- Email: [email protected]
Aharonov–Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment
Accepted Manuscript Aharonov–Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment I.C. Fonseca, K. Bakke PII: DOI: Reference:
S0003-4916(15)00362-0 http://dx.doi.org/10.1016/j.aop.2015.09.027 YAPHY 66972
To appear in:
Annals of Physics
Received date: 25 September 2014 Accepted date: 22 September 2015 Please cite this article as: I.C. Fonseca, K. Bakke, Aharonov–Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment, Annals of Physics (2015), http://dx.doi.org/10.1016/j.aop.2015.09.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Research Highlights
Highlights
• Scalar Aharonov-Bohm effect for a particle possessing a magnetic quadrupole moment • Aharonov-Anandan quantum phase for a particle with a magnetic quadrupole moment • Dependence of the energy levels on the Aharonov-Anandan quantum phase • Landau quantization associated with a particle possessing a magnetic quadrupole moment
1
*Manuscript Click here to view linked References
Aharonov-Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment I. C. Fonseca and K. Bakke∗ Departamento de F´ısica, Universidade Federal da Para´ıba, Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, PB, Brazil.
Abstract The arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment is investigated. It is shown that an Aharonov-Anandan quantum phase [Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987)] can be obtained in the quantum dynamics of a moving particle with a magnetic quadrupole moment. In particular, it is obtained an analogue of the scalar Aharonov-Bohm effect for a neutral particle [J. Anandan, Phys. Lett. A 138, 347 (1989)]. Besides, by confining the quantum particle to a hard-wall confining potential, the dependence of the energy levels on the geometric quantum phase is discussed and, as a consequence, persistent currents can arise from this dependence. Finally, an analogue of the Landau quantization is discussed. PACS numbers: 03.65.Vf, 14.80.Hv, 03.65.Ge Keywords: magnetic quadrupole moment, Aharonov-Anandan phase, scalar Aharonov-Bohm effect, geometric phase, persistent currents, Landau quantization
∗
Electronic address: [email protected]
1
I.
INTRODUCTION
From an adiabatic cyclic evolution, Berry [1] showed that the wave function of a quantum particle acquires a phase shift which has a fundamental importance in the studies of quantum interference effects. On the other hand, Aharonov and Anandan [2] showed that geometric quantum phases can be measured in any cyclic evolution. In recent years, geometric quantum phases have been widely discussed in the literature [3–15]. The best famous effect associated with geometric quantum phases is the Aharonov-Bohm effect [16]. Other well-known quantum effects are the scalar Aharonov-Bohm effect [9–11], the dual of the Aharonov-Bohm effect [17, 18], the Aharonov-Casher (AC) effect [19] and the He-McKellarWilkens (HMW) effect [20, 21]. It is worth mentioning that these quantum effects are also associated with particle systems possessing multipole moments. For instance, the AharonovBohm effect [16] and the He-McKellar-Wilkens effect [20, 21] are associated with particles possessing an electric monopole and a neutral particle possessing a permanent electric dipole moment, respectively. By extending to an electric quadrupole moment, Chen [15] obtained the geometric quantum phase, and the analogue of the scalar Aharonov-Bohm effect has been obtained in Ref. [22]. Associated with the magnetic multipole expansion, the dual of the Aharonov-Bohm effect [17, 18] is obtained by assuming the existence of a magnetic monopole, and the Aharonov-Casher effect [19] arises in neutral particle systems possessing a magnetic dipole moment. Recently, Dirac monopoles have been observed in synthetic magnetic field [23]. By dealing with the magnetic multipole expansion in an analogous way to the electric multipole extension, Chen [15] obtained a geometric quantum phase for a moving particle possessing a magnetic quadrupole moment. The magnetic quadrupole moment has been the subject of a great deal of discussion in the literature [15, 24–30], for instance, by dealing with P - and T -odds effects in atoms [26] and chiral anomaly [28]. However, the quantum dynamics of a moving particle possessing a magnetic quadrupole moment has not been explored in topics such as the scalar AharonovBohm effect [3, 4, 9–12, 22] and the Landau quantization [31–34]. In particular, the geometric phase obtained in Ref. [15] for a moving magnetic quadrupole moment is given by an electric field induced by a time-dependent magnetic field. In this work, we show that geometric quantum phases can be obtained for a moving particle possessing a magnetic quadrupole moment from the assumption of the existence of magnetic monopoles [17, 18, 20, 21, 35]. 2
The aim of this work is to investigate the quantum dynamics of a moving particle which possesses a magnetic quadrupole moment which interacts with electric and magnetic fields. We start by introducing the quantum dynamics of a moving particle possessing a magnetic quadrupole moment. In what follows, from the assumption of the existence of magnetic monopoles, we show that Aharonov-Anandan quantum phases [2] can be obtained in the magnetic quadrupole system. In particular, we obtain an analogue of the scalar AharonovBohm effect for a neutral particle [3, 4, 9–11]. Moreover, we investigate quantum effects associated with the Aharonov-Anandan quantum phase [2] when the quantum particle is confined to a hard-wall confining potential. Finally, we discuss the analogue of the Landau quantization associated with a magnetic quadrupole system. The structure of this work is as follows: in section II, we introduce the quantum dynamics for a moving particle possessing a magnetic quadrupole moment. Based on the assumption that magnetic monopoles exist and the field produced by these charges, we obtain the geometric quantum phases corresponding to the Aharonov-Anandan quantum phases [2]. As a particular case, we obtain the scalar Aharonov-Bohm effect associated with a particle possessing a magnetic quadrupole moment; in section III, we confine the quantum particle to a hard-wall confining potential and show that the energy levels of the bound states depend on the geometric quantum phase associated with the interaction between the magnetic quadrupole moment and an electric field; in section IV, we discuss the analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment; in section V, we present our conclusions.
II.
AHARONOV-ANANDAN QUANTUM PHASES
In this section, we discuss the arising of Aharonov-Anandan quantum phases [2] for a moving particle with a magnetic quadrupole moment. We start by introducing the quantum dynamics of a moving particle possessing a magnetic quadrupole moment which interacts with external fields. In the following, by using the Dirac phase factor method [36, 37], we obtain the geometric quantum phases. From Refs. [27, 30], we can consider a magnetic quadrupole moment as a spinless particle (such as an atom), then, the potential energy is defined by analogy with the classical dynamics of an electric quadrupole moment (in the
3
rest frame of the particle) and it is given by Um = −
X
Mij ∂i Bj ,
(1)
i,j
~ is the magnetic field and Mij is the magnetic quadrupole moment tensor, whose where B characteristic is that it is a symmetric and a traceless tensor [27, 30]. Henceforth, let us consider a moving particle possessing a magnetic quadrupole moment, ~ ′ . Thereby, the then, we have that the particle interacts with a different magnetic field B Lagrangian of this system in the frame of the moving particle is given by L = 21 mv 2 + P ′ ij Mij ∂i Bj . By applying the Lorentz transformation of the electromagnetic field, we have ~ for v ≪ c (SI units). Now, ~ ′ must be replaced with B ~′ = B ~ − 12 ~v × E that the magnetic field B c
~ and B ~ are the electric and magnetic fields in the laboratory frame, we have that the fields E ~ ·B ~ + 12 ~v · M ~ ×E ~ , respectively. In that way, the Lagrangian becomes L = 21 m v 2 + M c P P where we define Mi = j Mij ∂j by analogy with the vector Qi = j Qij ∂j defined in Ref.
