Bound states for a Coulomb-type potential induced by the interaction between a moving electric quadrupole moment and a magnetic field
aa r X i v : . [ qu a n t - ph ] M a y Bound states for a Coulomb-type potential induced by theinteraction between a moving electric quadrupole moment and amagnetic field
K. Bakke ∗ Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, PB, Brazil.
Abstract
We discuss the arising of bound states solutions of the Schr¨odinger equation due to the presenceof a Coulomb-type potential induced by the interaction between a moving electric quadrupolemoment and a magnetic field. Furthermore, we study the influence of the Coulomb-type potentialon the harmonic oscillator by showing a quantum effect characterized by the dependence of theangular frequency on the quantum numbers of the system, whose meaning is that not all values ofthe angular frequency are allowed.
PACS numbers: 03.65.Ge, 03.65.VfKeywords: electric quadrupole moment, Coulomb-type potential, bound states, harmonic oscillator ∗ Electronic address: kbakke@fisica.ufpb.br . INTRODUCTION The interaction between electric and magnetic fields and multipole moments has attracteda great deal of studies, such as the arising of geometric quantum phases [1–11], the holonomicquantum computation [12–14] and the Landau quantization [15–17]. In particular, the elec-tric quadrupole moment has been investigated in systems of strongly magnetized Rydbergatoms [18], since these systems are characterized by a large permanent electric quadrupolemoment. The electric quadrupole moment has also been investigated in molecular systems[19, 20] and atomic systems [21, 22]. Besides, recent studies of the interaction between amoving electric quadrupole moment and external fields [9, 10] have shown a difference be-tween the field configuration that yields the arising of geometric phases for an electric charge[1, 2, 23], an electric dipole moment [3, 8] and a moving electric quadrupole moment. Inshort, this difference arises from the structure of the gauge symmetry involving each system,for instance, the Aharonov-Bohm effect [23] and the electric Aharonov-Bohm effect [1, 2]are based on the gauge symmetry called U (1) gauge group, the He-McKellar-Wilkens effect[3] and the scalar Aharonov-Bohm effect for a neutral particle with a permanent electricdipole moment [14] are based on the SU (2) gauge symmetry, while the electric quadrupolemoment system can be described by U (1) gauge symmetry [17]. Moreover, the field configu-ration that gives rise to the appearance of geometric phases for a moving electric quadrupolemoment depends on the structure of the electric quadrupole tensor [9, 10]. This dependenceof the structure of the electric quadrupole tensor has also been pointed out in Refs. [10, 17]by dealing with the Landau quantization and the confinement analogous to a quantum dot.In this work, we explore this dependence of the field configuration that interacts with theelectric quadrupole moment on the structure of the electric quadrupole tensor. We considera moving electric quadrupole moment and study the arising of bound states solutions of theSchr¨odinger equation due to the presence of a Coulomb-type potential induced by the inter-action between a moving electric quadrupole moment and a magnetic field. Furthermore, westudy the influence of the Coulomb-type potential on the harmonic oscillator by showing aquantum effect characterized by the dependence of the harmonic oscillator frequency on thequantum numbers of the system, which means that not all values of the angular frequencyare allowed.The structure of this paper is: in section II, we start by making a brief review of the2uantum dynamics of a moving electric quadrupole moment interacting with external fields.In the following, we obtain the energy levels corresponding to having a quantum particlesubject to an attractive Coulomb-type potential induced by the interaction between a movingelectric quadrupole moment and a magnetic field; in section III, we study the influence of theCoulomb-type potential on the harmonic oscillator; in section IV, we present our conclusions. II. COULOMB-TYPE POTENTIAL INDUCED BY THE INTERACTION BE-TWEEN A MOVING ELECTRIC QUADRUPOLE MOMENT AND A MAGNETICFIELD
