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The effect of singular potentials on the harmonic oscillator
Annals of Physics 325 (2010) 2529–2541
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The effect of singular potentials on the harmonic oscillator C. Filgueiras a, E.O. Silva b, W. Oliveira c, F. Moraes a,* a
Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900 João Pessoa, PB, Brazil International Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitário Lagoa Nova, 59.072-970 Natal, RN, Brazil c Departamento de Física, Universidade Federal de Juiz de Fora, 36.036-330 Juiz de Fora, MG, Brazil b
a r t i c l e
i n f o
Article history: Received 4 May 2010 Accepted 21 May 2010 Available online 26 May 2010 Keywords: Self-adjoint extension Harmonic oscillator Landau levels Calogero model Noncommutative plane
a b s t r a c t We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a d-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The harmonic oscillator models a very large number of physical systems. It appears as the interaction between atoms in the elastic crystal, as the effective potential acting on electrons moving in a uniform magnetic field (Landau levels), in the quantization of fields, etc. Localized interactions are less ubiquitous but nonetheless important. They appear as singularities, like a d-function, for instance, which can be very handy when modelling very short-ranged interactions [1]. These are the so-called contact interactions, which appear in such diversified physical systems as nanoscale quantum devices [2] and in the optics of thin dielectric layers [3], for example. In quantum mechanics, singularities and pathological potentials, in general, are often dealt with by some kind of regularization. A common approach to ensure that the wave function in the presence of a singularity is square-integrable (and therefore might be associated to a bound state) is to force it to vanish on the singularity. More appropriately, an analysis based on the self-adjoint extension method [4], broadens the boundary condition
* Corresponding author. Tel.: +55 83 32167532; fax: +55 83 32167542. E-mail address: moraes@fisica.ufpb.br (F. Moraes). 0003-4916/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.05.012
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possibilities that still give bound states. The physics of the problem determines which of these possibilities is the right one, leaving no ambiguities, like done in Refs. [5,6] for a cosmic string. In a recent article [7], two of us used self-adjoint extension to study the quantum dynamics of a free particle on a conical surface as a toy model. This was motivated by a previous work [8] where we studied a gravitational analogue of the bound-state Aharonov–Bohm effect. Instead of a magnetic flux we considered a curvature flux provided by a cosmic string. This object, geometrically, corresponds to a Minkowiski space–time with a conical singularity, that is, a line element given by 2
2
2
ds ¼ c2 dt dz dq2 a2 q2 dh2 ;
ð1Þ
where a is related to the linear mass density of the string. For this choice of coordinates the conical singularity lies on the z-axis. Notice that the ordinary cone has its geometry described by the t = const., z = const. section of the cosmic string space–time. In [8] we solved the Schrödinger equation for a free particle in the background given by (1) and found a bound state without the need of confinement, a requirement for the usual bound-state Aharonov–Bohm effect to appear [9]. In [7] we studied the case of a particle moving on a cone under the influence of a pathological potential which goes with 1/q2. The curvature singularity of the cone enters the Schrödinger equation as the geometric potential [10] 2
U geo ¼
h ðH2 KÞ; 2M
ð2Þ
where H and K are, respectively, the mean and the Gaussian curvature of the surface. This is necessary due to the embedding of the surface in three-dimensional space [10]. It is the Gaussian curvature the one that contributes with the d-function [7]:
K¼
1 a dðqÞ
a
q
:
ð3Þ
The mean curvature leads to the pathological potential since [7]
H2 ¼
1 a2 : 4a2 q2
ð4Þ
In this work we address the problem of a harmonic oscillator on a plane with a single d-function singularity located at the origin of a polar coordinate system. Besides the obvious connection with references [8,7], this problem is related to applications like the study of Landau levels in a medium with a topological defect, the Calogero model, and the dynamics of a charged particle on the noncommutative plane in the presence of a flux tube, all of which we discuss here. It is important to notice that the influence of topology on Landau levels has been addressed over the years in the context of spaces with topological defects [11,12]. The difference between this work and the previous ones is that we are including the coupling between the singularity and the eigenfunctions, that is, we are including a coupling between the eigenfunctions and a short-ranged potential (modeled by a d-function interaction). We deal with this problem via the self-adjoint extension approach [4].
2. Two-dimensional harmonic oscillator in the presence of a d-function singularity The Hamiltonian of the harmonic oscillator in two-dimensional space is given by
Ho ¼
p2y p2x x2 x2 x2y y2 þ þ x þ ; 2M 2M 4 4
where the factor
1 4
ð5Þ
is written for convenience. In polar coordinates (r, u) it can be written as
" # h 1 @ @ 1 @2 x2 r 2 þ r þ 2 Ho ¼ ; @r r @ u2 2M r @r 4 2
ð6Þ
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where we considered xx = xy = x. We include here a d interaction, that is,
U short ¼ r
dðrÞ ; r
ð7Þ
where the subscript short means that (7) is a short-ranged potential modelled by the d interaction and r is the coupling constant. Besides the systems mentioned at the introduction, short-ranged potentials appear in several contexts in Aharonov–Bohm-like problems as the A–B problem for the spin-half particles [13], the coupling between eigenfunctions and conical defects/cosmic strings [14] and the coupling between eigenfunctions and torsion [15]. Our problem here is to find the bound states for the Hamiltonian
H ¼ Ho þ U short :
ð8Þ
As we said, we do not exclude the singularity point r = 0 but, instead, we follow Kay and Studer [16] and model the problem by boundary conditions. What we mean is: we temporarily forget the d-function potential and find which boundary conditions are allowed for Ho. This is the scope of the self-adjoint extension, which consists in determining the complete domain of an operator, that is, its complete set of eigenfunctions. But the self-adjoint extension provides us with an infinity of possible boundary conditions and therefore it cannot give us the true physics of the problem. Nevertheless, once fixed the physics at r = 0 [6,7,5], we are able to fit any arbitrary parameter coming from the self-adjoint extension and then we have a complete description of the problem. Since we have a singular point we must guarantee that the Hamiltonian is self-adjoint in the region of movement for, even if Hyo ¼ Ho , their domains might be different. The von Neumann–Krein method [4] is used to find the self-adjoint extensions where needed. An operator Ho with domain D(Ho) is self-adjoint if DðHyo Þ ¼ DðHo Þ and Hyo ¼ Ho . To proceed with the self-adjoint extension method we define the deficiency subspaces N± as [4]
Nþ ¼ fw 2 DðHyo Þ; Hyo w ¼ zþ w; Im zþ > 0g; N ¼ fw 2 DðHyo Þ; Hyo w ¼ z w; Im z < 0g; with dimensions n+ and n, respectively, which are called deficiency indices of H. A necessary and sufficient condition for H to be self-adjoint is that n+ = n = 0. On the other hand, if n+ = n P 1 then H has an infinite number of self-adjoint extensions parametrized by a unitary n n matrix, where n = n+ = n. Since the potential in this case is purely radial it is natural to assume separation of variables. Therefore the Hilbert space of square integrable eigenfunctions of H is decomposed as L2 (0, +1) L2 (S1). 2 The operator @@u2 is essentially self-adjoint in L2 (S1), with spectrum l2, where l 2 Z. For the radial eigenfunctions, special care is needed due to the singularity at r = 0. We rewrite the expression (6) as
" # 2 e o ¼ 1 @ r @ l þ c2 r2 ; H r @r @r r2
ð9Þ
where c2 ¼ 2Mh2 x2 . Thus, the eigenvalue equation is given by
e o W ¼ E0 W; H 0
ð10Þ
2
where E = 2ME/⁄ . We assume the eigenfunctions of (9) to be of the form
WðrÞ ¼ r l UðrÞ:
ð11Þ
Substituting (11) in (10), we produce
( " Ho UðrÞ ¼
# ) 1 d 2 2 þ c r UðrÞ ¼ E0 UðrÞ; þ ð1 þ 2lÞ 2 r dr dr 2
d
ð12Þ
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which defines the operator Ho . The general solution of the eigenvalue equation (12) is [17] cr 2
cr 2
UðrÞ ¼ Ae 2 Uðd; 1 þ l; cr2 Þ þ De 2 Mðd; 1 þ l; cr2 Þ;
ð13Þ
where U and M denote the confluent hypergeometric functions of the second kind and first kind, 0 respectively [17], d ¼ 1þl 4Ec and A and D are constants. In the case of non-confined quantum systems 2 a necessary condition for W(r) to be square-integrable is
lim WðrÞ ¼ 0;
r!1
cr 2
cr 2
which is fulfilled by rl Ae 2 Uðd; 1 þ l; cr2 Þ but not by r l De 2 Mðd; 1 þ l; cr2 Þ. For this reason we make D = 0. Therefore we are left with cr 2
WðrÞ ¼ Arl e 2 Uðd; 1 þ l; cr 2 Þ:
ð14Þ
2
In order to guarantee that W(r) 2 L (0, +1) we need to study the behavior of W(r) as r ? 0 which implies analysing the possible self-adjoint extensions. Now, in order to construct the self-adjoint extensions, we need to find the solution to
Ho y U ¼ ijU :
ð15Þ
in order to find N+ and N as defined in (9). Since Hyo ¼ Ho and, from (11) and (14), the solution to this equation is given by cr 2
U ðrÞ ¼ e 2 U d ; 1 þ l; cr 2 ;
ð16Þ
where d ¼ 1þl 4ic. 2 Let us consider U(d±, 1 + l, cr2) as r ? 0 [17]:
U d ; 1 þ l; cr 2 !
