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Vortex lattices in quantum mechanics
Physica A 303 (2002) 469–480
www.elsevier.com/locate/physa
Vortex lattices in quantum mechanics Tsunehiro Kobayashi Department of General Education for the Hearing Impaired, Tsukuba College of Technology, Ibaraki 305-0005, Japan Received 11 June 2001; received in revised form 27 August 2001
Abstract Vortex lattices are constructed in terms of linear combinations of solutions for Schr/odinger equation with a constant potential. The vortex lattices are mapped on the spaces with twodimensional rotationally symmetric potentials by using conformal mappings and the di1erences of the mapped vortex patterns are examined. The existence of vortex dipole and quadrupole is c 2002 Elsevier Science B.V. All rights reserved. also pointed out. Keywords: Vortex; Hydrodynamics; Quantum mechanics
Vortices play an interesting roles in various aspects of present-day physics such as vortex matters (vortex lattices) in condensed matters [1,2], quantum Hall e1ects [3–7], various vortex patterns of non-neutral plasma [8–11] and Bose–Einstein gases [12–16]. The vortices are well known objects in hydrodynamics and has been investigated in many aspects [17–21]. In quantum mechanics, hydrodynamical approach was vigorously investigated in the early stage of the development of quantum mechanics [22–34]. The fundamental properties of vortices in quantum mechanics were extensively examined by Hirschfelder and others [35 – 40] and the motions of vortex lines were also studied [41,42]. Recently an analysis of vortices has been carried out on the basis of non-linear Schr/odinger equations [43,44]. The vortex dynamics will hold an important position in quantum mechanical problems. In vortex dynamics, however, the construction of various vortex patterns from fundamental dynamics is still not very clear. Recently Kobayashi and Shimbori have proposed a way to investigate vortex patterns from the solutions of Schr/odinger equations [45]. By using the conformal mappings, the article has shown that the solutions with zero energy E-mail address: [email protected] (T. Kobayashi). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 5 0 1 - 5
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eigenvalue in the two-dimensional parabolic potential barriers (2D PPB) expressed by V = − m2 (x2 + y2 )=2 [46] can be extended to all two-dimensional rotationally sym metric potentials of the forms Va () = − a2 ga 2(a−1) with = x2 + y2 except a = 0 and the solutions are inDnitely degenerate as same as those in the 2D PPB [45]. It has also been shown that the inDnite degeneracy can be an origin of a variety of vortex patterns which appear at nodal points of wave functions and this idea can be applied in three dimensions. In the article, a few examples of simple vortex patterns are presented. In this paper, some patterns of vortex lattices that are constructed from simple linear combinations of the inDnitely degenerate states will be studied. Let us start from the arguments in two dimensions [45]. In two dimensions, the eigenvalue problems with the energy eigenvalue E are explicitly written as ˝2 − + Va () (x; y) = E (x; y) ; (1) 2m where = @2 =@x2 + @2 =@y2 ; the rotationally symmetric potentials are generally given by Va () = − a2 ga 2(a−1) ; with = x2 + y2 , a ∈ R (a = 0), and m and ga are, respectively, the mass of the particle and the coupling constant. Note here that the eigenvalues E should generally be taken as complex numbers that are allowed in conjugate spaces of Gel’fand triplets [47]. Note also that Va represents repulsive potentials for (ga ¿ 0; a ¿ 1) and (ga ¡ 0; a ¡ 1) and attractive potentials for (ga ¿ 0; a ¡ 1) and (ga ¡ 0; a ¿ 1). Let us consider the conformal mappings a = z a
with z = x + iy :
(2)
We use the notations ua and va deDned by a = ua + iva that are written as ua = a cos a’;
va = a sin a’ ;
where ’ = arctan(y=x). In the (ua ; va ) plane Eqs. (1) are written down as ˝2 2 2(a−1)=a a a − a − ga (ua ; va ) = E (ua ; va ) ; 2m where a = @2 =@ua2 + @2 =@va2 : We can rewrite the equations as ˝2 − a − ga (ua ; va ) = a−2 Ea2(1−a)=a (ua ; va ) : 2m
(3)
(4)
(5)
It is surprising that the equations become same for all values of a except a = 0 when the energy eigenvalue E have zero value. That is to say, for E = 0 the equations have the same form as that for the free particle with the constant potentials ga as ˝2 − a − ga (ua ; va ) = 0 : (6) 2m