In this section, we discuss the arising of bound states solutions of the Schr¨odinger equationthat describes the quantum dynamics of a moving electric quadrupole moment interactingwith a magnetic field. We show that the interaction between a moving electric quadrupolemoment and a magnetic field can give rise to a Coulomb-type potential, where both scat-tering and bound state solutions of the Schr¨odinger equation can be obtained. We startwith the description of the quantum dynamics of a moving electric quadrupole moment in-teracting with magnetic and electric fields as proposed in Ref. [9]. By following Ref. [9],we consider an electric quadrupole moment as a scalar particle, then, the potential energyof a multipole expansion in the classical dynamics of an electric quadrupole moment (in therest frame of the particle) is given by U = q Φ − ~d · ~ ∇ Φ + P i,j Q ij ∂ i ∂ j Φ · · · , where q isthe electric charge, ~d is the electric dipole moment, Q ij is the electric quadrupole moment(the tensor Q ij is a symmetric and traceless tensor) and Φ is the electric potential. Byconsidering q = 0, ~d = 0 and ~E = − ~ ∇ Φ ( ~E is the electric field), the potential energy can berewritten as U = − P i,j Q ij ∂ i E j .If we consider a moving particle (the electric quadrupole moment), we have that theparticle interacts with a different electric field ~E ′ . Thereby, the Lagrangian function of thissystem in the frame of the moving particle is given by L = mv − P ij Q ij ∂ i E ′ j . By applyingthe Lorentz transformation of the electromagnetic field, we have that the electric field ~E ′ must be replaced by ~E ′ = ~E + c ~v × ~B up to O (cid:16) v c (cid:17) . Now, we have that the field ~E and ~B are the electric and magnetic fields in the laboratory frame, respectively. In this way, the3agrangian function becomes L = 12 m v + ~Q · ~E − c ~v · (cid:16) ~Q × ~B (cid:17) , (1)where we defined the components of the vector ~Q by Q i = P j Q ij ∂ j ( Q ij is a symmetricand traceless tensor) as done in Ref. [9]. With the canonical momentum being ~p = m ~v − c (cid:16) ~Q × ~B (cid:17) , the classical Hamiltonian of this system becomes H = 12 m (cid:20) ~p + 1 c ( ~Q × ~B ) (cid:21) − ~Q · ~E. (2)In order to proceed with the quantization of the Hamiltonian, we replace the canonicalmomentum ~p by the Hermitian operator ˆ p = − i ~ ~ ∇ . In this way, the quantum dynamics ofa moving electric quadrupole moment can be described by the Schr¨odinger equation [9, 10] i ~ ∂ψ∂t = 12 m (cid:20) ˆ p + 1 c ( ~Q × ~B ) (cid:21) ψ − ~Q · ~E ψ. (3)From now on, we work with the units ~ = c = 1. Thereby, let us consider the non-nullcomponents of the tensor Q ij being Q ρz = Q zρ = Q, (4)where Q is a constant ( Q > ~B = λ m ρ ρ, (5)where λ m is a magnetic charge density, ρ = p x + y and ˆ ρ is an unit vector in the radialdirection. The field configuration given in Eq. (5) was proposed in Ref. [15] in orderto study the possibility of achieving the Landau quantization for neutral particles with apermanent electric dipole moment. This field configuration proposed in Ref. [15] is basedon the He-McKellar-Wilkens effect [3], where the wave function of a neutral particle witha permanent electric dipole moment acquires a geometric phase when the neutral particleencircles a line of magnetic monopoles. In recent years, this radial magnetic field has beenachieved through noninertial effects [16]. In this work, we show that the field configuration(5) cannot yield the Landau quantization for a moving magnetic quadrupole moment, butcan give rise to bound states solutions analogous to having a quantum particle confined inthe Coulomb potential. 