p 1 cl r2l : sinðp þ lpÞ Cðd lÞCð1 þ lÞ Cðd ÞCð1 lÞ
ð17Þ
R R Defining W±(r) = rlU±(r), let us find under which condition jW ðrÞj2 r dr ¼ jU ðrÞjr 1þ2l dr has a finite contribution from the near origin region. Taking (16) and (17) into account, we have
lim jU ðrÞj2 r 1þ2l ! C 1 r1þ2l þ C 2 r 12l þ C 3 r ;
ð18Þ
r!0
where C1, C2 and C3 are constants. Looking at Eq. (18), we see that W±(r) is square-integrable only when 1 < l < 1. In this case, since N+ is expanded by W+(r) only, we have that its dimension n+ = 1. The same applies to N and W(r) resulting in n = 1. Therefore, H possesses self-adjoint extensions parametrized by a unitary matrix U(1) = ein, with n 2 R ðmod 2pÞ. Thus, according to the von Neumann–Krein theory, the domain of Hyo is given by
DðHyo Þ ¼ DðHo Þ Nþ N :
ð19Þ
The essence of the self-adjoint extension approach is to extend the domain DðHo Þ to match DðHyo Þ and therefore make Ho self-adjoint. We get then
Dn ðHo Þ ¼ DðHyo Þ ¼ DðHo Þ Nþ N ;
ð20Þ
which means that, for each n, we have a possible domain for Ho . This is mathematics. The physical situation determines the value of n. The Hilbert space (20) contains vectors of the form
cr 2
UðrÞ ¼ vðrÞ þ C e 2 Uðdþ ; c; cr 2 Þ þ ein Uðd ; c; cr2 Þ ;
ð21Þ
where C is an arbitrary complex number, vð0Þ ¼ v_ ð0Þ ¼ 0 and rlv(r) 2 L2 (0, +1). The behavior of the wave functions (21) for a range of n was addressed in [18]. However, as we will see below, if we fix the physics of the problem at r = 0, there is no need for such analysis because the value of n is automatically selected.
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2533
In order to find the spectrum of (9) we consider the limit r ? 0 of W(r), that is, we substitute (17) and (21) into (14), which leads to
A
1 1 cl r 2l Cðd lÞCð1 þ lÞ CðdÞCð1 lÞ 1 ein 1 ein ¼C þ cl r 2l þ : Cðdþ lÞCð1 þ lÞ Cðd lÞCð1 þ lÞ Cðdþ ÞCð1 lÞ Cðd ÞCð1 lÞ
ð22Þ
Now, equating the coefficients of the same powers of r, we get
A 1 ein ¼C þ Cðd lÞ Cðdþ lÞ Cðd lÞ
ð23Þ
A 1 ein ¼C þ ; CðdÞ Cðdþ Þ Cðd Þ
ð24Þ
and
whose quotient leads to
Cðd lÞ ¼ CðdÞ
e
i
n 2
Cðdþ Þ i
i
n
e 2 Cðdþ lÞ
n
e2 þ Cðd Þ i
n
:
ð25Þ
þ Cðde2lÞ
The left hand side of this equation is a function of the energy E0 while its right hand side is a constant (even though it depends on the extension parameter n which is fixed by the physics of the problem). Therefore we have the equation
C C
0
0
¼ const:
1l 2
4Ec
1þl 2
4Ec
ð26Þ
As we said above we have achieved the energy levels with an arbitrary parameter n. Different values of it means inequivalent quantizations [19], that is, each physical problem selects a specific n. In what follows we give the details on how n may be determined for the generic case of a harmonic oscillator with a d-function singularity. This will lead to the applications of the next section. We are substituting the true problem H = H0 + Ushort(r) by the idealized problem, that is, H = H0 plus self-adjoint extension. So, consider the static problem of both true and idealized problem, respectively
H0 þ U short ðrÞ /true ðrÞ ¼ 0;
ð27Þ
H0 /n ðrÞ ¼ 0;
ð28Þ
where these operators are in the form given by Eq. (12). Eq. (7) becomes
U short ðrÞ ¼
2M r dðr aÞ : 2 r h
ð29Þ
At the end, the limit a = 0 is taken. So, to find out the value of n we state that
d/n =dr d/true =dr ¼ : /true ðrÞ r!a /n ðrÞ r!a
ð30Þ
As static solution for the idealized problem we take the relation (21). From the true problem above and for r 6 a, we have
( "
# ) 1 d 2 2 þ c r þ U short ðrÞ Utrue ðrÞ ¼ 0: þ ð1 þ 2lÞ 2 r dr dr d
2
ð31Þ
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Integrating this expression from 0 to a, we arrive at
Z
a
1 r 1þ2l
0
Z a d d/ ðrÞ 1þ2l dðr aÞ r 1þ2l true r /true ðrÞr 1þ2l dr: dr ¼ r dr dr a 0
ð32Þ
We omitted the integral containing the harmonic term because of the following:
Z
a
c2 r2 r1þ2l /true ðrÞdr /true ðr ¼ aÞ
Z
0
a
c2 r2 r1þ2l dr ! 0;
ð33Þ
0
as a ? 0. So, from (32), we get
ðd/true ðrÞ=dr Þjr!a r : /true ðr ! aÞ a
ð34Þ
Now, for the idealized problem, from (21) and from the expansion for U given by (17), we have
d/n =dr ¼h /n ðrÞ r!a
2lcl a12l 1
Cðbþ ÞCð1þlÞ
h
in
1
Cðbþ ÞCð1lÞ
þ Cðdþ eÞCð1lÞ
i
i h i: 2 l 2 l CðdðþcaÞCÞð1lÞ þ ein Cðb Þ1Cð1þlÞ CðdðcaÞCÞð1lÞ
ð35Þ
Inserting this in (34) we arrive at
h
2lcl a12l 1 Cðbþ ÞCð1þlÞ
h
1
Cðbþ ÞCð1lÞ
in
þ Cðdþ eÞCð1lÞ
i
r
i h i¼ : 2 l 2 l a CðdðþcaÞCÞð1lÞ þ ein Cðb Þ1Cð1þlÞ CðdðcaÞCÞð1lÞ
ð36Þ
This relation selects an approximate value for the self-adjoint extension parameter in terms of the physics of the problem. In order to determine the energy levels we substitute (14) in (21) and compute dUdrðrÞ =UðrÞ. Then, 2lcl a12l CðdÞCð1lÞ
h
1
CðbÞCð1þlÞ
2lcl a12l
h
1
Cðbþ ÞCð1lÞ
in
þ Cðdþ eÞCð1lÞ
i
i¼h i h i: cl a2l ðca2 Þl ðca2 Þl 1 1 in CðdÞ Cð1lÞ Cðbþ ÞCð1þlÞ Cðdþ ÞCð1lÞ þ e Cðb ÞCð1þlÞ Cðd ÞCð1lÞ
ð37Þ
Taken into account (36) the spectrum is given entirely by the physics of the problem, instead of the extension parameter:
h
2lcl a12l CðdÞCð1lÞ 1 CðbÞCð1þlÞ
r
i¼ : cl a2l a CðdÞ Cð1lÞ
ð38Þ
Rearranging this equation, we have
0 4Ec cl ð2l þ 1Þ Cð1 þ lÞ 2l
¼ a : 1l E0 r Cð1 lÞ C 2 4c
C
1þl 2
Making a = 0, the relation (38) yields C gives us the spectrum
E0 ¼ 2cð2n þ jlj þ 1Þ:
1þl 2
ð39Þ
0 0 4Ec ¼ 1, for l = jlj, and C 1l 4Ec ¼ 1, for l = jlj, which 2
ð40Þ
These are the well known energy levels of the two-dimensional harmonic oscillator with an important difference: due to the singular nature of the problem the range of l is given by 1 < l < 1. If we had imposed that the eigenfunction must be regular at the origin (U(r) v(r) 0) we would achieve the same spectrum (40), but l could take any integer value [11,12]. In this sense, the self-adjoint extension prevents us from obtaining a spectrum incompatible with the irregular nature of the Hamiltonian [20] when we have a short-ranged potential as Eq. (7). Remember that the true boundary condition is that the wave function must be square-integrable through all space and it does not matter if it is irregular at the origin [14,16].