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It should be noticed that in the case of a = 1 where the original potential is a constant g1 , the energy does not need to be zero but can take arbitrary real numbers, because the right-hand side of (5) has no dependence. In the a = 1 case, therefore, we should take g1 + E instead of ga . Here, let us brieJy comment on the conformal mappings a = z a . We see that the transformation maps the part of the (x; y) plane described by 0 6 ¡ ∞; 0 6 ’ ¡ =a on the upper half-plane of the (ua ; va ) plane for a ¿ 1 and the lower half-plane for a ¡ − 1. Note here that the maps on the part of the (ua ; va ) plane with the angle ’a = ’ − can be carried out by using the conformal mappings a () = z a e−i :
(7)
In the maps the variables are given by ua () = ua cos + va sin ;
va () = va cos − ua sin :
(8)
The relations ua (0) = ua and va (0) = va are obvious. Note that the angle represents the freedom of the angle of the incoming stationary wave. It is trivial that the equations for all a have the particular solutions ± ±ika ua 0 (ua ) = Na e
;
± ±ika va 0 (va ) = Na e
(9) for ga ¿ 0
(10)
and ±ka ua ± ; 0 (ua ) = Ma e
(11)
±ka va ± for ga ¡ 0 (12) 0 (va ) = Ma e where ka = 2m|ga |=˝, and Na and Ma are in general complex numbers. General solutions should be written by the linear combinations of (9) and (10) for ga ¿ 0 and those of (11) and (12) for ga ¡ 0. Examples for the 2D PPB with ga ¿ 0 are presented in Ref. [46]. In the following investigations, we shall concentrate our attention on the solutions of (9) and (10) that are expressed in terms of plane waves. Hereafter, we use the notations ua () and va () with − ¡ 6 , which are used in the conformal mappings of (7). We see that
e±ika ua ()
and
e±ika va ()
are the solutions of (6). (For details, see Ref. [45].) We notice here the degeneracy of the solutions. The origin of the inDnite degeneracy can easily be understood in the case of the 2D PPB [46]. It is known that energy eigenvalues of 1D PPB are given by pure imaginary values ∓i(n + 1=2)˝ with n = 0; 1; 2; : : : [48–53]. From this result, we see that the energy eigenvalues of the 2D PPB, which are composed of the sum of the 1D-PPB eigenvalues with the opposite signs such as ∓i(nx − ny )˝ with nx and ny = 0; 1; 2; : : : ; include the zero energy for nx = ny . It is apparent that all the states with the energy eigenvalues ∓i(nx − ny )˝ are inDnitely degenerate [46]. This means that all the rotationally symmetric potentials
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Va () have the same degeneracy for the zero energy states. By putting the wave function f± (ua ; va ) 0± (ua ) into (6) where f± (ua ; va ) are polynomial functions of ua and va , we obtain the equation @
a ± 2ika f± (ua ; va ) = 0 : (13) @ua A few examples of the functions f are given by [46] f0± (ua ; va ) = 1 ; f1± (ua ; va ) = 4ka va ; f2± (ua ; va ) = 4(4ka2 va2 + 1 ± 4ika ua ) :
(14)
We can obtain the general forms of the polynomials in the 2D PPB, which are generally √ written by the multiple of the polynomials of degree n, Hn± ( 2k2 x), such that fn± (u2 ; v2 ) = Hn± ( 2k2 x)Hn∓ ( 2k2 y) ; (15) where x and y in the right-hand side should be considered as the functions of u2 and v2 [46]. Since the form of Eqs. (6) is the same for all a, the solutions can be written by the same polynomial functions that are given in (15) for the PPB. That is to say, we can obtain the polynomials for arbitrary a by replacing u2 and v2 with ua and va in (15). Note that the polynomials Hn± ( ) with = m=˝x are deDned by the solutions for the eigenstates with E = ∓ i(n + 1=2)˝ in 1D PPB of the type V (x) = − m2 x2 =2 and they are written in terms of the Hermite polynomials Hn ( ) as Hn± ( ) = e±in=4 Hn (e∓i=4 ) :
(16)
(For details, see Refs. [51,52].) Note here that these wave functions in the two dimensions are the generalized functions of the conjugate spaces S(R2 )× in Gel’fand triplets, of which nuclear space is given by Schwarz space S(R2 ) (a set of rapidly decreasing functions like Gaussians; a linear subspace of Lebesgue space L2 (R2 )), such that S(R2 )× ⊃ L2 (R2 ) ⊃ S(R2 ). (For details, see Ref. [47].) The opera× × tion of Hˆ to the generalized functions ∈ S(R2 )× , where Hˆ is the extension of × the Hamiltonian Hˆ deDned in S(R2 ) to S(R2 )× , is deDned by |Hˆ ≡ Hˆ | × for ∀ ∈ S(R2 ), and the functions which make the matrix elements |Hˆ for ∀ ∈ S(R2 ) Dnite can be the generalized functions of S(R2 )× [47,52]. In the present × case, since |Hˆ for ∀ ∈ S(R2 ) are Dnite for all wave functions expressed by arbitrary Dnite-degree polynomials f(ua ; va ) multiplied by oscillating functions such that = f(ua ; va )ei (ua ; va ) where (ua ; va ) is a non-singular real-function, the solutions obtained above can be the generalized functions of S(R2 )× . Notice that ∈ S(R2 )× are generally2 not normalizable as the same as plane waves in scattering processes, i.e., dua dva | | = ∞. Actually we easily see that the wave functions cannot be normalized in terms of Dirac’s delta functions except the lowest polynomial.