4rom the interaction between the field configuration given in Eq. (5) and the magneticquadrupole moment (4), then, the Schr¨odinger equation in cylindrical coordinates becomes i ∂ψ∂t = − m (cid:20) ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ ∂ ∂ϕ + ∂ ∂z (cid:21) ψ − i Qλ m mρ ∂ψ∂ϕ + Q λ m m ψ. (6)Note that the operators ˆ p z = − i∂ z and ˆ L z = − i∂ ϕ commute with the Hamiltonian of theright-hand side of (6), then, a particular solution of Eq. (6) can be written in terms of theeigenvalues of the operator ˆ p z , and ˆ L z : ψ ( t, ρ, ϕ, z ) = e − i E t e i l ϕ e ikz R ( ρ ) , (7)where l = 0 , ± , ± , . . . , k is a constant, and R ( ρ ) is a function of the radial coordinate.Thereby, substituting the solution (7) into Eq. (6), we obtain the following radial equation: R ′′ + 1 ρ R ′ − l ρ R − δρ R + ζ R = 0 , (8)where we have defined the following parameters ζ = 2 m E − k − Q λ m δ = Q λ m l. Now, let us discuss the asymptotic behavior of the radial equation (8). For ρ → ∞ , wehave R ′′ + ζ R = 0 . (10)Therefore, we can find either scattering states (cid:0) R ∼ = e iζρ (cid:1) or bound states ( R ∼ = e − τρ ) if weconsider ζ = − τ [24–26]. Note that the fourth term on the left-hand side of Eq. (8) playsthe role of a Coulomb-like potential. This term stems from the interaction between themagnetic field (5) and the electric quadrupole moment defined in Eq. (4). Our intention isto obtain bound state solutions, then, the term proportional to δ behaves like an attractivepotential by considering the negative values of δ , that is, δ = − | δ | [24–26]. This occurs byconsidering either λ m > l < λ m < l > Q being always apositive number). Observe that these condition forbids the quantum number l to have thevalue l = 0, that is, for l = 0 there are no bound states solutions. Thereby, we rewrite Eq.(8) in the form: R ′′ + 1 ρ R ′ − l ρ R + | δ | ρ R − τ R = 0 . (11)5ext, by making a change of variables given by r = 2 τ ρ , we have in Eq. (11): R ′′ + 1 r R ′ − l r R + | δ | τ r R − R = 0 . (12)We can obtain a regular solution for the second-order differential equation (12) at the originby writing R ( r ) = e − r r | l | F ( r ) . (13)Substituting (13) into (12), we obtain the following second-order differential equation r F ′′ + [2 | l | + 1 − r ] F ′ + (cid:20) | δ | τ − | l | − (cid:21) F = 0 , (14)which corresponds to the Kummer equation or the confluent hypergeometric equation [27].In this way, the function F ( r ) corresponds to the Kummer function of first kind which isdefined as F ( r ) = F (cid:18) | l | + 12 − | δ | τ , | l | + 1 , r (cid:19) . (15)A finite radial wave function can be obtained by imposing the condition where the con-fluent hypergeometric series becomes a polynomial of degree n (where n = 0 , , , . . . ).This occurs when | l | + − | δ | τ = − n [27, 28]. In this way, from Eq. (9) we can take( − τ ) = 2 m E − k − Q λ m and the parameter δ , thus, we obtain E n, l = − m ( Q λ m l ) [ n + | l | + 1 / + k m + Q λ m m . (16)Equation (16) correspond to the energy levels for bound states yielded by a Coulomb-like potential induced by the interaction between the electric quadrupole moment given inEq. (4) and the magnetic field defined in Eq. (5). Observe that the energy levels (16) aredefined for l = 0 as we have discussed above. For l = 0, there are no bound states solutionsbecause the parameter δ defined in Eq. (9), which gives rise to a term that plays the role ofa Coulomb-like potential, vanishes. Besides, the energy levels of the bound states (16) couldnot be achieved if we have considered the non-null components of the tensor Q ij being, forinstance, Q ρϕ = Q ϕρ = 0 and the field given in Eq. (5). In this case, the term (cid:16) ~Q × ~B (cid:17) · ~p present in Eq. (3) does not induce a Coulomb-type term, thus, no bound states solutionsanalogous to having a quantum particle confined to a Coulomb potential can be achieved.Hence, the arising of the Coulomb-type potential in Eq. (6) from the interaction between6he magnetic field (5) and the electric quadrupole moment depends on the structure of thetensor Q ij . An analogous case has been pointed out in Ref. [17] for the Landau quantizationfor a moving electric quadrupole moment.Further, it should be interesting to explore the quantum dynamics of a moving electricquadrupole moment interacting with external field in the presence of linear topological de-fects [29]. As shown in Ref. [29] that both curvature and torsion can modify electric andmagnetic fields, in turn, one should expect that the spectrum of energy (16) can also bemodified by the presence of defects. III. INFLUENCE OF THE COULOMB-LIKE POTENTIAL ON THE HARMONICOSCILLATOR