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2535
3. Applications 3.1. Landau levels for a particle in the elastic continuum with a disclination In this subsection we study the dynamics of a charged particle in a continuous elastic medium with a topological defect, disclination, in the presence of uniform magnetic field ~ B. This problem was addressed in [11] without considering the coupling with the defect, which is done here via a d interaction associated to curvature. The continuous elastic medium with the topological defect is conceptually obtained by removing a wedge of material and is described by a non-Euclidean metric similar to (1): 2
2
ds ¼ dz þ dq2 þ a2 q2 dh2 :
ð41Þ
Here, a = 1 + k/2p and k is the angle that defines the wedge. Following Ref. [11], we consider the gauge condition A/ ¼ B2qa, which corresponds to a uniform magnetic field along the z axis. The total Hamiltonian for the system is given by
" # 2 1 @ h @ 1 @2 ihqB @ q2 B2 q2 þ 2 þ 2 2 þ H¼ ; q 2 @q 2a m @/ 2m q @ q a q @/ 8ma2
ð42Þ
where we have already factored out the motion along the z axis. The Schrödinger equation can be written as
"
# 2 d 1 d m2 b2 q2 þ þ UðqÞ ¼ E0 UðqÞ; dq2 q dq q2 4
ð43Þ
2 2
where m ¼ a‘ ; b2 ¼ hq2 aB2 ; E0 ¼ 2mE þ hqB‘ a2 and E is the total energy of the system. Following Allen et al. [14], h2 the particle is coupled to the defect via the Ricci scalar R of the background metric which describes the medium:
U short ¼ gR:
ð44Þ
Here, g is a coupling constant, which has no effect in the eigenvalues and R is essentially (3) up to a factor of two. The d-function singularity of Ushort affects the boundary condition and, consequently, the range of the quantum numbers. Evoking the results from the previous section and considering a negatively charged particle (q = jqj), the Landau levels are given by
E¼
x h ð2n þ m þ jmj þ 1Þ; 2a
ð45Þ
where 1 < m < 1 (see the previous section) and x ¼ jqjB is the cyclotron frequency (we take the speed m of light c = 1). Now we follow Ref. [11] and add the self-interaction [21] on the particle by adding the potential
U¼ Here,
q2 jðaÞ : 4p q
ð46Þ
is the dielectric constant of the material and
jðaÞ ¼ 2
Z 0
1
a1 cothða1 xÞ cothðxÞ sinhðxÞ
dx
ð47Þ
measures the distortion caused by the defect in the medium. The Schrödinger equation is now written as
"
# 2 d 1 d m2 b 2 2 þ þ x q þ UðqÞ ¼ E0 UðqÞ; dq2 q dq q2 q 2
ð48Þ
jðaÞ where b ¼ mq and x ¼ 2b. In Ref. [11] the eigenfunction was factored taking into account that 2p h2 U(q) ? 0 as q ? 0 and q ? 1 [11,22]:
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C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541 q2
1
WðqÞ ¼ C qtjmj ex 2 Fðx2 qÞ:
ð49Þ
This is the case when the core of the defect is impenetrable. When it is not, the Hamiltonian is singular because of the Coulomb interaction. Then we write U(q) in this case as q2
1
WðqÞ ¼ C qm ex 2 Fðx2 qÞ;
ð50Þ 1 4
which is in agreement with the previous section. Putting r ¼ x q and following [11], F must obey 2
r
d F dr
2
þ ða 2r 2 Þ
dF þ ðgr þ bÞF ¼ 0; dr
ð51Þ
where
a ¼ 2m þ 1;
ð52Þ
and
g¼
E0
2ðm þ 1Þ:
x
ð53Þ
In terms of a power series, 1 X
FðrÞ ¼
C k rk ;
ð54Þ
k¼0
with the recurrence relation
C kþ2 ¼
b ðg 2kÞ C kþ1 Ck: ðk þ 2Þðk þ a þ 1Þ ðk þ 2Þðk þ a þ 1Þ
ð55Þ
We obtain special exact solutions if
g ¼ 2n
ð56Þ
C nþ1 ¼ 0;
ð57Þ
and
simultaneously, where n is a maximum value of k. Then, the eigenvalues are
E¼
x h ðn þ 2mn þ 1Þ: 2a
ð58Þ
The index n on m stands for the fact that (55) limits the value that m can admit (which depends on which n we truncate the series (54)). From (55)–(57), with C0 = 1, we have 2
m1 ¼
b 1 ; 4 2
ð59Þ
for n = 1. Then, remembering that 1 < m < 1, b must satisfy 2
0 6 b < 6:
ð60Þ
For n = 2,
m2 ¼
3 2 1 ðb 2Þ 20 2
ð61Þ
and 2
0 6 b < 12:
ð62Þ
This procedure may be used to find the corresponding relations for n P 3. The results in this subsection show a reduced degeneracy of the Landau levels, as compared to the previous result [11], due to the condition 1 < m < 1, which appears naturally in the self-adjoint exten-
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2537
sion procedure. Notice that, if we choose to study the problem of Landau levels on a cone, we get essentially the same answer as above. The difference comes in the addition of the contribution to the energy due to the free motion of the particle along the z axis, which should be done here and 2 a2 Þ not in the conical surface case. Also, m needs to be redefined to m2 ¼ a‘ 2 ð1 , in order to have the 4a2 mean curvature contribution (4) accounted for in the Hamiltonian. 3.2. Calogero model Consider the rational Calogero model which describes a quantum system with N identical particles interacting with each other through a long-range inverse-square, a Coulomb and a harmonic interactions on the line[23]. The Hamiltonian for this system is given by
" # N X X a2 1=4 X2 @2 b 2 ffi; H¼ þ ðxi xj Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 P 2 8 @x xi x j i i
ð63Þ
where a and X are constants, xi is the coordinate of the ith particle and units have been chosen such that 2m/⁄2 = 1. Writing the wave function of this Hamiltonian as
W¼
Y
ðxi xj Þaþ1=2 /ðrÞPk ðxÞ;
ð64Þ
i
where x (x1, x2, . . . , xN),
r2 ¼
1X ðxi xj Þ2 N i
ð65Þ
and Pk is a homogeneous, translationally invariant, polynomial of degree k which satisfies the equation
" # N X X @2 1 @ @ P k ¼ 0: þ 2ða þ 1=2Þ ðxi xj Þ @xi @xj @x2i i
ð66Þ
Inserting (64) into HW = Ew and using (65) and (66), we arrive at 2
d /ðrÞ dr
2
0
ð1 þ 2lÞ
1 d/ðrÞ b /ðrÞ þ x2 r 2 /ðrÞ ¼ E/ðrÞ; r dr r
ð67Þ
0
where x2 ¼ 18 X2 N, b ¼ pbffiffiNffi and
1 2
1 2
l ¼ k þ ðN 3Þ þ NðN 1Þða þ 1=2Þ: x2 r2
In the next step, we put /ðrÞ ¼ e 2
z
d F 2
dz
þ ðc 2z2 Þ
2
1
ð68Þ 1
Fðx2 rÞ and z ¼ x2 r. Then F obeys
dF 0 þ ðgz þ b ÞF ¼ 0; dz
ð69Þ
where g ¼ xE 2ðl þ 1Þ and c = 2l + 1. So, we can evoke the results from the last subsection such that the eigenvalues are given by
E ¼ 2xðn þ ln þ 1Þ;
ð70Þ
where 1 < ln < 1 and l has been rewritten as ln due to its dependence on n. In this case, we have 2
ð71Þ
2
ð72Þ
0 6 b < 6N; for n = 1,
0 6 b < 12N;
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for n = 2, and so on. Note that we have one more condition to be satisfied because of 1 < ln < 1:
1 1 1 < k þ ðN 3Þ þ NðN 1Þða þ 1=2Þ < 1: 2 2
ð73Þ
The analysis of this inequality is the same as in the case b = 0 and can be found in [23]. The self-adjoint extension of (63) was addressed in [23], for b = 0 and x – 0 (harmonic), and in [24], for b – 0 and x = 0 (Coulomb), cases separately. In both cases the result is given in terms of the self-adjoint extension parameter. Here, we consider both cases together and use the divergence of the Coulomb interaction at the origin as a way to fix the self-adjoint extension parameter. 3.3. Charged particle moving on the noncommutative plane in the presence of a magnetic flux tube The study of noncommutative theories was proposed initially by Heisenberg and Snyder [25] and has gained strong impulse recently due to its natural appearance in distinct arenas of physics such as string theory with a constant background magnetic field in the presence of D-branes, and quantum Hall effect, where the noncommutativity is induced among the coordinates of the particle, in a plane, due to the presence of a strong magnetic field. It is opportune to mention here that this noncommutativity, in the first case mentioned above, was eliminated constructing a mechanical system which reproduces classical dynamics of the string [26]. Besides their origin in string theories and branes, noncommutative (NC) field theories have been studied extensively in other branches of physics [27–30]. There are two major trends involving noncommutative theories. One is the study of noncommutative spaces, where the noncommutativity is defined by the following relation among the coordinate operators