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The extension to three dimensions can easily be carried out in the cases with potentials that are separable into the (x; y) plane and the z direction such that V (x; y; z) = Va () + V (z) : When the energy eigenvalues of the z direction is given by Ez , we obtain the same equation as (6) for E − Ez = 0. (For the a = 1 case g1 + E − Ez ¿ 0 should be taken.) If we take the free motion with the momentum pz for the z direction, Ez = pz2 =2m should be taken. It is important that the total energy E is in general not equal to zero in the three dimensions. Note that wave functions for the separable potentials are written by the product such as (x; y; z) = (x; y) (z). Hereafter we shall not explicitly write (z) in the wave functions. Let us study vortices that appear in the linear combinations composed of the solutions with the polynomials of (15). Before going into the details we brieJy describe vortices in quantum mechanical hydrodynamics. The probability density (t; x; y) and the probability current j(t; x; y) of a wave function (t; x; y) in non-relativistic quantum mechanics are deDned by (t; x; y) ≡ | (t; x; y)|2 ;
(17)
j(t; x; y) ≡ Re[ (t; x; y)∗ (−i˝∇) (t; x; y)]=m :
(18)
They satisfy the equation of continuity @=@t + ∇ · j = 0: Following the analogue of the hydrodynamical approach [17–21], the Juid can be represented by the density and the Juid velocity C. They satisfy Euler’s equation of continuity @ + ∇ · (C) = 0 : (19) @t Comparing this equation with the continuity equation, the following deDnition for the quantum velocity of the state (t; x; y) is led in the hydrodynamical approach; C≡
j(t; x; y) : | (t; x; y)|2
(20)
Notice that and j in the present cases do not depend on time t. Now it is obvious that vortices appear at the zero points of the density, that is, the nodal points of the wave function. At the vortices, of course, the current j must not vanish. We should here remember that the solutions of (6) degenerate inDnitely. This fact indicates that we can construct wave functions having the nodal points at arbitrary positions in terms of linear combinations of the inDnitely degenerate solutions [45,46]. The strength of vortex is characterized by the circulation $ that is represented by the integral round a closed contour C encircling the vortex such that $= C ds (21) C
and it is quantized as $ = 2l˝=m ;
(22)
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where the circulation number l is an integer [36,39,42]. It should be stressed that we can perform the investigation of vortices for all the cases except a = 0 in the (ua ; va ) plane, because fundamental properties of vortices such as the numbers of vortices in the original plane and the mapped plane and the strengths of vortices do not change by the conformal mappings. Vortex patterns in the (x; y) plane can be obtained by the inverse transformations of the conformal mappings. Let us here show that vortex lines and vortex lattices can be constructed from simple linear combinations of the low lying polynomial solutions. And also the mapped patterns of those lines and lattices are investigated by the conformal mappings for a = 2 (PPB case; Va ˙ 2 ) and for a = 1=2 (Coulomb type; Va ˙ −1 ). In the following discussions the suNces a of ua , va and ka are omitted. (I) Vortex lines. Let us consider the linear combination of two degree 1 solutions such that (u; v) = veiku − ue−ikv ;
(23)
where the complex constant corresponding to the overall factor of the wave function is ignored, because the wave function belongs to the conjugate space of Gel’fand triplet and it is not normalizable. This means that the wave function represents a stationary Jow such as in scattering processes. The nodal points of the probability density |(u; v)|2 = u2 + v2 − 2uv cos k(u + v) appear at points satisfying the conditions u = ± v;
cos k(u + v) = ± 1 :
(24)
We have the nodal points at u = v = n=k
for n = integers :
(25)
In the (u; v) plane the positions of vortices can be on a line of u = v. After some elementary but tedious calculations we see that the circulation numbers of vortex strengths are given by l = − 1 for n = positive integers and l = 1 for n = negative ones. Note that the origin at u = v = 0 has no vortex. We can directly see the result by showing the fact that the strength of vortex $ becomes zero for the closed circle around the origin. We can interpret this result as follows; at the origin there exist a pair of vortices having the opposite circulation numbers, that is, they, respectively, belong to the vortex line with l = − 1 and that with l = 1. We may say that it is a vortex dipole. For the case of a = 1 (constant potential) we can take as u = x and v = y. In the case of a = 2 (PPB) we have u = x2 − y2 and v = 2xy. The relations for the nodal points are written down as 1 n 1 √ y= √ x; y=± for n = positive integers ; 2 2+1 ( 2 + 1)k 1 |n| 1 √ y=− √ x; y=± for n = negative integers : 2 ( 2 − 1)k 2−1 (26) We see that a vortex quadrupole composed of two vortex dipole appears at the origin.
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Fig. 1. Positions of vortices for n = ± 1; ±2; ±3 in the constant potential (a = 1), which are denoted by and stands for the vortex dipole at the origin.
◦
Fig. 2. Positions of vortices for n = ± 1; ±2; ±3 in the PPB (a = 2), which are denoted by for the vortex quadrupole at the origin.