[15], where Qij is the electric quadrupole moment tensor. Besides, the canonical momentum 1 ~ ~ of this system is given by p~ = m ~v + c2 M × E , then, the Hamiltonian of this system h i2 1 ~ × E) ~ ~ · B. ~ Let us proceed with the quantization of the p~ − c12 (M becomes H = 2m −M
Hamiltonian, therefore, we replace the canonical momentum p~ with the Hermitian operator ~ Thereby, the quantum dynamics of a moving magnetic quadrupole moment can pˆ = −i~∇. be described by the Schr¨odinger equation 2 1 ~ 1 ∂ψ ~ ~ ·B ~ ψ. pˆ − 2 (M × E) ψ − M = i~ ∂t 2m c
(2)
Now, we are able to study the arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment which interacts with electric and magnetic fields. From now on, we work with the units ~ = c = 1. Thereby, let us consider the following non-null components of the tensor Mij : Mρρ = Mϕϕ = M;
Mzz = −2M,
(3)
where M is a constant (M > 0). In recent years, Chen [15] showed, by considering the interaction between an electric quadrupole moment and a magnetic field produced by an electric charge density given by J~ = − B0 ϕˆ (where ϕˆ is a unit vector in the azimuthal direction), ρ
that the wave function of a quantum particle acquires a geometric phase. Therefore, let us 4
¯ ¯ m is a constant. Hence, ˆ where λ consider a magnetic current density given by J~m = − λρm ϕ,
the magnetic quadrupole is moving in a medium instead of a vacuum. The presence of this magnetic current density generates an electric field given by: ¯ m ln ρ zˆ, ~ =λ E
(4)
where zˆ is a unit vector in the z-direction and ρ =
p x2 + y 2 . It is worth mentioning that
the electric field given in Eq. (4) was suggested in Ref. [15], with the intention of achieving
a field configuration that gives rise to a geometric phase for a moving magnetic quadrupole moment. However, the electric field suggested in Ref. [15] is produced by a time-dependent magnetic field, which differs from our proposal. Then, by using the Dirac phase factor method [36, 37], we can write the solution to the Schr¨odinger equation (2) as ψ = eiφ1 ψ0 ,
(5)
where ψ0 is the solution to the Schr¨odinger equation in the absence of fields, that is, i
1 ∂ψ0 =− ∇2 ψ0 , ∂t 2m
(6)
and φ1 is the geometric quantum phase acquired by the wave function of the particle, which is given by φ1 =
I
~ eff (x) · d~r = A
I
¯m. ~ ~ M × E · d~r = −2π M λ
(7)
The geometric quantum phase (7) is obtained in a closed path without using the adiabatic approximation, therefore, it corresponds to the Aharonov-Anandan quantum phase associated with a moving particle possessing a magnetic quadrupole moment that interacts with the electric field (4). Observe that this quantum phase does not depend on the velocity of the quantum particle; thus, the quantum phase (7) is a nondispersive quantum phase [38–40]. Note that the presence of the magnetic current density J~m can disturb the system. However, we should observe that the magnetic current density vanishes for large values of ρ. Hence, as pointed out in Ref. [15], the closed path in which the particle takes must be a large contour in such a way that we can neglect the magnetic current density J~m and any disturbing effect that stems from the presence of the magnetic current density. Next, let us discuss a different case of the interaction between a magnetic quadrupole moment and external fields. In this case, we also consider the magnetic quadrupole moment 5
to be given in Eq. (3) that interacts with a magnetic field given by [33] ~ = λm ρ ρˆ, B 2
(8)
where λm corresponds to a magnetic charge density. Then, by using the Dirac phase factor method [36, 37] as in Eq. (5), the interaction between the magnetic field (8) and the magnetic quadrupole moment (3) yields the arising of a geometric quantum phase given by Z τ ~ ·B ~ dt = λm M τ , φSAB = M 2 0
(9)
where τ is the time spent by the quantum particle traveling a closed path [3, 4, 22] and ψ0 is also the solution to Eq. (6). The phase shift given in Eq. (9) corresponds to the scalar Aharonov-Bohm effect [3, 4, 9–11, 22] for a moving particle with a magnetic quadrupole moment. Note that this quantum phase does not depend on the velocity of the quantum particle; thus, the quantum phase (9) is a nondispersive quantum phase [38–40]. Moreover, we have obtained the geometric phase (9) without using the adiabatic approximation, therefore, the geometric phase (9) is also an Aharonov-Anandan quantum phase [2].
III.
CONFINEMENT TO A HARD-WALL POTENTIAL
In this section, let us discuss a case where this quantum particle is confined to a hardwall confining potential and investigate the quantum effects associated with the AharonovAnandan quantum phase obtained in Eq. (7). We show that the energy levels depend on the geometric phase (7) and, as a consequence, persistent currents can arise in the confined region. From Eqs. (2) and (7), we can write ~ eff = M ~ ×E ~ = − φ1 ϕ, A ˆ φ0 ρ
(10)
where ϕˆ is a unit vector in the azimuthal direction and φ0 = 2π. In this way, the Schr¨odinger equation (2) becomes 2 2 i φ1 ∂ψ ∂ φ1 1 1 ∂ 1 ∂2 ∂2 1 ∂ψ ψ− =− + + + + ψ. i ∂t 2m ∂ρ2 ρ ∂ρ ρ2 ∂ϕ2 ∂z m φ0 ρ2 ∂ϕ 2m φ0 ρ
(11)
ˆ z = −i∂ϕ commute with the Hamiltonian of Observe that the operators pˆz = −i∂z and L the right-hand side of Eq. (11). Hence, the solution to Eq. (11) can be written in terms of the eigenvalues of the operators above: ψ = e−iEt ei l ϕ eikz R (ρ) , 6
(12)
where l = 0, ±1, ±2, . . . and k is a constant.