h
i x0i ; x0j ¼ ihhij I;
ð74Þ
where hij is a antisymmetric parameter. In the second one, which we use here, the noncommutativity of field operators can be imposed by a small deformation in the canonical structure of the field theories [31]. This leads to a tiny violation of the microcausality principle. Thus, in this trend, the noncommutative space can be realized by a generalized Heisenberg operator algebra [31] given by
h
i x0i ; x0j ¼ ihhij I; h i x0i ; p0j ¼ ihdij I; h i p0i ; p0j ¼ ihbij I;
ð75Þ
where hij, bij are the parameters of the noncommutativity. This algebra can be constructed by changing the standard product of the classical observable by the star product (Weyl-Moyal):
i ðf gÞðxÞ ¼ exp hij @ i @ j f ðxÞgðyÞjy¼x : 2
ð76Þ
An important fact is that there is a mapping between the quantum dynamics of a particle in such a noncommutative space and the quantum dynamics in ordinary space given by the following general transformations [32]:
1 ~ x ~ x0 ¼ ~ p ^~ h; 2 1 ~ p0 ¼ ~ x ^~ b; pþ ~ 2
ð77Þ
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2539
where ~ x0 and ~ p0 satisfy (75). Note that ~ x and ~ p satisfy the usual commutation relations
½xi ; xj ¼ 0; ½xi ; pj ¼ ihdij I;
ð78Þ
½pi ; pj ¼ 0: In this section we study the bound state noncommutative Aharonov–Bohm effect, which provides us with short-ranged potentials if the particle has spin. As usual, we consider the Hamiltonian of the system having the same form as in the commutative case but with the commutation relations given by Eq. (75). Then, in the non-relativistic limit, we have
H0 ð~ x0 ; ~ p0 Þ ¼
~ p02 : 2M
ð79Þ
Let us consider the Aharonov–Bohm problem, that is, the coupling [13]
eA0/ ¼
/ r0
ð80Þ
and
eB0 ¼ /
dðr 0 Þ ; r0
ð81Þ
where / is the magnetic flux. With these relations in hands, the Hamiltonian (79) is rewritten as
H0 ¼
" # 1 /2 2i/ @ /s dðr0 Þ ~ þ p02 þ 02 02 ; 0 2M r @u 2M r 0 r
ð82Þ
where s = +1(s = 1) for spin up (down). Considering the simple case b1 = b2 = h1 = h2 = 0, b3 = b and h3 = h, the relation (77) with the conditions h 1 and b 1 yields
" # 1 1 @ @ 1 @2 ib @ b2 r2 þ 2 þ r H¼ þ 2M r @r @r r @ u2 2M @ u 8M " # 1 /2 2/ i @ br /s dðr 0 Þ þ þ þ 0 : 2M r2 r r @u 2 2M r 0
ð83Þ ð84Þ
The 1/r0 term in this equation is a Coulomb term and its expansion to first order in h was obtained in [33] and it is given by 1/r0 1/r + Lzh/4r3, where Lz ¼ i @@u is the angular momentum in the z direction with eigenvalue denoted by m. Now, we have to deal with the delta potential present in (81), which is not a simple task in the context of a noncommutativity theory. Since the delta potentials are idealizations for a real physical situation consisting in quantum mechanics in the presence of a short-ranged potential, we consider here the expression for it given by [34]
dh ¼
1
ph
r02
e h
1
ph
r2
e h :
ð85Þ
So, we replace the last term in (84) by the short-ranged potential
U short ðrÞ ¼
r 1 rh2 1 Lz h ; þ e r 4r 3 2M ph
ð86Þ
where r = /s. In what follows, we will consider three different situations. First, if h = 0 we recover the harmonic oscillator problem in the presence of a short-ranged potential
U short ðrÞ ¼
/s dðrÞ : 2M r
ð87Þ
Using the above results, the eigenvalues will be given by
E¼
b ð2n þ jm þ /j m / þ 1Þ; 2M
ð88Þ
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for 1 < m + / < 1. Second, for h – 0 and m – 0, besides Ushort(r) given by (86) we have the potential 1/ r4 which implies that the eigenfunction does not have a closed form. But, if we ignore this term, the relation (89) below, with l = m + /, gives a first approximation for the problem, since h 1. Like Eq. (39), we get
" # 0 12l 4Ec l l Cð1 þ lÞ 2plh
¼c þh ; 0 Cð1 lÞ rR C 1l 4Ec 2
C
1þl 2
ð89Þ
pffiffiffi which is exact for the case m = 0 (third situation). In all cases, we considered that a ¼ h, which gives R R 2 2 a 1 R ¼ p1ffiffih 0 er =h dr ¼ 0 ex dx ¼ 0:7468241330. Since h 1, this spectrum might not be much different from the oscillator levels (88). 4. Conclusions The easiest way of dealing with singularities in quantum mechanics is by imposing that the eigenfunction vanishes at the singularity. Although convenient, this does not necessarily give the best description (or even the correct one) of the physical phenomenon studied. Care must be taken to insure that the Hamiltonian is self-adjoint. This can be achieved by extending the domain of H to equal that of H . By doing this, a family of boundary conditions might appear, including the one that requires the eigenfunction to vanish at the singularity. With this in hands, physics itself determines the appropriate boundary condition. In this article, we have done this for both a d-function and a Coulomb singularity. We focused on the two-dimensional harmonic oscillator with a singularity at the origin for its broad application throughout physics. We applied our results to the specific cases of Landau levels in the elastic continuum plus a topological defect, the Calogero model and the quantum dynamics of a particle on the noncommutative plane. Acknowledgements This work was partially supported by CNPq and CAPES (NANOBIOTEC). E.O.S. thanks the Brazilian Ministry of Science and Technology for a post-doctoral fellowship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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Contents lists available at ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
The effect of singular potentials on the harmonic oscillator C. Filgueiras a, E.O. Silva b, W. Oliveira c, F. Moraes a,* a
Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900 João Pessoa, PB, Brazil International Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitário Lagoa Nova, 59.072-970 Natal, RN, Brazil c Departamento de Física, Universidade Federal de Juiz de Fora, 36.036-330 Juiz de Fora, MG, Brazil b
a r t i c l e
i n f o
Article history: Received 4 May 2010 Accepted 21 May 2010 Available online 26 May 2010 Keywords: Self-adjoint extension Harmonic oscillator Landau levels Calogero model Noncommutative plane
a b s t r a c t We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a d-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The harmonic oscillator models a very large number of physical systems. It appears as the interaction between atoms in the elastic crystal, as the effective potential acting on electrons moving in a uniform magnetic field (Landau levels), in the quantization of fields, etc. Localized interactions are less ubiquitous but nonetheless important. They appear as singularities, like a d-function, for instance, which can be very handy when modelling very short-ranged interactions [1]. These are the so-called contact interactions, which appear in such diversified physical systems as nanoscale quantum devices [2] and in the optics of thin dielectric layers [3], for example. In quantum mechanics, singularities and pathological potentials, in general, are often dealt with by some kind of regularization. A common approach to ensure that the wave function in the presence of a singularity is square-integrable (and therefore might be associated to a bound state) is to force it to vanish on the singularity. More appropriately, an analysis based on the self-adjoint extension method [4], broadens the boundary condition
* Corresponding author. Tel.: +55 83 32167532; fax: +55 83 32167542. E-mail address: moraes@fisica.ufpb.br (F. Moraes). 0003-4916/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.05.012
