre functions of integer indices m and n while the/(r) = r°. Thus the central problem in a search for a new solutions is connected with construction of the counterpart of associate Legendre functions described by fractional value of the angular momentum. Let us recall first the explicit expression for the Legendre polynomials

 

 

 

where the variable x = cos(9) and the index n in the Eq.(vv) is always integer. Suppose that, we allow it to be fractional (e.g n-*s= 1/2) and write, in somewhat formal way, the expression for a fractional Р\/г{х) = Ps(x) Legendre-reminiscent function:

p,W = w =          Чтт-^ттг^2-1)1'2. (7)

2                 Г(1/2)2|/2 dxm

where the half-integer derivative we assume to be given by the Riemann-Louville formula (3).

Consequently, by analogy with the integer case we can define the associate to P,(x) Legendre-like function P' {x) as

 

 

 

In view of Eqs (4-8) we can define the analogons the spherical function (5) and write the fractional-counterpart of the solution (4) as

4»,(r,Є,ф) = гІ/2У; (6,ф) = r1,2Pss(9)ей* . (9)

The most important fact is that taking the щ given by Eq. (9) we have V2v|/j(9^) = 0 for ф * 0...2пя, i.e. the fractional harmonic (9) fulfils Laplace equation everywhere except of the HIVS half-plane. This is because of the characteristic to fractional harmonics multivalueness of \|/,. The constructed by mathematical tries analogon of associate Legendre polynomial is an eigenfunction of Laplace operator (2) as well as eigenfunction of the "z" component of angular momentum everywhere except of the HIVS region.

The Eqs (7) and (8) suggest that the Pss function can have many analogons. Really for e.g. 1/2 <a< 1 we can easily write down an expression for a new fractional Legendre function in the following form [5]

 

ад = ~(,2-іГ, (io)

dx

and consequently in view of Eq. (8) an expression for associate to (10) Legendre function

РаРМ = са(!-^)р'24^(^-і)а> (ID

dx v

where p < a. It is worth to note that some associate Legendre functions of the type

= ca(i - *V->« j-y - i)» = ca ix2Jf.a)/2. (12)

as well as the Pap for which both quantum numbers are half-integers а = л + 1/2, p = m + l/2,p<acan be obtained from generalized Rodrigues formula (6) with help of conventional differentiation (i.e. they are expressed by derivatives of integral order). With any associate Legendre function Pa p are associated fractional spherical harmonics

Гар(Є,ф) = Ра,р(Є(ф)еір<>. (13) Similarly as the function \ys given by Eq. (9) functions

у0^в,ф) = г%р(Є,фу0*, (14) are the solutions of Laplace equation everywhere outside the HIVS region.

The harmonics with integer indices have a special status since they form a complete set of functions that allows to express any regular function of spherical coordinates as a linear combination of these functions. The set (12-14) of fractional harmonics is overcomplete. For the full description of the problem under consideration it is enough to limit considerations to the functions of half-integer indices. Evidently for the functions given by Eq. (14) we have У3\)/0'3(г,6,ф) = 0 for ф * 0...2ля, i.e. the fractional harmonics (14) fulfill Laplace equation everywhere except of the HIVS half-plane. Due to their multivalueness the fractional spherical harmonics show discontinuity at ф = 0 and ф = 2ил. However, using the half-integer harmonics we can create the function

АІ (Є, ф) = >Y» (Є, ф) -     (Є, ф) = 2P0)(3 (Є, ф) 8іп / 2), (15)

with a = p = 1/2, which fulfills the relation ЛДб, ф = 0) = Aj(Q, ф = 2я) but still is multivalued. In turn, using the /^(б, ф) we can create single valued function

Я.р0(Є,ф) = 2Р0іР(Є,ф)8іп(ф/2), (16)

where we have taken absolute value of віп(ф/2). In view of the relations (15) functions ^•Лб,ф) fulfill Laplace equation outside the HIVS region. For ф = 0...2л (i.e. at the HIVS half-plane) Х,Дб,ф) are continuous but their derivatives d/d^[ka(Q, Ф)] show a finite jump discontinuity. This property of Яар(8, ф) functions allows us to construct the solution of the HIVS problem. In the close vicinity of ф = 0 and ф = the derivative <АУф[|5Іп(ф/2)|] can be written as

</2 j БІп(ф/2) (     fCQSdt/2) ♦-»<>-
d§2         1-со8/21 ф-»0*

This means that this derivative can be written as

^^(ф) + 0(2л), (18) с/ф

where 5(ф) is the Dirac delta function. In a similar way behave the derivatives of the half-integer functions |sin(n+l/2) ф)|. Within the HIVS half-plane the functions |sin(«+l/2) ф| vanish and only the term proportional to the Dirac delta function must be accounted for, while outside the HIVS region the Dirac delta function gives no contribution. This means, that for the half-integer functions given by Eq. (16) we have the identity.

V2г°Я.ра(Є,ф) = 2p(rsin Є)-2Рар(6,ф)5(ф), (20)

where а = n+1/2, P = m + 1/2, p < a).

The full solution of the Schroedinger equation (2) can be constructed as a linear combination of the half-integer functions гаА.ра(Є,ф). To prove that let us recall the fractional version of the Taylor formula [6]

ю=£УгЩь-аГ'+*-ы- (21)


 

liquid crystals [11]. In these systems both finite and half-infinite vortex sheets have been observed. We have considered a general case of HIVS system with nonuniform singular potential V(r,Q). Application of the fractional calculus allows us to uncover hidden symmetry of the system. To construct analytical solution we adopt lineac superposition of spherical harmonics with half-integer quantum numbers. The coefficients of this combination are directly related to the coefficients of the fractional Taylor expansion of the singular potential V(r,Q). The analytical solution of the HIVS problem presented above can be extended onto more complicated structures of half-infinite vortex sheets.

 

Financial support from the State Committee for Scientific Research /KBN/ via University grant is gratefully acknowledged.

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Butzer P. L., Westphal U. In an Access to Fractional Differentiation via Fractional Difference Quotients // Lcct. Notes. Math. 1975. Vol. 457. Springer. Berlin. Butzer P. L., Westphal U. Applications of Fractional Calculus in Physics. Edited by Hilfer R. World Scientific. Singapore. 2000.

11.   Landau L. D., Lifshitz E. M. In Theory of Elasticity. Pergamon. Oxford. 1986.

КВАНТОВА ВИХРОВА ЗАВІСА 3. Бонк

Інститут фізики, Ченстоховський педагогічний університет вул.Армії Крайовей, 13/15, 42-200, Ченстохова, Республіка Польща e-mail: z. bak(a),wsp. czest.pl

Топологічні дефекти існують як у класичних, так і у квантових системах. Ми розглянули природу стаціонарних напівскінчених вихрових завіс у квантовій системі. З використанням диференціювання дробового порядку побудовали нове зображення сферичних гармонік з дробовими квантовими числами та показали, що розв'язок квантової вихрової завіси можна розкласти за сферичними гармоніками з півцілими квантовими числами.

Ключові слова: квантова вихрова завіса, диференціювання дробового порядку, сферичні гармоніки з півцілими квантовими числами.

Стаття надійшла до редколегії 20.06.2002 Прийнята до друку 17.10.2002

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