Substituting the solution (12) into the
Schr¨odinger equation (11), we obtain τ2 1 R′′ + R′ − 2 R + β 2 R = 0, ρ ρ
(13)
where we have defined the following parameters in Eq. (13): τ = l+ 2
φ1 ; φ0
(14) 2
β = 2mE − k . From now on, let us take k = 0 in order to have a planar system. Note that the radial equation (13) corresponds to the Bessel differential equation [41], whose general solution is R (ρ) = A Jτ (βρ) + B Nτ (βρ), where the functions Jτ (βρ) and Nτ (βρ) are the Bessel functions of the first and second kinds. Henceforth, we consider the wave function of the particle to be well-behaved at the origin and vanishes at a fixed radius ρ0 . Since the function Nτ (βρ) diverges at the origin; thus, we must take B = 0 and write the solution to Eq. (13) as R (ρ) = A J|τ | (βρ) [42]. Thereby, by assuming βρ0 ≫ 0, then, we can write [41, 42] r 2 |τ | π π J|τ | (βρ0 ) → cos βρ0 − . − πβρ0 2 4 Therefore, from Eqs. (14) and (15) and by imposing R (ρ0 ) = 0, we obtain 2 π φ1 3π 1 nπ + l + + En, l ≈ , 2mρ20 2 φ0 4
(15)
(16)
where n = 0, 1, 2, . . . is the quantum number associated with the radial modes. Hence, the spectrum of energy given in Eq. (16) is the energy levels of a particle possessing a magnetic quadrupole moment which interacts with the electric field (4) confined to a hardwall confining potential. Again, note that the presence of the magnetic current density J~m can disturb the system and bound states can or cannot be achieved. Therefore, the discrete set of energy levels obtained in Eq. (16) can or cannot be obtained. As we have pointed out the previous section, we have that the magnetic current density vanishes for large values of ρ. Hence, the discrete set of energy levels given in Eq. (16) can be obtained by considering a particular case where the value of ρ0 is large enough in such a way that we can neglect any disturbing effect that stems from the presence of the magnetic current density. In this way, 7
it is possible to make an analogy between the present study and the study of the Landau levels in the presence of an anti-dot potential made in Ref. [43]. In Ref. [43], it is shown that the anti-dot potential (a scattering potential) generates a set of discrete bound states around the anti-dot potential for each Landau level. In the present case, the presence of the magnetic current density J~m can disturb the system as a scattering potential, then, for large values of ρ0 we have a set of bound states, whose energy levels are given in Eq. (16), around this scattering potential. Observe that the energy levels (16) depend on the Aharonov-Anandan geometric phase given in Eq. (7) whose periodicity is φ0 = 2π, that is, we have that En, l (φ1 + φ0 ) = En, l+1 (φ1 ). Besides, we have the presence of persistent currents which stem from the dependence of the energy levels (16) on the geometric quantum phase φ1 . By using the Byers-Yang relation [44, 45], we have I=−
X ∂En, l n, l
∂φ1
|τ | π 3π τ 1 X nπ + ≈− + . 4mρ20 n, l 2 4 |τ |
(17)
We can also see that the persistent current (17) is a periodic function of the geometric quantum phase φ1 given in Eq. (7). Note that we have taken the asymptotic limit in order to obtain the energy levels (16), therefore one can consider the intensity of the magnetic current density to be neglected at ρ = ρ0 in agreement with Ref. [15].
IV.
ANALOGUE OF THE LANDAU QUANTIZATION
In this section, we discuss the conditions that must be imposed on the external field in order to achieve an analogue of the Landau quantization [31] for a moving particle possessing a magnetic quadrupole moment. The Landau quantization [31] takes place when the motion of a charged particle in a plane perpendicular to a uniform magnetic field acquires distinct orbits and the energy spectrum of this system becomes discrete and infinitely degenerate. It is important in studies of two-dimensional surfaces [46–48], the quantum Hall effect [49] and Bose-Einstein condensation [50, 51]. Recently, the Landau quantization for a moving electric quadrupole moment has been proposed in Ref. [34] by imposing that the electric quadrupole tensor must be symmetric and traceless tensor and there exists the presence of h i ~ ~ ~ ~ ~ is associated with a uniform effective magnetic field given by Beff = ∇ × Q × B , where Q ~ is the magnetic field in the laboratory frame. the electric quadrupole tensor [15, 34] and B 8
In order to achieve an analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment, the magnetic quadrupole tensor and the field configuration in the laboratory frame must satisfy the following conditions: the magnetic quadrupole tensor Mij must be a symmetric and traceless matrix as established in Refs. [27, 30]; there is a uniform effective magnetic field perpendicular to the plane of motion of the particle given by h i ~ eff = ∇ ~ × M ~ ×E ~ , B
(18)
~ satisfies the electrostatic conditions. where the electric field E Now, let us consider the magnetic quadrupole moment tensor to be defined by the following components: Mρz = Mzρ = M,
(19)
where M is also a constant (M > 0). Note that the magnetic quadrupole moment defined by the components given in Eq. (19) satisfies the condition in which the magnetic quadrupole tensor Mij is a symmetric and traceless matrix. On the other hand, in order to satisfy the last two conditions established above, let us consider an electric field given by 2 ~ = λ ρ ρˆ, E 2
(20)
~ eff = where λ is a constant. In this way, we have an effective vector potential given by A ~ ×E ~ = λ M ρ ϕˆ and, consequently, the effective magnetic field (18) is uniform in the zM direction, that is, perpendicular to the plane of motion of the quantum particle. Therefore, the conditions for achieving the Landau quantization are satisfied. Thereby, the Schr¨odinger equation (2) becomes 2 M λ ∂ψ M 2 λ2 2 ∂ 1 1 ∂ 1 ∂2 ∂2 ∂ψ ψ − i =− + + + + ρ ψ. i ∂t 2m ∂ρ2 ρ ∂ρ ρ2 ∂ϕ2 ∂z m ∂ϕ 2m
(21)
ˆ z = −i∂ϕ commute with the We can also observe that the operators pˆz = −i∂z and L Hamiltonian of the right-hand side of Eq. (21). Hence, the solution to Eq. (21) can be written in terms of the eigenvalues of these operators as in Eq. (12), that is, ψ = e−iEt ei l ϕ eikz R (ρ), where l = 0, ±1, ±2, . . . and k is a constant. Substituting this solution into the Schr¨odinger equation (21), we obtain 1 l2 2mE − k 2 R = −R′′ − R′ + 2 R + 2 M λ l R + M 2 λ2 ρ2 R. ρ ρ 9
(22)
From now on, let us take k = 0 in order to have a planar system. In what follows, let us make a change of variables in Eq. (22) given by: ξ = M λ ρ2 . Thus, we have ξ R′′ + R′ −
l2 ξ R − R + µ R = 0, 4ξ 4
(23)
where we have defined the parameter µ=
l mE − . 2Mλ 2
(24)
The solution to Eq. (23) is given by choosing the wave function regular at the origin; thus, it can be written in the form: |l|
ξ
R (ξ) = ξ 2 e− 2 F (ξ) . Therefore, substituting (25) into (23), we obtain |l| 1 ′′ ′ F = 0. − ξ F + [|l| + 1 − ξ] F + µ − 2 2
(25)
(26)
The second order differential equation (26) is the Kummer equation or the confluent hypergeometric equation [41], where F (ξ) = 1 F1 |l|2 + 12 − µ, |l| + 1, ξ is the Kummer
function of first kind. Now, let us impose that the confluent hypergeometric series becomes
a polynomial of degree n, (n = 0, 1, 2, . . .), then, the radial wave function becomes finite
En, l
|l| 2
+ 21 − µ = −n. Hence, by using Eq. (24), we have |l| l 2M λ 1 n+ . (27) = + + m 2 2 2
everywhere [31]. This happens when
Hence, the energy levels given in Eq. (27) correspond to the Landau levels for a moving particle possessing a magnetic quadrupole moment. The analogue of the cyclotron frequency is, thus, given by ω =
2M λ . m
Based on the analogues of the Landau quantization for neutral particles, the quantum Hall effect has been discussed in recent years for neutral particles with a permanent magnetic dipole moment [32] and in a Lorentz symmetry breaking setup [52]. Hence, an interesting point of discussion would be the possibility of obtaining an analogue of the quantum Hall conductivity [49] for a neutral particle possessing a magnetic quadrupole moment.
V.