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possibilities that still give bound states. The physics of the problem determines which of these possibilities is the right one, leaving no ambiguities, like done in Refs. [5,6] for a cosmic string. In a recent article [7], two of us used self-adjoint extension to study the quantum dynamics of a free particle on a conical surface as a toy model. This was motivated by a previous work [8] where we studied a gravitational analogue of the bound-state Aharonov–Bohm effect. Instead of a magnetic flux we considered a curvature flux provided by a cosmic string. This object, geometrically, corresponds to a Minkowiski space–time with a conical singularity, that is, a line element given by 2
2
2
ds ¼ c2 dt dz dq2 a2 q2 dh2 ;
ð1Þ
where a is related to the linear mass density of the string. For this choice of coordinates the conical singularity lies on the z-axis. Notice that the ordinary cone has its geometry described by the t = const., z = const. section of the cosmic string space–time. In [8] we solved the Schrödinger equation for a free particle in the background given by (1) and found a bound state without the need of confinement, a requirement for the usual bound-state Aharonov–Bohm effect to appear [9]. In [7] we studied the case of a particle moving on a cone under the influence of a pathological potential which goes with 1/q2. The curvature singularity of the cone enters the Schrödinger equation as the geometric potential [10] 2
U geo ¼
h ðH2 KÞ; 2M
ð2Þ
where H and K are, respectively, the mean and the Gaussian curvature of the surface. This is necessary due to the embedding of the surface in three-dimensional space [10]. It is the Gaussian curvature the one that contributes with the d-function [7]:
K¼
1 a dðqÞ
a
q
:
ð3Þ
The mean curvature leads to the pathological potential since [7]
H2 ¼
1 a2 : 4a2 q2
ð4Þ
In this work we address the problem of a harmonic oscillator on a plane with a single d-function singularity located at the origin of a polar coordinate system. Besides the obvious connection with references [8,7], this problem is related to applications like the study of Landau levels in a medium with a topological defect, the Calogero model, and the dynamics of a charged particle on the noncommutative plane in the presence of a flux tube, all of which we discuss here. It is important to notice that the influence of topology on Landau levels has been addressed over the years in the context of spaces with topological defects [11,12]. The difference between this work and the previous ones is that we are including the coupling between the singularity and the eigenfunctions, that is, we are including a coupling between the eigenfunctions and a short-ranged potential (modeled by a d-function interaction). We deal with this problem via the self-adjoint extension approach [4].
2. Two-dimensional harmonic oscillator in the presence of a d-function singularity The Hamiltonian of the harmonic oscillator in two-dimensional space is given by
Ho ¼
p2y p2x x2 x2 x2y y2 þ þ x þ ; 2M 2M 4 4
where the factor
1 4
ð5Þ
is written for convenience. In polar coordinates (r, u) it can be written as
" # h 1 @ @ 1 @2 x2 r 2 þ r þ 2 Ho ¼ ; @r r @ u2 2M r @r 4 2
ð6Þ
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2531
where we considered xx = xy = x. We include here a d interaction, that is,
U short ¼ r
dðrÞ ; r
ð7Þ
where the subscript short means that (7) is a short-ranged potential modelled by the d interaction and r is the coupling constant. Besides the systems mentioned at the introduction, short-ranged potentials appear in several contexts in Aharonov–Bohm-like problems as the A–B problem for the spin-half particles [13], the coupling between eigenfunctions and conical defects/cosmic strings [14] and the coupling between eigenfunctions and torsion [15]. Our problem here is to find the bound states for the Hamiltonian
H ¼ Ho þ U short :
ð8Þ
As we said, we do not exclude the singularity point r = 0 but, instead, we follow Kay and Studer [16] and model the problem by boundary conditions. What we mean is: we temporarily forget the d-function potential and find which boundary conditions are allowed for Ho. This is the scope of the self-adjoint extension, which consists in determining the complete domain of an operator, that is, its complete set of eigenfunctions. But the self-adjoint extension provides us with an infinity of possible boundary conditions and therefore it cannot give us the true physics of the problem. Nevertheless, once fixed the physics at r = 0 [6,7,5], we are able to fit any arbitrary parameter coming from the self-adjoint extension and then we have a complete description of the problem. Since we have a singular point we must guarantee that the Hamiltonian is self-adjoint in the region of movement for, even if Hyo ¼ Ho , their domains might be different. The von Neumann–Krein method [4] is used to find the self-adjoint extensions where needed. An operator Ho with domain D(Ho) is self-adjoint if DðHyo Þ ¼ DðHo Þ and Hyo ¼ Ho . To proceed with the self-adjoint extension method we define the deficiency subspaces N± as [4]
Nþ ¼ fw 2 DðHyo Þ; Hyo w ¼ zþ w; Im zþ > 0g; N ¼ fw 2 DðHyo Þ; Hyo w ¼ z w; Im z < 0g; with dimensions n+ and n, respectively, which are called deficiency indices of H. A necessary and sufficient condition for H to be self-adjoint is that n+ = n = 0. On the other hand, if n+ = n P 1 then H has an infinite number of self-adjoint extensions parametrized by a unitary n n matrix, where n = n+ = n. Since the potential in this case is purely radial it is natural to assume separation of variables. Therefore the Hilbert space of square integrable eigenfunctions of H is decomposed as L2 (0, +1) L2 (S1). 2 The operator @@u2 is essentially self-adjoint in L2 (S1), with spectrum l2, where l 2 Z. For the radial eigenfunctions, special care is needed due to the singularity at r = 0. We rewrite the expression (6) as
" # 2 e o ¼ 1 @ r @ l þ c2 r2 ; H r @r @r r2
ð9Þ
where c2 ¼ 2Mh2 x2 . Thus, the eigenvalue equation is given by
e o W ¼ E0 W; H 0
ð10Þ
2
where E = 2ME/⁄ . We assume the eigenfunctions of (9) to be of the form
WðrÞ ¼ r l UðrÞ:
ð11Þ
Substituting (11) in (10), we produce
( " Ho UðrÞ ¼
# ) 1 d 2 2 þ c r UðrÞ ¼ E0 UðrÞ; þ ð1 þ 2lÞ 2 r dr dr 2
d
ð12Þ
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which defines the operator Ho . The general solution of the eigenvalue equation (12) is [17] cr 2
cr 2
UðrÞ ¼ Ae 2 Uðd; 1 þ l; cr2 Þ þ De 2 Mðd; 1 þ l; cr2 Þ;
ð13Þ
where U and M denote the confluent hypergeometric functions of the second kind and first kind, 0 respectively [17], d ¼ 1þl 4Ec and A and D are constants. In the case of non-confined quantum systems 2 a necessary condition for W(r) to be square-integrable is
lim WðrÞ ¼ 0;
r!1
cr 2
cr 2
which is fulfilled by rl Ae 2 Uðd; 1 þ l; cr2 Þ but not by r l De 2 Mðd; 1 þ l; cr2 Þ. For this reason we make D = 0. Therefore we are left with cr 2
WðrÞ ¼ Arl e 2 Uðd; 1 þ l; cr 2 Þ:
ð14Þ
2
In order to guarantee that W(r) 2 L (0, +1) we need to study the behavior of W(r) as r ? 0 which implies analysing the possible self-adjoint extensions. Now, in order to construct the self-adjoint extensions, we need to find the solution to
Ho y U ¼ ijU :
ð15Þ
in order to find N+ and N as defined in (9). Since Hyo ¼ Ho and, from (11) and (14), the solution to this equation is given by cr 2
U ðrÞ ¼ e 2 U d ; 1 þ l; cr 2 ;
ð16Þ
where d ¼ 1þl 4ic. 2 Let us consider U(d±, 1 + l, cr2) as r ? 0 [17]:
U d ; 1 þ l; cr 2 !