In the following, we discuss the influence of the Coulomb-like potential induced the in-teraction between the electric quadrupole moment given in Eq. (4) and the magnetic fieldgiven in Eq. (5) on the two-dimensional harmonic oscillator V ( ρ ) = mωρ . By followingthe steps from Eq. (6) to Eq. (9), we have the following radial equation R ′′ + 1 ρ R ′ − l ρ R − δρ R − m ω ρ R + ζ R = 0 , (17)Now, we do not need to consider ζ = − τ as in the previous section, thus, we canconsider a general case involving the parameter ζ defined in Eq. (9). Again, the presenceof the Coulomb-type term in Eq. (17) imposes that the angular quantum number is definedfor l = 0. Thereby, let us consider a new change of variables given by: ξ = √ mω ρ . Thus,we have R ′′ + 1 ξ R ′ − l ξ R + αξ R − ξ R + ζ mω R = 0 , (18)where we have defined the following parameter α = Q λ m l √ mω = δ √ mω . (19)Hence, a regular radial wave function at the origin can be obtained by writing the solutionof the second order differential equation (18) in the form: R ( ξ ) = e − ξ ξ | l | H ( ξ ) . (20)7ubstituting (20) into (18), we obtain H ′′ + (cid:20) | l | + 1 ξ − ξ (cid:21) H ′ + (cid:20) g + αξ (cid:21) H = 0 , (21)where g = ζ mω − − | l | . The function H ( ξ ), which is solution of the second order differentialequation (21), is known as the Heun biconfluent function [24, 25, 30–32]: H ( ξ ) = H (cid:20) | l | , , ζ mω , α, ξ (cid:21) . (22)In order to proceed with our discussion about bound states solutions, let us use theFrobenius method [33, 34]. Thereby, the solution of Eq. (22) can be written as a powerseries expansion around the origin: H ( ξ ) = ∞ X j =0 a j ξ j . (23)Substituting the series (23) into (22), we obtain the recurrence relation: a j +2 = α ( j + 2) ( j + 1 + θ ) a j +1 − ( g − j )( j + 2) ( j + 1 + θ ) a j , (24)where θ = 2 | l | + 1. By starting with a = 1 and using the relation (24), we can calculateother coefficients of the power series expansion (23). For instance, a = αθ = Q λ m l √ mω (2 | l | + 1) ; a = α θ (1 + θ ) − g θ ) (25)= ( Q λ m l ) mω (2 | l | + 1) (2 | l | + 2) − g | l | + 2) . Bound state solutions correspond to finite solutions, therefore, we can obtain boundstate solution by imposing that the power series expansion (23) or the Heun Biconfluentseries becomes a polynomial of degree n . Through the expression (24), we can see that thepower series expansion (23) becomes a polynomial of degree n if we impose the conditions[24, 25, 31–33]: g = 2 n and a n +1 = 0 , (26)where n = 1 , , , . . . , and g = ζ mω − | l | −
2. From the condition g = 2 n , we can obtain theexpression for the energy levels for bound states: E n, l = ω [ n + | l | + 1] + Q λ m m + k m . (27)8quation (27) is the energy levels of the two-dimensional harmonic oscillator under theinfluence of the Coulomb-type potential induced by the interaction between the electricquadrupole moment given in Eq. (4) and the magnetic field given in Eq. (5). Note that theinfluence of the Coulomb-like potential makes that the ground state to be defined by thequantum number n = 1 instead of the quantum number n = 0 as it is well-known. Moreover,we have that the spectrum of energy (27) is defined for all values of the quantum number l that differ from zero, that is, for l = 0. In the case l = 0, there is no influence of theCoulomb-like potential induced by the interaction between the electric quadrupole momentand the magnetic field because the parameter α vanishes (as we can see in Eqs. (19) and (9)).However, at first glance, the energy levels (27) do not depend on the parameter δ defined inEq. (9), which gives rise to a Coulomb-type potential. In this way, we would not have thecomplete information of the spectrum of energy. However, observe that we have not analysedthe condition a