CONCLUSIONS
We have investigated the arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment which interacts with external 10
fields. Based on the assumption that magnetic monopoles exist, we have shown two field configurations in the laboratory frame that can give rise to Aharonov-Anandan quantum phases [2]. In particular, we have discussed the scalar Aharonov-Bohm effect associated with a particle with a magnetic quadrupole moment. In all cases discussed here, we have seen that the geometric phases do not depend on the velocity of the particle which characterize nondispersive quantum phases [38–40]. Moreover, we have confined the quantum particle to a hard-wall confining potential and investigated the quantum effects associated with the Aharonov-Anandan quantum phase. We have obtained that the energy levels depend on the geometric quantum phase and have a periodicity equal to φ0 = 2π. From this dependence on the geometric quantum phase of the energy levels, we have seen that persistent currents can arise in the system, where the persistent currents are also a periodic function of the geometric quantum phase. Finally, we have discussed which conditions must be imposed in order to achieve an analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment. We have shown that there are three conditions: one is that the magnetic quadrupole tensor must be a symmetric and a traceless matrix, while another is that there exists a h i ~ eff = ∇ ~ × M ~ ×E ~ , where the electric field E ~ uniform effective magnetic field given by B must satisfy the electrostatic conditions. As an example, we have considered the magnetic quadrupole tensor to be a non-diagonal matrix as given in Eq. (19) and an electric field proportional to ρ2 , then, all conditions for achieving the Landau quantization are satisfied and the energy levels corresponding to the analogue of the Landau levels are obtained. Recently, geometric quantum phase for an electric quadrupole moment have been investigated in the noncommutative quantum mechanics [53]. This would be an interesting point of investigation of both Anandan quantum phase and the scalar Aharonov-Bohm effect associated with the magnetic quadrupole moment. Besides, this open new discussions about the Landau quantization for both electric [34] and magnetic quadrupole moment, since one can expect new contributions to the energy levels which stem from the noncommutative space [54]. On the other hand, it is worth observing that we have considered relativistic corrections 2 of the field up to O vc2 . An interesting case would be the analysis of this system under 2 the influence of the relativistic corrections which includes terms of order O vc2 . In this
case, the relativistic effects can give rise to an effective mass [55] which can be interesting in studies of position-dependent mass systems [56–59]. We hope to bring these discussions 11
in the near future.
Acknowledgments
The authors would like to thank the Brazilian agencies CNPq and CAPES for financial support.
[1] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). [2] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987). [3] J. Anandan, Phys. Lett. A 138, 347 (1989). [4] J. Anandan, Phys. Rev. Lett. 85, 1354 (2000). [5] J. Anandan, Phys. Lett. A 133, 171 (1988). [6] J. Anandan, Phys. Lett. A 147, 3 (1990). [7] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988). [8] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, The geometric phase in quantum systems: foundations, mathematical concepts and applications in molecular and condensed matter physics (Springer-Verlag, New York, 2003). [9] M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Springer-Verlag, in: Lecture Notes in Physics, Vol. 340, Berlin, 1989). [10] A. Zeilinger, in Fundamental Aspects of Quantum Theory, edited by V. Gorini and A. Frigero (Plenum, New York, 1985). [11] A. Zeilinger, J. Phys. Colloq. (France) 45, C3-213 (1984). [12] J. Hamilton, Aharonov-Bohm and other cyclic phenomena (Springer-Verlag, Berlin, 1997). [13] C. Furtado and C. A. de Lima Ribeiro, Phys. Rev A 69, 064104 (2004). [14] A. G. de Lima, H. Belich, and K. Bakke, Eur. Phys. J. Plus 128, 154 (2013). [15] C.-C. Chen, Phys. Rev. A 51, 2611 (1995). [16] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). [17] J. P. Dowling, C. Williams and J. D. Franson, Phys. Rev. Lett. 83, 2486 (1999). [18] C. Furtado and G. Duarte, Phys. Scr. 71, 7 (2005). [19] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984).
12
[20] X. G. He and B. H. J. McKellar, Phys. Rev. A 47, 3424 (1993). [21] M. Wilkens, Phys. Rev. Lett. 72, 5 (1994). [22] K. Bakke, Ann. Phys. (Berlin) 524, 338 (2012). [23] M. W. Ray, E. Ruokokoski, S. Kandel, M. M¨ott¨ onen and D. S. Hall, Nature 505, 657 (2014). [24] I. B.Khriplovich, Zh. Eksp. Teor. Fiz. 71, 51 (1976) [Sov. Phys. JETP 44, 25 (1976)]. [25] O. P. Sushkov, V. V. Flambaum and I. B. Khriplovich, Zh. Eksp. Teor. Fiz. 87, 1521 (1984) [Sov. Phys. JETP 60, 873 (1984)]. [26] I. B. Khriplovich, Parity Nonconseroation in Atomic Phenomena (Gordon and Breach, London, 1991). [27] V. F. Dmitriev, I. B. Khriplovich and V. B. Telitsin, Phys. Rev. C 50, 2358 (1994). [28] D. E. Kharzeev, H.-U. Yee and I. Zahed, Phys. Rev. D. 84, 037503 (2011). [29] V. F. Dmitriev, V. B. Telitsin, V. V. Flambaum and V. A. Dzuba, Phys. Rev. C 54, 3305 (1996). [30] H. S. Radt and R. P. Hurst, Phys. Rev. A 2, 696 (1970). [31] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, the nonrelativistic theory, 3rd Ed. (Pergamon, Oxford, 1977). [32] M. Ericsson and E. Sj¨oqvist, Phys. Rev. A 65, 013607 (2001). [33] L. R. Ribeiro et al, Phys. Lett. A 348, 135 (2006). [34] J. Lemos de Melo, K. Bakke and C. Furtado, Phys. Scr. 84, 045023 (2011). [35] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). [36] P. A. M. Dirac, Proc. R. Soc. A 133, 60 (1931). [37] M. V. Berry, Eur. J. Phys. 1, 240 (1980). [38] G. Badurek, H. Weinfruter, R. G¨ ahler, A. Kollmar, S. Wehinger, and A. Zeilinger, Phys. Rev. Lett. 71, 307 (1993). [39] M. Peshkin and H. J. Lipkin, Phys. Rev. Lett. 74, 2847 (1995). [40] M. Peshkin, Found. Phys. 29, 481 (1999). [41] M. Abramowitz and I. A. Stegum, Handbook of mathematical functions (Dover Publications Inc., New York, 1965). [42] S. Azevedo and F. Moraes, Phys. Lett. A 246, 374 (1998). [43] W.-C. Tan and J. C. Inkson, Semicond. Sci. Technol. 11, 1635 (1996). [44] N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961).
13
[45] W.-C. Tan and J. C. Inkson, Phys. Rev. B 60, 5626 (1999). [46] A. Comtet, Ann. Phys. (NY) 173, 185 (1987). [47] C. Grosche, Ann. Phys. (NY) 187, 110 (1988). [48] G.V. Dunne, Ann. Phys. (NY) 215, 233 (1992). [49] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1990). [50] B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 010402 (2001). [51] B. Paredes, P. Zoller, and J. I. Cirac, Solid State Commun. 127, 155 (2003). [52] L. R. Ribeiro. E. Passos and C. Furtado, J. Phys. G: Nucl. Part. Phys. 39, 105004 (2012). [53] H. Nizamidin, A. Anwar, S. Dulat and K. Li, Front. Phys. (2014), DOI: 10.1007/s11467-0140425-1 [54] L. R. Ribeiro, E. Passos, C. Furtado and J. R. Nascimento, Eur. Phys. J. C 56, 597 (2008). [55] H. Wei, R. Han and X. Wei, Phys. Rev. Lett. 75, 2071 (1995). [56] L. Dekar, L. Chetouani and T. F. Hammann, Phys. Rev. A 59, 107 (1999). [57] A. D. Alhaidari, Phys. Rev. A 66, 042116 (2002). [58] A. D. Alhaidari, Phys. Lett. A 322, 72 (2004). [59] L. Serra and E. Lipparini, Europhys. Lett. 40, 667 (1997).