p 1 cl r2l : sinðp þ lpÞ Cðd lÞCð1 þ lÞ Cðd ÞCð1 lÞ
ð17Þ
R R Defining W±(r) = rlU±(r), let us find under which condition jW ðrÞj2 r dr ¼ jU ðrÞjr 1þ2l dr has a finite contribution from the near origin region. Taking (16) and (17) into account, we have
lim jU ðrÞj2 r 1þ2l ! C 1 r1þ2l þ C 2 r 12l þ C 3 r ;
ð18Þ
r!0
where C1, C2 and C3 are constants. Looking at Eq. (18), we see that W±(r) is square-integrable only when 1 < l < 1. In this case, since N+ is expanded by W+(r) only, we have that its dimension n+ = 1. The same applies to N and W(r) resulting in n = 1. Therefore, H possesses self-adjoint extensions parametrized by a unitary matrix U(1) = ein, with n 2 R ðmod 2pÞ. Thus, according to the von Neumann–Krein theory, the domain of Hyo is given by
DðHyo Þ ¼ DðHo Þ Nþ N :
ð19Þ
The essence of the self-adjoint extension approach is to extend the domain DðHo Þ to match DðHyo Þ and therefore make Ho self-adjoint. We get then
Dn ðHo Þ ¼ DðHyo Þ ¼ DðHo Þ Nþ N ;
ð20Þ
which means that, for each n, we have a possible domain for Ho . This is mathematics. The physical situation determines the value of n. The Hilbert space (20) contains vectors of the form
cr 2
UðrÞ ¼ vðrÞ þ C e 2 Uðdþ ; c; cr 2 Þ þ ein Uðd ; c; cr2 Þ ;
ð21Þ
where C is an arbitrary complex number, vð0Þ ¼ v_ ð0Þ ¼ 0 and rlv(r) 2 L2 (0, +1). The behavior of the wave functions (21) for a range of n was addressed in [18]. However, as we will see below, if we fix the physics of the problem at r = 0, there is no need for such analysis because the value of n is automatically selected.
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2533
In order to find the spectrum of (9) we consider the limit r ? 0 of W(r), that is, we substitute (17) and (21) into (14), which leads to
A
1 1 cl r 2l Cðd lÞCð1 þ lÞ CðdÞCð1 lÞ 1 ein 1 ein ¼C þ cl r 2l þ : Cðdþ lÞCð1 þ lÞ Cðd lÞCð1 þ lÞ Cðdþ ÞCð1 lÞ Cðd ÞCð1 lÞ
ð22Þ
Now, equating the coefficients of the same powers of r, we get
A 1 ein ¼C þ Cðd lÞ Cðdþ lÞ Cðd lÞ
ð23Þ
A 1 ein ¼C þ ; CðdÞ Cðdþ Þ Cðd Þ
ð24Þ
and
whose quotient leads to
Cðd lÞ ¼ CðdÞ
e
i
n 2
Cðdþ Þ i
i
n
e 2 Cðdþ lÞ
n
e2 þ Cðd Þ i
n
:
ð25Þ
þ Cðde2lÞ
The left hand side of this equation is a function of the energy E0 while its right hand side is a constant (even though it depends on the extension parameter n which is fixed by the physics of the problem). Therefore we have the equation
C C
0
0
¼ const:
1l 2
4Ec
1þl 2
4Ec
ð26Þ
As we said above we have achieved the energy levels with an arbitrary parameter n. Different values of it means inequivalent quantizations [19], that is, each physical problem selects a specific n. In what follows we give the details on how n may be determined for the generic case of a harmonic oscillator with a d-function singularity. This will lead to the applications of the next section. We are substituting the true problem H = H0 + Ushort(r) by the idealized problem, that is, H = H0 plus self-adjoint extension. So, consider the static problem of both true and idealized problem, respectively
H0 þ U short ðrÞ /true ðrÞ ¼ 0;
ð27Þ
H0 /n ðrÞ ¼ 0;
ð28Þ
where these operators are in the form given by Eq. (12). Eq. (7) becomes
U short ðrÞ ¼
2M r dðr aÞ : 2 r h
ð29Þ
At the end, the limit a = 0 is taken. So, to find out the value of n we state that
d/n =dr d/true =dr ¼ : /true ðrÞ r!a /n ðrÞ r!a
ð30Þ
As static solution for the idealized problem we take the relation (21). From the true problem above and for r 6 a, we have
( "
# ) 1 d 2 2 þ c r þ U short ðrÞ Utrue ðrÞ ¼ 0: þ ð1 þ 2lÞ 2 r dr dr d
2
ð31Þ
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Integrating this expression from 0 to a, we arrive at
Z
a
1 r 1þ2l
0
Z a d d/ ðrÞ 1þ2l dðr aÞ r 1þ2l true r /true ðrÞr 1þ2l dr: dr ¼ r dr dr a 0
ð32Þ
We omitted the integral containing the harmonic term because of the following:
Z
a
c2 r2 r1þ2l /true ðrÞdr /true ðr ¼ aÞ
Z
0
a
c2 r2 r1þ2l dr ! 0;
ð33Þ
0
as a ? 0. So, from (32), we get
ðd/true ðrÞ=dr Þjr!a r : /true ðr ! aÞ a
ð34Þ
Now, for the idealized problem, from (21) and from the expansion for U given by (17), we have
d/n =dr ¼h /n ðrÞ r!a
2lcl a12l 1
Cðbþ ÞCð1þlÞ
h
in
1
Cðbþ ÞCð1lÞ
þ Cðdþ eÞCð1lÞ
i
i h i: 2 l 2 l CðdðþcaÞCÞð1lÞ þ ein Cðb Þ1Cð1þlÞ CðdðcaÞCÞð1lÞ
ð35Þ
Inserting this in (34) we arrive at
h
2lcl a12l 1 Cðbþ ÞCð1þlÞ
h
1
Cðbþ ÞCð1lÞ
in
þ Cðdþ eÞCð1lÞ
i
r
i h i¼ : 2 l 2 l a CðdðþcaÞCÞð1lÞ þ ein Cðb Þ1Cð1þlÞ CðdðcaÞCÞð1lÞ
ð36Þ
This relation selects an approximate value for the self-adjoint extension parameter in terms of the physics of the problem. In order to determine the energy levels we substitute (14) in (21) and compute dUdrðrÞ =UðrÞ. Then, 2lcl a12l CðdÞCð1lÞ
h
1
CðbÞCð1þlÞ
2lcl a12l
h
1
Cðbþ ÞCð1lÞ
in
þ Cðdþ eÞCð1lÞ
i
i¼h i h i: cl a2l ðca2 Þl ðca2 Þl 1 1 in CðdÞ Cð1lÞ Cðbþ ÞCð1þlÞ Cðdþ ÞCð1lÞ þ e Cðb ÞCð1þlÞ Cðd ÞCð1lÞ
ð37Þ
Taken into account (36) the spectrum is given entirely by the physics of the problem, instead of the extension parameter:
h
2lcl a12l CðdÞCð1lÞ 1 CðbÞCð1þlÞ
r
i¼ : cl a2l a CðdÞ Cð1lÞ
ð38Þ
Rearranging this equation, we have
0 4Ec cl ð2l þ 1Þ Cð1 þ lÞ 2l
¼ a : 1l E0 r Cð1 lÞ C 2 4c
C
1þl 2
Making a = 0, the relation (38) yields C gives us the spectrum
E0 ¼ 2cð2n þ jlj þ 1Þ:
1þl 2
ð39Þ
0 0 4Ec ¼ 1, for l = jlj, and C 1l 4Ec ¼ 1, for l = jlj, which 2
ð40Þ
These are the well known energy levels of the two-dimensional harmonic oscillator with an important difference: due to the singular nature of the problem the range of l is given by 1 < l < 1. If we had imposed that the eigenfunction must be regular at the origin (U(r) v(r) 0) we would achieve the same spectrum (40), but l could take any integer value [11,12]. In this sense, the self-adjoint extension prevents us from obtaining a spectrum incompatible with the irregular nature of the Hamiltonian [20] when we have a short-ranged potential as Eq. (7). Remember that the true boundary condition is that the wave function must be square-integrable through all space and it does not matter if it is irregular at the origin [14,16].