•,
•, and ◦ stands
In the case of a = 1=2 (Coulomb type), by using the relations u2 −v2 = x and 2uv = y, we obtain the conditions for the nodal points as follows; x = 0;
y=2
n2 2 k2
for n = non-zero integers :
(27)
Note that the origin is a singular point, where the source of the potential exists. Figures for a = 1; 2 and 1=2 are presented in Figs. 1, 2 and 3, which, respectively, represent the vortex pattern for the constant potential, the PPB, and the Coulomb type
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T. Kobayashi / Physica A 303 (2002) 469–480
Fig. 3. Positions of vortices for n = 1; 2; 3 in the Coulomb type potential (a = 1=2), which are denoted by , and stands for the source at the origin.
•
◦
one. Note here that the di1erences of the potentials clearly appear not only in the vortex patterns but the properties of the singularities at the origin as well. In hydrodynamics it is known that vortex lines for the constant potential (a = 1) are stationary but generally unstable for perturbations [17–21]. Note that the parallel vortex lines are constructed from the linear combinations of the lowest and the degree 1 polynomials [45]. (II) Vortex lattices. Let us consider the linear combination of a stationary wave and a plane wave such that (u; v) = cos ku − e−ikv :
(28)
The nodal points of the probability density |(u; v)|2 = 1 + cos2 ku − 2 cos ku cos kv appear at positions satisfying u = m=k;
v = n=k ;
(29) m
n
where both of m and n must be even or odd, that is, (−1) = (−1) must be fulDlled. These conditions produce a vortex lattice presented in Fig. 4, which was suggested in Ref. [54]. In the cases of the PPB (a = 2) and the Coulomb type (a = 1=2) vortices appear at the cross points of the following two functions; x2 − y2 = m=k;
xy = n=2k
x2 + y2 = (m2 + n2 )2 4 =k 4 ;
for the PPB ; y = 2mn2 =k 2
for the Coulomb type :
(30)
T. Kobayashi / Physica A 303 (2002) 469–480
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Fig. 4. Positions of vortices for m; n = 0; ±1; ±2; ±3; ±4 in a constant potential (a = 1), which are denoted and the distance between the neighbouring lines are taken by =k. by
•
Fig. 5. Positions of vortices for the PPB (a = 2), which are denoted by
•.
In the arbitrary values of a we obtain the circulation number l =−1 for the all vortices. Figures for a = 2 and 1=2 are given in Figs. 5 and 6, respectively. In these arguments we see the following points: (1) The construction of vortex lattices in experiments seems to be not very diNcult. In fact the vortex lattice of (II) can be produced from a stationary wave and a plane wave perpendicular to the stationary wave.
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Fig. 6. Positions of vortices for 0 6 |m|, |n| 6 3 in the Coulomb type potential (a = 1=2), which are denoted stands for the source at by , the distance between the neighbouring lines are taken by 2(=k)2 , and the origin.
•
◦
(2) The di1erences of potentials can be clearly seen from the vortex patterns. Especially the distances between two neighbouring vortices are a good object to identify the type of the potentials. That is to say, the vortices appear in an equal distance =k in the case of the constant potential (a = 1), whereas the distances become smaller in the regions far from the origin for a ¿ 1 and larger for a ¡ 1 in comparison with those near the origin. (3) The property of the singularity at the origin is also a good object to identify the potentials. Though it is at this moment diNcult to categorize the present experimental vortex patterns [8–16], we shall be able to understand fundamental dynamics of vortex phenomena from vortex patterns. It is also noticed that the present results can be applicable not only to quantum phenomena but also those in classical Juids by changing the parameters m, ˝ and ga in the original equation. Up to now we have not discussed on the stability of the vortex lattices. In order to investigate the time development of the patterns we have to take account of the fact that the solutions used here belong to the conjugate spaces of Gel’fand triplets. In the spaces the eigenstates generally have complex energy eigenvalues such as resonances and the eigenvalues are expressed by pairs of complex conjugates such that E = ' ± i, where '; ∈ R [47]. It is known that the + and − signs of the imaginary part in the eigenvalues, respectively, represent resonance-formation and resonance-decay processes. We see that this pairing property is the origin of the inDnite degeneracy of the solutions