14
S0003-4916(15)00362-0 http://dx.doi.org/10.1016/j.aop.2015.09.027 YAPHY 66972
To appear in:
Annals of Physics
Received date: 25 September 2014 Accepted date: 22 September 2015 Please cite this article as: I.C. Fonseca, K. Bakke, Aharonov–Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment, Annals of Physics (2015), http://dx.doi.org/10.1016/j.aop.2015.09.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Research Highlights
Highlights
• Scalar Aharonov-Bohm effect for a particle possessing a magnetic quadrupole moment • Aharonov-Anandan quantum phase for a particle with a magnetic quadrupole moment • Dependence of the energy levels on the Aharonov-Anandan quantum phase • Landau quantization associated with a particle possessing a magnetic quadrupole moment
1
*Manuscript Click here to view linked References
Aharonov-Anandan quantum phases and Landau quantization associated with a magnetic quadrupole moment I. C. Fonseca and K. Bakke∗ Departamento de F´ısica, Universidade Federal da Para´ıba, Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, PB, Brazil.
Abstract The arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment is investigated. It is shown that an Aharonov-Anandan quantum phase [Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987)] can be obtained in the quantum dynamics of a moving particle with a magnetic quadrupole moment. In particular, it is obtained an analogue of the scalar Aharonov-Bohm effect for a neutral particle [J. Anandan, Phys. Lett. A 138, 347 (1989)]. Besides, by confining the quantum particle to a hard-wall confining potential, the dependence of the energy levels on the geometric quantum phase is discussed and, as a consequence, persistent currents can arise from this dependence. Finally, an analogue of the Landau quantization is discussed. PACS numbers: 03.65.Vf, 14.80.Hv, 03.65.Ge Keywords: magnetic quadrupole moment, Aharonov-Anandan phase, scalar Aharonov-Bohm effect, geometric phase, persistent currents, Landau quantization
∗
Electronic address: [email protected]
1
I.
INTRODUCTION
From an adiabatic cyclic evolution, Berry [1] showed that the wave function of a quantum particle acquires a phase shift which has a fundamental importance in the studies of quantum interference effects. On the other hand, Aharonov and Anandan [2] showed that geometric quantum phases can be measured in any cyclic evolution. In recent years, geometric quantum phases have been widely discussed in the literature [3–15]. The best famous effect associated with geometric quantum phases is the Aharonov-Bohm effect [16]. Other well-known quantum effects are the scalar Aharonov-Bohm effect [9–11], the dual of the Aharonov-Bohm effect [17, 18], the Aharonov-Casher (AC) effect [19] and the He-McKellarWilkens (HMW) effect [20, 21]. It is worth mentioning that these quantum effects are also associated with particle systems possessing multipole moments. For instance, the AharonovBohm effect [16] and the He-McKellar-Wilkens effect [20, 21] are associated with particles possessing an electric monopole and a neutral particle possessing a permanent electric dipole moment, respectively. By extending to an electric quadrupole moment, Chen [15] obtained the geometric quantum phase, and the analogue of the scalar Aharonov-Bohm effect has been obtained in Ref. [22]. Associated with the magnetic multipole expansion, the dual of the Aharonov-Bohm effect [17, 18] is obtained by assuming the existence of a magnetic monopole, and the Aharonov-Casher effect [19] arises in neutral particle systems possessing a magnetic dipole moment. Recently, Dirac monopoles have been observed in synthetic magnetic field [23]. By dealing with the magnetic multipole expansion in an analogous way to the electric multipole extension, Chen [15] obtained a geometric quantum phase for a moving particle possessing a magnetic quadrupole moment. The magnetic quadrupole moment has been the subject of a great deal of discussion in the literature [15, 24–30], for instance, by dealing with P - and T -odds effects in atoms [26] and chiral anomaly [28]. However, the quantum dynamics of a moving particle possessing a magnetic quadrupole moment has not been explored in topics such as the scalar AharonovBohm effect [3, 4, 9–12, 22] and the Landau quantization [31–34]. In particular, the geometric phase obtained in Ref. [15] for a moving magnetic quadrupole moment is given by an electric field induced by a time-dependent magnetic field. In this work, we show that geometric quantum phases can be obtained for a moving particle possessing a magnetic quadrupole moment from the assumption of the existence of magnetic monopoles [17, 18, 20, 21, 35]. 2
The aim of this work is to investigate the quantum dynamics of a moving particle which possesses a magnetic quadrupole moment which interacts with electric and magnetic fields. We start by introducing the quantum dynamics of a moving particle possessing a magnetic quadrupole moment. In what follows, from the assumption of the existence of magnetic monopoles, we show that Aharonov-Anandan quantum phases [2] can be obtained in the magnetic quadrupole system. In particular, we obtain an analogue of the scalar AharonovBohm effect for a neutral particle [3, 4, 9–11]. Moreover, we investigate quantum effects associated with the Aharonov-Anandan quantum phase [2] when the quantum particle is confined to a hard-wall confining potential. Finally, we discuss the analogue of the Landau quantization associated with a magnetic quadrupole system. The structure of this work is as follows: in section II, we introduce the quantum dynamics for a moving particle possessing a magnetic quadrupole moment. Based on the assumption that magnetic monopoles exist and the field produced by these charges, we obtain the geometric quantum phases corresponding to the Aharonov-Anandan quantum phases [2]. As a particular case, we obtain the scalar Aharonov-Bohm effect associated with a particle possessing a magnetic quadrupole moment; in section III, we confine the quantum particle to a hard-wall confining potential and show that the energy levels of the bound states depend on the geometric quantum phase associated with the interaction between the magnetic quadrupole moment and an electric field; in section IV, we discuss the analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment; in section V, we present our conclusions.
II.
AHARONOV-ANANDAN QUANTUM PHASES
In this section, we discuss the arising of Aharonov-Anandan quantum phases [2] for a moving particle with a magnetic quadrupole moment. We start by introducing the quantum dynamics of a moving particle possessing a magnetic quadrupole moment which interacts with external fields. In the following, by using the Dirac phase factor method [36, 37], we obtain the geometric quantum phases. From Refs. [27, 30], we can consider a magnetic quadrupole moment as a spinless particle (such as an atom), then, the potential energy is defined by analogy with the classical dynamics of an electric quadrupole moment (in the
3
rest frame of the particle) and it is given by Um = −
X
Mij ∂i Bj ,
(1)
i,j
~ is the magnetic field and Mij is the magnetic quadrupole moment tensor, whose where B characteristic is that it is a symmetric and a traceless tensor [27, 30]. Henceforth, let us consider a moving particle possessing a magnetic quadrupole moment, ~ ′ . Thereby, the then, we have that the particle interacts with a different magnetic field B Lagrangian of this system in the frame of the moving particle is given by L = 21 mv 2 + P ′ ij Mij ∂i Bj . By applying the Lorentz transformation of the electromagnetic field, we have ~ for v ≪ c (SI units). Now, ~ ′ must be replaced with B ~′ = B ~ − 12 ~v × E that the magnetic field B c
~ and B ~ are the electric and magnetic fields in the laboratory frame, we have that the fields E ~ ·B ~ + 12 ~v · M ~ ×E ~ , respectively. In that way, the Lagrangian becomes L = 21 m v 2 + M c P P where we define Mi = j Mij ∂j by analogy with the vector Qi = j Qij ∂j defined in Ref.