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
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3. Applications 3.1. Landau levels for a particle in the elastic continuum with a disclination In this subsection we study the dynamics of a charged particle in a continuous elastic medium with a topological defect, disclination, in the presence of uniform magnetic field ~ B. This problem was addressed in [11] without considering the coupling with the defect, which is done here via a d interaction associated to curvature. The continuous elastic medium with the topological defect is conceptually obtained by removing a wedge of material and is described by a non-Euclidean metric similar to (1): 2
2
ds ¼ dz þ dq2 þ a2 q2 dh2 :
ð41Þ
Here, a = 1 + k/2p and k is the angle that defines the wedge. Following Ref. [11], we consider the gauge condition A/ ¼ B2qa, which corresponds to a uniform magnetic field along the z axis. The total Hamiltonian for the system is given by
" # 2 1 @ h @ 1 @2 ihqB @ q2 B2 q2 þ 2 þ 2 2 þ H¼ ; q 2 @q 2a m @/ 2m q @ q a q @/ 8ma2
ð42Þ
where we have already factored out the motion along the z axis. The Schrödinger equation can be written as
"
# 2 d 1 d m2 b2 q2 þ þ UðqÞ ¼ E0 UðqÞ; dq2 q dq q2 4
ð43Þ
2 2
where m ¼ a‘ ; b2 ¼ hq2 aB2 ; E0 ¼ 2mE þ hqB‘ a2 and E is the total energy of the system. Following Allen et al. [14], h2 the particle is coupled to the defect via the Ricci scalar R of the background metric which describes the medium:
U short ¼ gR:
ð44Þ
Here, g is a coupling constant, which has no effect in the eigenvalues and R is essentially (3) up to a factor of two. The d-function singularity of Ushort affects the boundary condition and, consequently, the range of the quantum numbers. Evoking the results from the previous section and considering a negatively charged particle (q = jqj), the Landau levels are given by
E¼
x h ð2n þ m þ jmj þ 1Þ; 2a
ð45Þ
where 1 < m < 1 (see the previous section) and x ¼ jqjB is the cyclotron frequency (we take the speed m of light c = 1). Now we follow Ref. [11] and add the self-interaction [21] on the particle by adding the potential
U¼ Here,
q2 jðaÞ : 4p q
ð46Þ
is the dielectric constant of the material and
jðaÞ ¼ 2
Z 0
1
a1 cothða1 xÞ cothðxÞ sinhðxÞ
dx
ð47Þ
measures the distortion caused by the defect in the medium. The Schrödinger equation is now written as
"
# 2 d 1 d m2 b 2 2 þ þ x q þ UðqÞ ¼ E0 UðqÞ; dq2 q dq q2 q 2
ð48Þ
jðaÞ where b ¼ mq and x ¼ 2b. In Ref. [11] the eigenfunction was factored taking into account that 2p h2 U(q) ? 0 as q ? 0 and q ? 1 [11,22]:
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C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541 q2
1
WðqÞ ¼ C qtjmj ex 2 Fðx2 qÞ:
ð49Þ
This is the case when the core of the defect is impenetrable. When it is not, the Hamiltonian is singular because of the Coulomb interaction. Then we write U(q) in this case as q2
1
WðqÞ ¼ C qm ex 2 Fðx2 qÞ;
ð50Þ 1 4
which is in agreement with the previous section. Putting r ¼ x q and following [11], F must obey 2
r
d F dr
2
þ ða 2r 2 Þ
dF þ ðgr þ bÞF ¼ 0; dr
ð51Þ
where
a ¼ 2m þ 1;
ð52Þ
and
g¼
E0
2ðm þ 1Þ:
x
ð53Þ
In terms of a power series, 1 X
FðrÞ ¼
C k rk ;
ð54Þ
k¼0
with the recurrence relation
C kþ2 ¼
b ðg 2kÞ C kþ1 Ck: ðk þ 2Þðk þ a þ 1Þ ðk þ 2Þðk þ a þ 1Þ
ð55Þ
We obtain special exact solutions if
g ¼ 2n
ð56Þ
C nþ1 ¼ 0;
ð57Þ
and
simultaneously, where n is a maximum value of k. Then, the eigenvalues are
E¼
x h ðn þ 2mn þ 1Þ: 2a
ð58Þ
The index n on m stands for the fact that (55) limits the value that m can admit (which depends on which n we truncate the series (54)). From (55)–(57), with C0 = 1, we have 2
m1 ¼
b 1 ; 4 2
ð59Þ
for n = 1. Then, remembering that 1 < m < 1, b must satisfy 2
0 6 b < 6:
ð60Þ
For n = 2,
m2 ¼
3 2 1 ðb 2Þ 20 2
ð61Þ
and 2
0 6 b < 12:
ð62Þ
This procedure may be used to find the corresponding relations for n P 3. The results in this subsection show a reduced degeneracy of the Landau levels, as compared to the previous result [11], due to the condition 1 < m < 1, which appears naturally in the self-adjoint exten-
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2537
sion procedure. Notice that, if we choose to study the problem of Landau levels on a cone, we get essentially the same answer as above. The difference comes in the addition of the contribution to the energy due to the free motion of the particle along the z axis, which should be done here and 2 a2 Þ not in the conical surface case. Also, m needs to be redefined to m2 ¼ a‘ 2 ð1 , in order to have the 4a2 mean curvature contribution (4) accounted for in the Hamiltonian. 3.2. Calogero model Consider the rational Calogero model which describes a quantum system with N identical particles interacting with each other through a long-range inverse-square, a Coulomb and a harmonic interactions on the line[23]. The Hamiltonian for this system is given by
" # N X X a2 1=4 X2 @2 b 2 ffi; H¼ þ ðxi xj Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 P 2 8 @x xi x j i i
ð63Þ
where a and X are constants, xi is the coordinate of the ith particle and units have been chosen such that 2m/⁄2 = 1. Writing the wave function of this Hamiltonian as
W¼
Y
ðxi xj Þaþ1=2 /ðrÞPk ðxÞ;
ð64Þ
i
where x (x1, x2, . . . , xN),
r2 ¼
1X ðxi xj Þ2 N i
ð65Þ
and Pk is a homogeneous, translationally invariant, polynomial of degree k which satisfies the equation
" # N X X @2 1 @ @ P k ¼ 0: þ 2ða þ 1=2Þ ðxi xj Þ @xi @xj @x2i i
ð66Þ
Inserting (64) into HW = Ew and using (65) and (66), we arrive at 2
d /ðrÞ dr
2
0
ð1 þ 2lÞ
1 d/ðrÞ b /ðrÞ þ x2 r 2 /ðrÞ ¼ E/ðrÞ; r dr r
ð67Þ
0
where x2 ¼ 18 X2 N, b ¼ pbffiffiNffi and
1 2
1 2
l ¼ k þ ðN 3Þ þ NðN 1Þða þ 1=2Þ: x2 r2
In the next step, we put /ðrÞ ¼ e 2
z
d F 2
dz
þ ðc 2z2 Þ
2
1
ð68Þ 1
Fðx2 rÞ and z ¼ x2 r. Then F obeys
dF 0 þ ðgz þ b ÞF ¼ 0; dz
ð69Þ
where g ¼ xE 2ðl þ 1Þ and c = 2l + 1. So, we can evoke the results from the last subsection such that the eigenvalues are given by
E ¼ 2xðn þ ln þ 1Þ;
ð70Þ
where 1 < ln < 1 and l has been rewritten as ln due to its dependence on n. In this case, we have 2
ð71Þ
2
ð72Þ
0 6 b < 6N; for n = 1,
0 6 b < 12N;
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for n = 2, and so on. Note that we have one more condition to be satisfied because of 1 < ln < 1:
1 1 1 < k þ ðN 3Þ þ NðN 1Þða þ 1=2Þ < 1: 2 2
ð73Þ