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and the inDnite degeneracy stems from the balance between the resonance-formation and resonance-decay processes. This fact seems to indicate that many vortex systems are possibly unstable for perturbations. Actually, the existence of vortex lattices has already been pointed out and it has also been noticed that those systems will decay from their edges, where the balance between the formation and decay processes is broken [54]. In such processes, we have to discuss on dynamics in many vortex systems and statistical mechanics should be extended into the conjugate spaces of Gel’fand triplets, where the freedom of the imaginary energy eigenvalues must be introduced [54 –56]. In the theory the inDnite degeneracy plays another important role, that is, it becomes the origin of new entropy. The new entropy brings essentially new processes in thermal non-equilibrium [55,56]. At present we still have many fundamental questions in the investigation of dynamics in the conjugate spaces of Gel’fand triplets. The study of vortex dynamics will bring us a new prospect and open a new dynamics for the systems essentially described by the states in Gel’fand triplets. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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Vortex lattices in quantum mechanics Tsunehiro Kobayashi Department of General Education for the Hearing Impaired, Tsukuba College of Technology, Ibaraki 305-0005, Japan Received 11 June 2001; received in revised form 27 August 2001
Abstract Vortex lattices are constructed in terms of linear combinations of solutions for Schr/odinger equation with a constant potential. The vortex lattices are mapped on the spaces with twodimensional rotationally symmetric potentials by using conformal mappings and the di1erences of the mapped vortex patterns are examined. The existence of vortex dipole and quadrupole is c 2002 Elsevier Science B.V. All rights reserved. also pointed out. Keywords: Vortex; Hydrodynamics; Quantum mechanics
Vortices play an interesting roles in various aspects of present-day physics such as vortex matters (vortex lattices) in condensed matters [1,2], quantum Hall e1ects [3–7], various vortex patterns of non-neutral plasma [8–11] and Bose–Einstein gases [12–16]. The vortices are well known objects in hydrodynamics and has been investigated in many aspects [17–21]. In quantum mechanics, hydrodynamical approach was vigorously investigated in the early stage of the development of quantum mechanics [22–34]. The fundamental properties of vortices in quantum mechanics were extensively examined by Hirschfelder and others [35 – 40] and the motions of vortex lines were also studied [41,42]. Recently an analysis of vortices has been carried out on the basis of non-linear Schr/odinger equations [43,44]. The vortex dynamics will hold an important position in quantum mechanical problems. In vortex dynamics, however, the construction of various vortex patterns from fundamental dynamics is still not very clear. Recently Kobayashi and Shimbori have proposed a way to investigate vortex patterns from the solutions of Schr/odinger equations [45]. By using the conformal mappings, the article has shown that the solutions with zero energy E-mail address: [email protected] (T. Kobayashi). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 5 0 1 - 5
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eigenvalue in the two-dimensional parabolic potential barriers (2D PPB) expressed by V = − m2 (x2 + y2 )=2 [46] can be extended to all two-dimensional rotationally sym metric potentials of the forms Va () = − a2 ga 2(a−1) with = x2 + y2 except a = 0 and the solutions are inDnitely degenerate as same as those in the 2D PPB [45]. It has also been shown that the inDnite degeneracy can be an origin of a variety of vortex patterns which appear at nodal points of wave functions and this idea can be applied in three dimensions. In the article, a few examples of simple vortex patterns are presented. In this paper, some patterns of vortex lattices that are constructed from simple linear combinations of the inDnitely degenerate states will be studied. Let us start from the arguments in two dimensions [45]. In two dimensions, the eigenvalue problems with the energy eigenvalue E are explicitly written as ˝2 − + Va () (x; y) = E (x; y) ; (1) 2m where = @2 =@x2 + @2 =@y2 ; the rotationally symmetric potentials are generally given by Va () = − a2 ga 2(a−1) ; with = x2 + y2 , a ∈ R (a = 0), and m and ga are, respectively, the mass of the particle and the coupling constant. Note here that the eigenvalues E should generally be taken as complex numbers that are allowed in conjugate spaces of Gel’fand triplets [47]. Note also that Va represents repulsive potentials for (ga ¿ 0; a ¿ 1) and (ga ¡ 0; a ¡ 1) and attractive potentials for (ga ¿ 0; a ¡ 1) and (ga ¡ 0; a ¿ 1). Let us consider the conformal mappings a = z a
with z = x + iy :
(2)
We use the notations ua and va deDned by a = ua + iva that are written as ua = a cos a’;
va = a sin a’ ;
where ’ = arctan(y=x). In the (ua ; va ) plane Eqs. (1) are written down as ˝2 2 2(a−1)=a a a − a − ga (ua ; va ) = E (ua ; va ) ; 2m where a = @2 =@ua2 + @2 =@va2 : We can rewrite the equations as ˝2 − a − ga (ua ; va ) = a−2 Ea2(1−a)=a (ua ; va ) : 2m
(3)
(4)
(5)
It is surprising that the equations become same for all values of a except a = 0 when the energy eigenvalue E have zero value. That is to say, for E = 0 the equations have the same form as that for the free particle with the constant potentials ga as ˝2 − a − ga (ua ; va ) = 0 : (6) 2m