[15], where Qij is the electric quadrupole moment tensor. Besides, the canonical momentum 1 ~ ~ of this system is given by p~ = m ~v + c2 M × E , then, the Hamiltonian of this system h i2 1 ~ × E) ~ ~ · B. ~ Let us proceed with the quantization of the p~ − c12 (M becomes H = 2m −M
Hamiltonian, therefore, we replace the canonical momentum p~ with the Hermitian operator ~ Thereby, the quantum dynamics of a moving magnetic quadrupole moment can pˆ = −i~∇. be described by the Schr¨odinger equation 2 1 ~ 1 ∂ψ ~ ~ ·B ~ ψ. pˆ − 2 (M × E) ψ − M = i~ ∂t 2m c
(2)
Now, we are able to study the arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment which interacts with electric and magnetic fields. From now on, we work with the units ~ = c = 1. Thereby, let us consider the following non-null components of the tensor Mij : Mρρ = Mϕϕ = M;
Mzz = −2M,
(3)
where M is a constant (M > 0). In recent years, Chen [15] showed, by considering the interaction between an electric quadrupole moment and a magnetic field produced by an electric charge density given by J~ = − B0 ϕˆ (where ϕˆ is a unit vector in the azimuthal direction), ρ
that the wave function of a quantum particle acquires a geometric phase. Therefore, let us 4
¯ ¯ m is a constant. Hence, ˆ where λ consider a magnetic current density given by J~m = − λρm ϕ,
the magnetic quadrupole is moving in a medium instead of a vacuum. The presence of this magnetic current density generates an electric field given by: ¯ m ln ρ zˆ, ~ =λ E
(4)
where zˆ is a unit vector in the z-direction and ρ =
p x2 + y 2 . It is worth mentioning that
the electric field given in Eq. (4) was suggested in Ref. [15], with the intention of achieving
a field configuration that gives rise to a geometric phase for a moving magnetic quadrupole moment. However, the electric field suggested in Ref. [15] is produced by a time-dependent magnetic field, which differs from our proposal. Then, by using the Dirac phase factor method [36, 37], we can write the solution to the Schr¨odinger equation (2) as ψ = eiφ1 ψ0 ,
(5)
where ψ0 is the solution to the Schr¨odinger equation in the absence of fields, that is, i
1 ∂ψ0 =− ∇2 ψ0 , ∂t 2m
(6)
and φ1 is the geometric quantum phase acquired by the wave function of the particle, which is given by φ1 =
I
~ eff (x) · d~r = A
I
¯m. ~ ~ M × E · d~r = −2π M λ
(7)
The geometric quantum phase (7) is obtained in a closed path without using the adiabatic approximation, therefore, it corresponds to the Aharonov-Anandan quantum phase associated with a moving particle possessing a magnetic quadrupole moment that interacts with the electric field (4). Observe that this quantum phase does not depend on the velocity of the quantum particle; thus, the quantum phase (7) is a nondispersive quantum phase [38–40]. Note that the presence of the magnetic current density J~m can disturb the system. However, we should observe that the magnetic current density vanishes for large values of ρ. Hence, as pointed out in Ref. [15], the closed path in which the particle takes must be a large contour in such a way that we can neglect the magnetic current density J~m and any disturbing effect that stems from the presence of the magnetic current density. Next, let us discuss a different case of the interaction between a magnetic quadrupole moment and external fields. In this case, we also consider the magnetic quadrupole moment 5
to be given in Eq. (3) that interacts with a magnetic field given by [33] ~ = λm ρ ρˆ, B 2
(8)
where λm corresponds to a magnetic charge density. Then, by using the Dirac phase factor method [36, 37] as in Eq. (5), the interaction between the magnetic field (8) and the magnetic quadrupole moment (3) yields the arising of a geometric quantum phase given by Z τ ~ ·B ~ dt = λm M τ , φSAB = M 2 0
(9)
where τ is the time spent by the quantum particle traveling a closed path [3, 4, 22] and ψ0 is also the solution to Eq. (6). The phase shift given in Eq. (9) corresponds to the scalar Aharonov-Bohm effect [3, 4, 9–11, 22] for a moving particle with a magnetic quadrupole moment. Note that this quantum phase does not depend on the velocity of the quantum particle; thus, the quantum phase (9) is a nondispersive quantum phase [38–40]. Moreover, we have obtained the geometric phase (9) without using the adiabatic approximation, therefore, the geometric phase (9) is also an Aharonov-Anandan quantum phase [2].
III.
CONFINEMENT TO A HARD-WALL POTENTIAL
In this section, let us discuss a case where this quantum particle is confined to a hardwall confining potential and investigate the quantum effects associated with the AharonovAnandan quantum phase obtained in Eq. (7). We show that the energy levels depend on the geometric phase (7) and, as a consequence, persistent currents can arise in the confined region. From Eqs. (2) and (7), we can write ~ eff = M ~ ×E ~ = − φ1 ϕ, A ˆ φ0 ρ
(10)
where ϕˆ is a unit vector in the azimuthal direction and φ0 = 2π. In this way, the Schr¨odinger equation (2) becomes 2 2 i φ1 ∂ψ ∂ φ1 1 1 ∂ 1 ∂2 ∂2 1 ∂ψ ψ− =− + + + + ψ. i ∂t 2m ∂ρ2 ρ ∂ρ ρ2 ∂ϕ2 ∂z m φ0 ρ2 ∂ϕ 2m φ0 ρ
(11)
ˆ z = −i∂ϕ commute with the Hamiltonian of Observe that the operators pˆz = −i∂z and L the right-hand side of Eq. (11). Hence, the solution to Eq. (11) can be written in terms of the eigenvalues of the operators above: ψ = e−iEt ei l ϕ eikz R (ρ) , 6
(12)
where l = 0, ±1, ±2, . . . and k is a constant.
Substituting the solution (12) into the
Schr¨odinger equation (11), we obtain τ2 1 R′′ + R′ − 2 R + β 2 R = 0, ρ ρ
(13)
where we have defined the following parameters in Eq. (13): τ = l+ 2
φ1 ; φ0
(14) 2
β = 2mE − k . From now on, let us take k = 0 in order to have a planar system. Note that the radial equation (13) corresponds to the Bessel differential equation [41], whose general solution is R (ρ) = A Jτ (βρ) + B Nτ (βρ), where the functions Jτ (βρ) and Nτ (βρ) are the Bessel functions of the first and second kinds. Henceforth, we consider the wave function of the particle to be well-behaved at the origin and vanishes at a fixed radius ρ0 . Since the function Nτ (βρ) diverges at the origin; thus, we must take B = 0 and write the solution to Eq. (13) as R (ρ) = A J|τ | (βρ) [42]. Thereby, by assuming βρ0 ≫ 0, then, we can write [41, 42] r 2 |τ | π π J|τ | (βρ0 ) → cos βρ0 − . − πβρ0 2 4 Therefore, from Eqs. (14) and (15) and by imposing R (ρ0 ) = 0, we obtain 2 π φ1 3π 1 nπ + l + + En, l ≈ , 2mρ20 2 φ0 4
(15)
(16)
where n = 0, 1, 2, . . . is the quantum number associated with the radial modes. Hence, the spectrum of energy given in Eq. (16) is the energy levels of a particle possessing a magnetic quadrupole moment which interacts with the electric field (4) confined to a hardwall confining potential. Again, note that the presence of the magnetic current density J~m can disturb the system and bound states can or cannot be achieved. Therefore, the discrete set of energy levels obtained in Eq. (16) can or cannot be obtained. As we have pointed out the previous section, we have that the magnetic current density vanishes for large values of ρ. Hence, the discrete set of energy levels given in Eq. (16) can be obtained by considering a particular case where the value of ρ0 is large enough in such a way that we can neglect any disturbing effect that stems from the presence of the magnetic current density. In this way, 7
it is possible to make an analogy between the present study and the study of the Landau levels in the presence of an anti-dot potential made in Ref. [43]. In Ref. [43], it is shown that the anti-dot potential (a scattering potential) generates a set of discrete bound states around the anti-dot potential for each Landau level. In the present case, the presence of the magnetic current density J~m can disturb the system as a scattering potential, then, for large values of ρ0 we have a set of bound states, whose energy levels are given in Eq. (16), around this scattering potential. Observe that the energy levels (16) depend on the Aharonov-Anandan geometric phase given in Eq. (7) whose periodicity is φ0 = 2π, that is, we have that En, l (φ1 + φ0 ) = En, l+1 (φ1 ). Besides, we have the presence of persistent currents which stem from the dependence of the energy levels (16) on the geometric quantum phase φ1 . By using the Byers-Yang relation [44, 45], we have I=−
X ∂En, l n, l
∂φ1
|τ | π 3π τ 1 X nπ + ≈− + . 4mρ20 n, l 2 4 |τ |
(17)
We can also see that the persistent current (17) is a periodic function of the geometric quantum phase φ1 given in Eq. (7). Note that we have taken the asymptotic limit in order to obtain the energy levels (16), therefore one can consider the intensity of the magnetic current density to be neglected at ρ = ρ0 in agreement with Ref. [15].