n +1 = 0 given in Eq. (26) yet. We have that this condition yields an expressioninvolving specific values of the angular frequency and the parameter δ , that is, it yields anexpression involving the angular frequency and the quantum numbers { n, l } [24, 31, 32]. Asimilar analysis has been done in Ref. [24], where a relation involving the harmonic oscillatorfrequency, the Lorentz symmetry-breaking parameters and the total angular momentumquantum number is obtained. In Ref. [32], a relation involving a coupling constant of aCoulomb-like potential, the cyclotron frequency and the total angular momentum quantumnumber in semiconductors threaded by a dislocation density is obtained. Moreover, a relationinvolving the mass of a relativistic particle, a scalar potential coupling constant and the totalangular momentum quantum number is achieved in Ref. [31].Thereby, in order to obtain a relation between the harmonic oscillator frequency and theCoulomb-type potential induced by the interaction between the electric quadrupole momentand a radial magnetic field, let us assume that the angular frequency ω can be adjustedin such a way that the condition a n +1 = 0 is satisfied. This means that not all values ofthe angular frequency ω are allowed, but some specific values of ω which depend on thequantum numbers { n, l } , thus, we label ω = ω n, l . In this way, the conditions established inEq. (26) are satisfied and a polynomial solution for the function H ( ζ ) given in Eq. (23) isobtained [31]. As an example, let us consider n = 1, which corresponds to the ground state,and analyse the condition a n +1 = 0. For n = 1, we have a = 0. The condition a = 0, thus,9ields ω , l = Q λ m l m (2 | l | + 1) . (28)In this way, the general expression for the energy levels (27) is given by: E n, l = ω n, l [ n + | l | + 1] + k m . (29)Hence, we have seen in Eq. (29) that the effects of the Coulomb-like potential inducedby the interaction between the magnetic field and the electric quadrupole moment on thespectrum of energy of the harmonic oscillator corresponds to a change of the energy levels,where the ground state is defined by the quantum number n = 1 and the angular frequencydepends on the quantum numbers { n, l } . This dependence of the cyclotron frequency on thequantum numbers { n, l } means that not all values of the cyclotron frequency are allowed,but a discrete set of values for the cyclotron frequency [24, 25, 31–33]. IV. CONCLUSIONS
We have seen that bound states solutions for the Schr¨odinger equation arise from theinteraction between a moving electric quadrupole moment and an external magnetic field.We have shown that the interaction between a moving electric quadrupole moment and aradial magnetic field can induce a Coulomb-like potential, where the spectrum of energy ofthe bound state solutions are defined for all values of the quantum number l that differsfrom zero. For l = 0, the term which plays the role of the attractive Coulomb-type potentialvanishes and no bound states exist.Furthermore, we have studied the influence of the Coulomb-like potential induced by theinteraction between a moving electric quadrupole moment and a radial magnetic field on theharmonic oscillator potential and shown that the ground state of the harmonic oscillator isdefined by the quantum number n = 1 instead of the well-known quantum number n = 0.Moreover, we have shown that the spectrum of energy of the harmonic oscillator is definedfor all values of the quantum number l that differ from zero, that is, for l = 0. For l = 0,there is no influence of the Coulomb-like potential on the harmonic oscillator. Other effect ofthe Coulomb-like potential induced by the interaction between a moving electric quadrupolemoment and a radial magnetic field on the harmonic oscillator is the dependence of the10ngular frequency on the quantum numbers { n, l } , which means that not all values of thecyclotron frequency are allowed. As example, we have calculated the angular frequency ofthe ground states n = 1. Acknowledgments
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