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It should be noticed that in the case of a = 1 where the original potential is a constant g1 , the energy does not need to be zero but can take arbitrary real numbers, because the right-hand side of (5) has no dependence. In the a = 1 case, therefore, we should take g1 + E instead of ga . Here, let us brieJy comment on the conformal mappings a = z a . We see that the transformation maps the part of the (x; y) plane described by 0 6 ¡ ∞; 0 6 ’ ¡ =a on the upper half-plane of the (ua ; va ) plane for a ¿ 1 and the lower half-plane for a ¡ − 1. Note here that the maps on the part of the (ua ; va ) plane with the angle ’a = ’ − can be carried out by using the conformal mappings a () = z a e−i :
(7)
In the maps the variables are given by ua () = ua cos + va sin ;
va () = va cos − ua sin :
(8)
The relations ua (0) = ua and va (0) = va are obvious. Note that the angle represents the freedom of the angle of the incoming stationary wave. It is trivial that the equations for all a have the particular solutions ± ±ika ua 0 (ua ) = Na e
;
± ±ika va 0 (va ) = Na e
(9) for ga ¿ 0
(10)
and ±ka ua ± ; 0 (ua ) = Ma e
(11)
±ka va ± for ga ¡ 0 (12) 0 (va ) = Ma e where ka = 2m|ga |=˝, and Na and Ma are in general complex numbers. General solutions should be written by the linear combinations of (9) and (10) for ga ¿ 0 and those of (11) and (12) for ga ¡ 0. Examples for the 2D PPB with ga ¿ 0 are presented in Ref. [46]. In the following investigations, we shall concentrate our attention on the solutions of (9) and (10) that are expressed in terms of plane waves. Hereafter, we use the notations ua () and va () with − ¡ 6 , which are used in the conformal mappings of (7). We see that
e±ika ua ()
and
e±ika va ()
are the solutions of (6). (For details, see Ref. [45].) We notice here the degeneracy of the solutions. The origin of the inDnite degeneracy can easily be understood in the case of the 2D PPB [46]. It is known that energy eigenvalues of 1D PPB are given by pure imaginary values ∓i(n + 1=2)˝ with n = 0; 1; 2; : : : [48–53]. From this result, we see that the energy eigenvalues of the 2D PPB, which are composed of the sum of the 1D-PPB eigenvalues with the opposite signs such as ∓i(nx − ny )˝ with nx and ny = 0; 1; 2; : : : ; include the zero energy for nx = ny . It is apparent that all the states with the energy eigenvalues ∓i(nx − ny )˝ are inDnitely degenerate [46]. This means that all the rotationally symmetric potentials
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Va () have the same degeneracy for the zero energy states. By putting the wave function f± (ua ; va ) 0± (ua ) into (6) where f± (ua ; va ) are polynomial functions of ua and va , we obtain the equation @
a ± 2ika f± (ua ; va ) = 0 : (13) @ua A few examples of the functions f are given by [46] f0± (ua ; va ) = 1 ; f1± (ua ; va ) = 4ka va ; f2± (ua ; va ) = 4(4ka2 va2 + 1 ± 4ika ua ) :
(14)
We can obtain the general forms of the polynomials in the 2D PPB, which are generally √ written by the multiple of the polynomials of degree n, Hn± ( 2k2 x), such that fn± (u2 ; v2 ) = Hn± ( 2k2 x)Hn∓ ( 2k2 y) ; (15) where x and y in the right-hand side should be considered as the functions of u2 and v2 [46]. Since the form of Eqs. (6) is the same for all a, the solutions can be written by the same polynomial functions that are given in (15) for the PPB. That is to say, we can obtain the polynomials for arbitrary a by replacing u2 and v2 with ua and va in (15). Note that the polynomials Hn± ( ) with = m=˝x are deDned by the solutions for the eigenstates with E = ∓ i(n + 1=2)˝ in 1D PPB of the type V (x) = − m2 x2 =2 and they are written in terms of the Hermite polynomials Hn ( ) as Hn± ( ) = e±in=4 Hn (e∓i=4 ) :
(16)
(For details, see Refs. [51,52].) Note here that these wave functions in the two dimensions are the generalized functions of the conjugate spaces S(R2 )× in Gel’fand triplets, of which nuclear space is given by Schwarz space S(R2 ) (a set of rapidly decreasing functions like Gaussians; a linear subspace of Lebesgue space L2 (R2 )), such that S(R2 )× ⊃ L2 (R2 ) ⊃ S(R2 ). (For details, see Ref. [47].) The opera× × tion of Hˆ to the generalized functions ∈ S(R2 )× , where Hˆ is the extension of × the Hamiltonian Hˆ deDned in S(R2 ) to S(R2 )× , is deDned by |Hˆ ≡ Hˆ | × for ∀ ∈ S(R2 ), and the functions which make the matrix elements |Hˆ for ∀ ∈ S(R2 ) Dnite can be the generalized functions of S(R2 )× [47,52]. In the present × case, since |Hˆ for ∀ ∈ S(R2 ) are Dnite for all wave functions expressed by arbitrary Dnite-degree polynomials f(ua ; va ) multiplied by oscillating functions such that = f(ua ; va )ei (ua ; va ) where (ua ; va ) is a non-singular real-function, the solutions obtained above can be the generalized functions of S(R2 )× . Notice that ∈ S(R2 )× are generally2 not normalizable as the same as plane waves in scattering processes, i.e., dua dva | | = ∞. Actually we easily see that the wave functions cannot be normalized in terms of Dirac’s delta functions except the lowest polynomial.