IV.
ANALOGUE OF THE LANDAU QUANTIZATION
In this section, we discuss the conditions that must be imposed on the external field in order to achieve an analogue of the Landau quantization [31] for a moving particle possessing a magnetic quadrupole moment. The Landau quantization [31] takes place when the motion of a charged particle in a plane perpendicular to a uniform magnetic field acquires distinct orbits and the energy spectrum of this system becomes discrete and infinitely degenerate. It is important in studies of two-dimensional surfaces [46–48], the quantum Hall effect [49] and Bose-Einstein condensation [50, 51]. Recently, the Landau quantization for a moving electric quadrupole moment has been proposed in Ref. [34] by imposing that the electric quadrupole tensor must be symmetric and traceless tensor and there exists the presence of h i ~ ~ ~ ~ ~ is associated with a uniform effective magnetic field given by Beff = ∇ × Q × B , where Q ~ is the magnetic field in the laboratory frame. the electric quadrupole tensor [15, 34] and B 8
In order to achieve an analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment, the magnetic quadrupole tensor and the field configuration in the laboratory frame must satisfy the following conditions: the magnetic quadrupole tensor Mij must be a symmetric and traceless matrix as established in Refs. [27, 30]; there is a uniform effective magnetic field perpendicular to the plane of motion of the particle given by h i ~ eff = ∇ ~ × M ~ ×E ~ , B
(18)
~ satisfies the electrostatic conditions. where the electric field E Now, let us consider the magnetic quadrupole moment tensor to be defined by the following components: Mρz = Mzρ = M,
(19)
where M is also a constant (M > 0). Note that the magnetic quadrupole moment defined by the components given in Eq. (19) satisfies the condition in which the magnetic quadrupole tensor Mij is a symmetric and traceless matrix. On the other hand, in order to satisfy the last two conditions established above, let us consider an electric field given by 2 ~ = λ ρ ρˆ, E 2
(20)
~ eff = where λ is a constant. In this way, we have an effective vector potential given by A ~ ×E ~ = λ M ρ ϕˆ and, consequently, the effective magnetic field (18) is uniform in the zM direction, that is, perpendicular to the plane of motion of the quantum particle. Therefore, the conditions for achieving the Landau quantization are satisfied. Thereby, the Schr¨odinger equation (2) becomes 2 M λ ∂ψ M 2 λ2 2 ∂ 1 1 ∂ 1 ∂2 ∂2 ∂ψ ψ − i =− + + + + ρ ψ. i ∂t 2m ∂ρ2 ρ ∂ρ ρ2 ∂ϕ2 ∂z m ∂ϕ 2m
(21)
ˆ z = −i∂ϕ commute with the We can also observe that the operators pˆz = −i∂z and L Hamiltonian of the right-hand side of Eq. (21). Hence, the solution to Eq. (21) can be written in terms of the eigenvalues of these operators as in Eq. (12), that is, ψ = e−iEt ei l ϕ eikz R (ρ), where l = 0, ±1, ±2, . . . and k is a constant. Substituting this solution into the Schr¨odinger equation (21), we obtain 1 l2 2mE − k 2 R = −R′′ − R′ + 2 R + 2 M λ l R + M 2 λ2 ρ2 R. ρ ρ 9
(22)
From now on, let us take k = 0 in order to have a planar system. In what follows, let us make a change of variables in Eq. (22) given by: ξ = M λ ρ2 . Thus, we have ξ R′′ + R′ −
l2 ξ R − R + µ R = 0, 4ξ 4
(23)
where we have defined the parameter µ=
l mE − . 2Mλ 2
(24)
The solution to Eq. (23) is given by choosing the wave function regular at the origin; thus, it can be written in the form: |l|
ξ
R (ξ) = ξ 2 e− 2 F (ξ) . Therefore, substituting (25) into (23), we obtain |l| 1 ′′ ′ F = 0. − ξ F + [|l| + 1 − ξ] F + µ − 2 2
(25)
(26)
The second order differential equation (26) is the Kummer equation or the confluent hypergeometric equation [41], where F (ξ) = 1 F1 |l|2 + 12 − µ, |l| + 1, ξ is the Kummer
function of first kind. Now, let us impose that the confluent hypergeometric series becomes
a polynomial of degree n, (n = 0, 1, 2, . . .), then, the radial wave function becomes finite
En, l
|l| 2
+ 21 − µ = −n. Hence, by using Eq. (24), we have |l| l 2M λ 1 n+ . (27) = + + m 2 2 2
everywhere [31]. This happens when
Hence, the energy levels given in Eq. (27) correspond to the Landau levels for a moving particle possessing a magnetic quadrupole moment. The analogue of the cyclotron frequency is, thus, given by ω =
2M λ . m
Based on the analogues of the Landau quantization for neutral particles, the quantum Hall effect has been discussed in recent years for neutral particles with a permanent magnetic dipole moment [32] and in a Lorentz symmetry breaking setup [52]. Hence, an interesting point of discussion would be the possibility of obtaining an analogue of the quantum Hall conductivity [49] for a neutral particle possessing a magnetic quadrupole moment.
V.