The analysis of this inequality is the same as in the case b = 0 and can be found in [23]. The self-adjoint extension of (63) was addressed in [23], for b = 0 and x – 0 (harmonic), and in [24], for b – 0 and x = 0 (Coulomb), cases separately. In both cases the result is given in terms of the self-adjoint extension parameter. Here, we consider both cases together and use the divergence of the Coulomb interaction at the origin as a way to fix the self-adjoint extension parameter. 3.3. Charged particle moving on the noncommutative plane in the presence of a magnetic flux tube The study of noncommutative theories was proposed initially by Heisenberg and Snyder [25] and has gained strong impulse recently due to its natural appearance in distinct arenas of physics such as string theory with a constant background magnetic field in the presence of D-branes, and quantum Hall effect, where the noncommutativity is induced among the coordinates of the particle, in a plane, due to the presence of a strong magnetic field. It is opportune to mention here that this noncommutativity, in the first case mentioned above, was eliminated constructing a mechanical system which reproduces classical dynamics of the string [26]. Besides their origin in string theories and branes, noncommutative (NC) field theories have been studied extensively in other branches of physics [27–30]. There are two major trends involving noncommutative theories. One is the study of noncommutative spaces, where the noncommutativity is defined by the following relation among the coordinate operators
h
i x0i ; x0j ¼ ihhij I;
ð74Þ
where hij is a antisymmetric parameter. In the second one, which we use here, the noncommutativity of field operators can be imposed by a small deformation in the canonical structure of the field theories [31]. This leads to a tiny violation of the microcausality principle. Thus, in this trend, the noncommutative space can be realized by a generalized Heisenberg operator algebra [31] given by
h
i x0i ; x0j ¼ ihhij I; h i x0i ; p0j ¼ ihdij I; h i p0i ; p0j ¼ ihbij I;
ð75Þ
where hij, bij are the parameters of the noncommutativity. This algebra can be constructed by changing the standard product of the classical observable by the star product (Weyl-Moyal):
i ðf gÞðxÞ ¼ exp hij @ i @ j f ðxÞgðyÞjy¼x : 2
ð76Þ
An important fact is that there is a mapping between the quantum dynamics of a particle in such a noncommutative space and the quantum dynamics in ordinary space given by the following general transformations [32]:
1 ~ x ~ x0 ¼ ~ p ^~ h; 2 1 ~ p0 ¼ ~ x ^~ b; pþ ~ 2
ð77Þ
C. Filgueiras et al. / Annals of Physics 325 (2010) 2529–2541
2539
where ~ x0 and ~ p0 satisfy (75). Note that ~ x and ~ p satisfy the usual commutation relations
½xi ; xj ¼ 0; ½xi ; pj ¼ ihdij I;
ð78Þ
½pi ; pj ¼ 0: In this section we study the bound state noncommutative Aharonov–Bohm effect, which provides us with short-ranged potentials if the particle has spin. As usual, we consider the Hamiltonian of the system having the same form as in the commutative case but with the commutation relations given by Eq. (75). Then, in the non-relativistic limit, we have
H0 ð~ x0 ; ~ p0 Þ ¼
~ p02 : 2M
ð79Þ
Let us consider the Aharonov–Bohm problem, that is, the coupling [13]
eA0/ ¼
/ r0
ð80Þ
and
eB0 ¼ /
dðr 0 Þ ; r0
ð81Þ
where / is the magnetic flux. With these relations in hands, the Hamiltonian (79) is rewritten as
H0 ¼
" # 1 /2 2i/ @ /s dðr0 Þ ~ þ p02 þ 02 02 ; 0 2M r @u 2M r 0 r
ð82Þ
where s = +1(s = 1) for spin up (down). Considering the simple case b1 = b2 = h1 = h2 = 0, b3 = b and h3 = h, the relation (77) with the conditions h 1 and b 1 yields
" # 1 1 @ @ 1 @2 ib @ b2 r2 þ 2 þ r H¼ þ 2M r @r @r r @ u2 2M @ u 8M " # 1 /2 2/ i @ br /s dðr 0 Þ þ þ þ 0 : 2M r2 r r @u 2 2M r 0
ð83Þ ð84Þ
The 1/r0 term in this equation is a Coulomb term and its expansion to first order in h was obtained in [33] and it is given by 1/r0 1/r + Lzh/4r3, where Lz ¼ i @@u is the angular momentum in the z direction with eigenvalue denoted by m. Now, we have to deal with the delta potential present in (81), which is not a simple task in the context of a noncommutativity theory. Since the delta potentials are idealizations for a real physical situation consisting in quantum mechanics in the presence of a short-ranged potential, we consider here the expression for it given by [34]
dh ¼
1
ph
r02
e h
1
ph
r2
e h :
ð85Þ
So, we replace the last term in (84) by the short-ranged potential
U short ðrÞ ¼
r 1 rh2 1 Lz h ; þ e r 4r 3 2M ph
ð86Þ
where r = /s. In what follows, we will consider three different situations. First, if h = 0 we recover the harmonic oscillator problem in the presence of a short-ranged potential
U short ðrÞ ¼
/s dðrÞ : 2M r
ð87Þ
Using the above results, the eigenvalues will be given by
E¼
b ð2n þ jm þ /j m / þ 1Þ; 2M
ð88Þ
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for 1 < m + / < 1. Second, for h – 0 and m – 0, besides Ushort(r) given by (86) we have the potential 1/ r4 which implies that the eigenfunction does not have a closed form. But, if we ignore this term, the relation (89) below, with l = m + /, gives a first approximation for the problem, since h 1. Like Eq. (39), we get
" # 0 12l 4Ec l l Cð1 þ lÞ 2plh
¼c þh ; 0 Cð1 lÞ rR C 1l 4Ec 2
C
1þl 2
ð89Þ
pffiffiffi which is exact for the case m = 0 (third situation). In all cases, we considered that a ¼ h, which gives R R 2 2 a 1 R ¼ p1ffiffih 0 er =h dr ¼ 0 ex dx ¼ 0:7468241330. Since h 1, this spectrum might not be much different from the oscillator levels (88). 4. Conclusions The easiest way of dealing with singularities in quantum mechanics is by imposing that the eigenfunction vanishes at the singularity. Although convenient, this does not necessarily give the best description (or even the correct one) of the physical phenomenon studied. Care must be taken to insure that the Hamiltonian is self-adjoint. This can be achieved by extending the domain of H to equal that of H . By doing this, a family of boundary conditions might appear, including the one that requires the eigenfunction to vanish at the singularity. With this in hands, physics itself determines the appropriate boundary condition. In this article, we have done this for both a d-function and a Coulomb singularity. We focused on the two-dimensional harmonic oscillator with a singularity at the origin for its broad application throughout physics. We applied our results to the specific cases of Landau levels in the elastic continuum plus a topological defect, the Calogero model and the quantum dynamics of a particle on the noncommutative plane. Acknowledgements This work was partially supported by CNPq and CAPES (NANOBIOTEC). E.O.S. thanks the Brazilian Ministry of Science and Technology for a post-doctoral fellowship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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