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The extension to three dimensions can easily be carried out in the cases with potentials that are separable into the (x; y) plane and the z direction such that V (x; y; z) = Va () + V (z) : When the energy eigenvalues of the z direction is given by Ez , we obtain the same equation as (6) for E − Ez = 0. (For the a = 1 case g1 + E − Ez ¿ 0 should be taken.) If we take the free motion with the momentum pz for the z direction, Ez = pz2 =2m should be taken. It is important that the total energy E is in general not equal to zero in the three dimensions. Note that wave functions for the separable potentials are written by the product such as (x; y; z) = (x; y) (z). Hereafter we shall not explicitly write (z) in the wave functions. Let us study vortices that appear in the linear combinations composed of the solutions with the polynomials of (15). Before going into the details we brieJy describe vortices in quantum mechanical hydrodynamics. The probability density (t; x; y) and the probability current j(t; x; y) of a wave function (t; x; y) in non-relativistic quantum mechanics are deDned by (t; x; y) ≡ | (t; x; y)|2 ;
(17)
j(t; x; y) ≡ Re[ (t; x; y)∗ (−i˝∇) (t; x; y)]=m :
(18)
They satisfy the equation of continuity @=@t + ∇ · j = 0: Following the analogue of the hydrodynamical approach [17–21], the Juid can be represented by the density and the Juid velocity C. They satisfy Euler’s equation of continuity @ + ∇ · (C) = 0 : (19) @t Comparing this equation with the continuity equation, the following deDnition for the quantum velocity of the state (t; x; y) is led in the hydrodynamical approach; C≡
j(t; x; y) : | (t; x; y)|2
(20)
Notice that and j in the present cases do not depend on time t. Now it is obvious that vortices appear at the zero points of the density, that is, the nodal points of the wave function. At the vortices, of course, the current j must not vanish. We should here remember that the solutions of (6) degenerate inDnitely. This fact indicates that we can construct wave functions having the nodal points at arbitrary positions in terms of linear combinations of the inDnitely degenerate solutions [45,46]. The strength of vortex is characterized by the circulation $ that is represented by the integral round a closed contour C encircling the vortex such that $= C ds (21) C
and it is quantized as $ = 2l˝=m ;
(22)
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where the circulation number l is an integer [36,39,42]. It should be stressed that we can perform the investigation of vortices for all the cases except a = 0 in the (ua ; va ) plane, because fundamental properties of vortices such as the numbers of vortices in the original plane and the mapped plane and the strengths of vortices do not change by the conformal mappings. Vortex patterns in the (x; y) plane can be obtained by the inverse transformations of the conformal mappings. Let us here show that vortex lines and vortex lattices can be constructed from simple linear combinations of the low lying polynomial solutions. And also the mapped patterns of those lines and lattices are investigated by the conformal mappings for a = 2 (PPB case; Va ˙ 2 ) and for a = 1=2 (Coulomb type; Va ˙ −1 ). In the following discussions the suNces a of ua , va and ka are omitted. (I) Vortex lines. Let us consider the linear combination of two degree 1 solutions such that (u; v) = veiku − ue−ikv ;
(23)
where the complex constant corresponding to the overall factor of the wave function is ignored, because the wave function belongs to the conjugate space of Gel’fand triplet and it is not normalizable. This means that the wave function represents a stationary Jow such as in scattering processes. The nodal points of the probability density |(u; v)|2 = u2 + v2 − 2uv cos k(u + v) appear at points satisfying the conditions u = ± v;
cos k(u + v) = ± 1 :
(24)
We have the nodal points at u = v = n=k
for n = integers :
(25)
In the (u; v) plane the positions of vortices can be on a line of u = v. After some elementary but tedious calculations we see that the circulation numbers of vortex strengths are given by l = − 1 for n = positive integers and l = 1 for n = negative ones. Note that the origin at u = v = 0 has no vortex. We can directly see the result by showing the fact that the strength of vortex $ becomes zero for the closed circle around the origin. We can interpret this result as follows; at the origin there exist a pair of vortices having the opposite circulation numbers, that is, they, respectively, belong to the vortex line with l = − 1 and that with l = 1. We may say that it is a vortex dipole. For the case of a = 1 (constant potential) we can take as u = x and v = y. In the case of a = 2 (PPB) we have u = x2 − y2 and v = 2xy. The relations for the nodal points are written down as 1 n 1 √ y= √ x; y=± for n = positive integers ; 2 2+1 ( 2 + 1)k 1 |n| 1 √ y=− √ x; y=± for n = negative integers : 2 ( 2 − 1)k 2−1 (26) We see that a vortex quadrupole composed of two vortex dipole appears at the origin.
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Fig. 1. Positions of vortices for n = ± 1; ±2; ±3 in the constant potential (a = 1), which are denoted by and stands for the vortex dipole at the origin.
◦
Fig. 2. Positions of vortices for n = ± 1; ±2; ±3 in the PPB (a = 2), which are denoted by for the vortex quadrupole at the origin.