CONCLUSIONS
We have investigated the arising of geometric quantum phases in the wave function of a moving particle possessing a magnetic quadrupole moment which interacts with external 10
fields. Based on the assumption that magnetic monopoles exist, we have shown two field configurations in the laboratory frame that can give rise to Aharonov-Anandan quantum phases [2]. In particular, we have discussed the scalar Aharonov-Bohm effect associated with a particle with a magnetic quadrupole moment. In all cases discussed here, we have seen that the geometric phases do not depend on the velocity of the particle which characterize nondispersive quantum phases [38–40]. Moreover, we have confined the quantum particle to a hard-wall confining potential and investigated the quantum effects associated with the Aharonov-Anandan quantum phase. We have obtained that the energy levels depend on the geometric quantum phase and have a periodicity equal to φ0 = 2π. From this dependence on the geometric quantum phase of the energy levels, we have seen that persistent currents can arise in the system, where the persistent currents are also a periodic function of the geometric quantum phase. Finally, we have discussed which conditions must be imposed in order to achieve an analogue of the Landau quantization for a moving particle possessing a magnetic quadrupole moment. We have shown that there are three conditions: one is that the magnetic quadrupole tensor must be a symmetric and a traceless matrix, while another is that there exists a h i ~ eff = ∇ ~ × M ~ ×E ~ , where the electric field E ~ uniform effective magnetic field given by B must satisfy the electrostatic conditions. As an example, we have considered the magnetic quadrupole tensor to be a non-diagonal matrix as given in Eq. (19) and an electric field proportional to ρ2 , then, all conditions for achieving the Landau quantization are satisfied and the energy levels corresponding to the analogue of the Landau levels are obtained. Recently, geometric quantum phase for an electric quadrupole moment have been investigated in the noncommutative quantum mechanics [53]. This would be an interesting point of investigation of both Anandan quantum phase and the scalar Aharonov-Bohm effect associated with the magnetic quadrupole moment. Besides, this open new discussions about the Landau quantization for both electric [34] and magnetic quadrupole moment, since one can expect new contributions to the energy levels which stem from the noncommutative space [54]. On the other hand, it is worth observing that we have considered relativistic corrections 2 of the field up to O vc2 . An interesting case would be the analysis of this system under 2 the influence of the relativistic corrections which includes terms of order O vc2 . In this
case, the relativistic effects can give rise to an effective mass [55] which can be interesting in studies of position-dependent mass systems [56–59]. We hope to bring these discussions 11
in the near future.
Acknowledgments
The authors would like to thank the Brazilian agencies CNPq and CAPES for financial support.
[1] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). [2] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987). [3] J. Anandan, Phys. Lett. A 138, 347 (1989). [4] J. Anandan, Phys. Rev. Lett. 85, 1354 (2000). [5] J. Anandan, Phys. Lett. A 133, 171 (1988). [6] J. Anandan, Phys. Lett. A 147, 3 (1990). [7] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988). [8] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, The geometric phase in quantum systems: foundations, mathematical concepts and applications in molecular and condensed matter physics (Springer-Verlag, New York, 2003). [9] M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Springer-Verlag, in: Lecture Notes in Physics, Vol. 340, Berlin, 1989). [10] A. Zeilinger, in Fundamental Aspects of Quantum Theory, edited by V. Gorini and A. Frigero (Plenum, New York, 1985). [11] A. Zeilinger, J. Phys. Colloq. (France) 45, C3-213 (1984). [12] J. Hamilton, Aharonov-Bohm and other cyclic phenomena (Springer-Verlag, Berlin, 1997). [13] C. Furtado and C. A. de Lima Ribeiro, Phys. Rev A 69, 064104 (2004). [14] A. G. de Lima, H. Belich, and K. Bakke, Eur. Phys. J. Plus 128, 154 (2013). [15] C.-C. Chen, Phys. Rev. A 51, 2611 (1995). [16] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). [17] J. P. Dowling, C. Williams and J. D. Franson, Phys. Rev. Lett. 83, 2486 (1999). [18] C. Furtado and G. Duarte, Phys. Scr. 71, 7 (2005). [19] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984).
12
[20] X. G. He and B. H. J. McKellar, Phys. Rev. A 47, 3424 (1993). [21] M. Wilkens, Phys. Rev. Lett. 72, 5 (1994). [22] K. Bakke, Ann. Phys. (Berlin) 524, 338 (2012). [23] M. W. Ray, E. Ruokokoski, S. Kandel, M. M¨ott¨ onen and D. S. Hall, Nature 505, 657 (2014). [24] I. B.Khriplovich, Zh. Eksp. Teor. Fiz. 71, 51 (1976) [Sov. Phys. JETP 44, 25 (1976)]. [25] O. P. Sushkov, V. V. Flambaum and I. B. Khriplovich, Zh. Eksp. Teor. Fiz. 87, 1521 (1984) [Sov. Phys. JETP 60, 873 (1984)]. [26] I. B. Khriplovich, Parity Nonconseroation in Atomic Phenomena (Gordon and Breach, London, 1991). [27] V. F. Dmitriev, I. B. Khriplovich and V. B. Telitsin, Phys. Rev. C 50, 2358 (1994). [28] D. E. Kharzeev, H.-U. Yee and I. Zahed, Phys. Rev. D. 84, 037503 (2011). [29] V. F. Dmitriev, V. B. Telitsin, V. V. Flambaum and V. A. Dzuba, Phys. Rev. C 54, 3305 (1996). [30] H. S. Radt and R. P. Hurst, Phys. Rev. A 2, 696 (1970). [31] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, the nonrelativistic theory, 3rd Ed. (Pergamon, Oxford, 1977). [32] M. Ericsson and E. Sj¨oqvist, Phys. Rev. A 65, 013607 (2001). [33] L. R. Ribeiro et al, Phys. Lett. A 348, 135 (2006). [34] J. Lemos de Melo, K. Bakke and C. Furtado, Phys. Scr. 84, 045023 (2011). [35] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). [36] P. A. M. Dirac, Proc. R. Soc. A 133, 60 (1931). [37] M. V. Berry, Eur. J. Phys. 1, 240 (1980). [38] G. Badurek, H. Weinfruter, R. G¨ ahler, A. Kollmar, S. Wehinger, and A. Zeilinger, Phys. Rev. Lett. 71, 307 (1993). [39] M. Peshkin and H. J. Lipkin, Phys. Rev. Lett. 74, 2847 (1995). [40] M. Peshkin, Found. Phys. 29, 481 (1999). [41] M. Abramowitz and I. A. Stegum, Handbook of mathematical functions (Dover Publications Inc., New York, 1965). [42] S. Azevedo and F. Moraes, Phys. Lett. A 246, 374 (1998). [43] W.-C. Tan and J. C. Inkson, Semicond. Sci. Technol. 11, 1635 (1996). [44] N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961).
13
[45] W.-C. Tan and J. C. Inkson, Phys. Rev. B 60, 5626 (1999). [46] A. Comtet, Ann. Phys. (NY) 173, 185 (1987). [47] C. Grosche, Ann. Phys. (NY) 187, 110 (1988). [48] G.V. Dunne, Ann. Phys. (NY) 215, 233 (1992). [49] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1990). [50] B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 010402 (2001). [51] B. Paredes, P. Zoller, and J. I. Cirac, Solid State Commun. 127, 155 (2003). [52] L. R. Ribeiro. E. Passos and C. Furtado, J. Phys. G: Nucl. Part. Phys. 39, 105004 (2012). [53] H. Nizamidin, A. Anwar, S. Dulat and K. Li, Front. Phys. (2014), DOI: 10.1007/s11467-0140425-1 [54] L. R. Ribeiro, E. Passos, C. Furtado and J. R. Nascimento, Eur. Phys. J. C 56, 597 (2008). [55] H. Wei, R. Han and X. Wei, Phys. Rev. Lett. 75, 2071 (1995). [56] L. Dekar, L. Chetouani and T. F. Hammann, Phys. Rev. A 59, 107 (1999). [57] A. D. Alhaidari, Phys. Rev. A 66, 042116 (2002). [58] A. D. Alhaidari, Phys. Lett. A 322, 72 (2004). [59] L. Serra and E. Lipparini, Europhys. Lett. 40, 667 (1997).
14