•,
•, and ◦ stands
In the case of a = 1=2 (Coulomb type), by using the relations u2 −v2 = x and 2uv = y, we obtain the conditions for the nodal points as follows; x = 0;
y=2
n2 2 k2
for n = non-zero integers :
(27)
Note that the origin is a singular point, where the source of the potential exists. Figures for a = 1; 2 and 1=2 are presented in Figs. 1, 2 and 3, which, respectively, represent the vortex pattern for the constant potential, the PPB, and the Coulomb type
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Fig. 3. Positions of vortices for n = 1; 2; 3 in the Coulomb type potential (a = 1=2), which are denoted by , and stands for the source at the origin.
•
◦
one. Note here that the di1erences of the potentials clearly appear not only in the vortex patterns but the properties of the singularities at the origin as well. In hydrodynamics it is known that vortex lines for the constant potential (a = 1) are stationary but generally unstable for perturbations [17–21]. Note that the parallel vortex lines are constructed from the linear combinations of the lowest and the degree 1 polynomials [45]. (II) Vortex lattices. Let us consider the linear combination of a stationary wave and a plane wave such that (u; v) = cos ku − e−ikv :
(28)
The nodal points of the probability density |(u; v)|2 = 1 + cos2 ku − 2 cos ku cos kv appear at positions satisfying u = m=k;
v = n=k ;
(29) m
n
where both of m and n must be even or odd, that is, (−1) = (−1) must be fulDlled. These conditions produce a vortex lattice presented in Fig. 4, which was suggested in Ref. [54]. In the cases of the PPB (a = 2) and the Coulomb type (a = 1=2) vortices appear at the cross points of the following two functions; x2 − y2 = m=k;
xy = n=2k
x2 + y2 = (m2 + n2 )2 4 =k 4 ;
for the PPB ; y = 2mn2 =k 2
for the Coulomb type :
(30)
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477
Fig. 4. Positions of vortices for m; n = 0; ±1; ±2; ±3; ±4 in a constant potential (a = 1), which are denoted and the distance between the neighbouring lines are taken by =k. by
•
Fig. 5. Positions of vortices for the PPB (a = 2), which are denoted by
•.
In the arbitrary values of a we obtain the circulation number l =−1 for the all vortices. Figures for a = 2 and 1=2 are given in Figs. 5 and 6, respectively. In these arguments we see the following points: (1) The construction of vortex lattices in experiments seems to be not very diNcult. In fact the vortex lattice of (II) can be produced from a stationary wave and a plane wave perpendicular to the stationary wave.
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Fig. 6. Positions of vortices for 0 6 |m|, |n| 6 3 in the Coulomb type potential (a = 1=2), which are denoted stands for the source at by , the distance between the neighbouring lines are taken by 2(=k)2 , and the origin.
•
◦
(2) The di1erences of potentials can be clearly seen from the vortex patterns. Especially the distances between two neighbouring vortices are a good object to identify the type of the potentials. That is to say, the vortices appear in an equal distance =k in the case of the constant potential (a = 1), whereas the distances become smaller in the regions far from the origin for a ¿ 1 and larger for a ¡ 1 in comparison with those near the origin. (3) The property of the singularity at the origin is also a good object to identify the potentials. Though it is at this moment diNcult to categorize the present experimental vortex patterns [8–16], we shall be able to understand fundamental dynamics of vortex phenomena from vortex patterns. It is also noticed that the present results can be applicable not only to quantum phenomena but also those in classical Juids by changing the parameters m, ˝ and ga in the original equation. Up to now we have not discussed on the stability of the vortex lattices. In order to investigate the time development of the patterns we have to take account of the fact that the solutions used here belong to the conjugate spaces of Gel’fand triplets. In the spaces the eigenstates generally have complex energy eigenvalues such as resonances and the eigenvalues are expressed by pairs of complex conjugates such that E = ' ± i, where '; ∈ R [47]. It is known that the + and − signs of the imaginary part in the eigenvalues, respectively, represent resonance-formation and resonance-decay processes. We see that this pairing property is the origin of the inDnite degeneracy of the solutions
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and the inDnite degeneracy stems from the balance between the resonance-formation and resonance-decay processes. This fact seems to indicate that many vortex systems are possibly unstable for perturbations. Actually, the existence of vortex lattices has already been pointed out and it has also been noticed that those systems will decay from their edges, where the balance between the formation and decay processes is broken [54]. In such processes, we have to discuss on dynamics in many vortex systems and statistical mechanics should be extended into the conjugate spaces of Gel’fand triplets, where the freedom of the imaginary energy eigenvalues must be introduced [54 –56]. In the theory the inDnite degeneracy plays another important role, that is, it becomes the origin of new entropy. The new entropy brings essentially new processes in thermal non-equilibrium [55,56]. At present we still have many fundamental questions in the investigation of dynamics in the conjugate spaces of Gel’fand triplets. The study of vortex dynamics will bring us a new prospect and open a new dynamics for the systems essentially described by the states in Gel’fand triplets. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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