Landau Quantization for an electric quadrupole moment of Position-Dependent Mass Quantum Particles interacting with Electromagnetic fields
aa r X i v : . [ qu a n t - ph ] J a n Landau Quantization for An electric quadrupole moment of Position-Dependent MassQuantum Particles interacting with Electromagnetic fields
Zeinab Algadhi ∗ and Omar Mustafa † Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey,Tel.: +90 392 6301378; fax: +90 3692 365 1604.
Abstract:
Analogous to Landau quantization related to a neutral particle possessing an elec-tric quadrupole moment, we generalize such a Landau quantization to include position-dependentmass (PDM) neutral particles. Using cylindrical coordinates, the exact solvability of PDM neutralparticles with an electric quadrupole moment moving in electromagnetic fields is reported. Theinteraction between the electric quadrupole moment of a PDM neutral particle and a magnetic fieldin the absence of an electric field is analyzed for two different radial cylindrical PDM settings. Next,two particular cases of radial electric fields ( −→ E = λρ b ρ and −→ E = λρ b ρ ) are considered to investigatetheir influence on the Landau quantization (of this system using the same models of PDM settings).The exact eigenvalues and eigenfunctions for each case are analytically obtained. PACS numbers : Keywords:
Electric quadrupole moment, Landau quantization, position-dependent mass Hamil-tonian, cylindrical coordinates, electromagnetic fields.
I. INTRODUCTION
The interaction of multipole moments with the electromagnetic fields has attracted a lot of attention and pro-duced fundamental quantum effects. For example, the Aharonov-Bohm effect [1–4] for a charged particle, the scalarAharonov-Bohm and He- Mckellar- Willens effects [5–11], for bound states [12], and Landau quantization [13–16] foran electric dipole moment of a neutral particle. Furthermore, recent studies have investigated the interaction betweenthe quadrupole moments of neutral particle and external fields in several quantum systems such as geometric quantumphases [17], noncommutative quantum mechanics [19], nuclear structure [20, 21], atomic systems [22–27], molecularsystems [28–30], and Landau quantization [18, 31–33].In particular, the study of the Landau quantization has recently been applied to the quantum dynamics of anelectric quadrupole moment [18, 31]. It investigated the possibility of achieving the Landau quantization for neutralparticles, resulting from the coupling of the electric quadrupole moment with a magnetic field, making a similarminimal coupling with a constant magnetic field [1, 2]. Moreover, they have discussed the conditions necessary for thefield configuration in order to achieve the Landau quantization for neutral particles possessing an electric quadrupolemoment [17, 18]. It is shown [18] that the field configuration in the quadrupole system is dependent on the structureof the quadrupole tensor ( i.e., diagonal or non-diagonal), and has to be different in each case. However, all theprevious studies have used different methods for quantum systems of multipole moments with constant mass. Suchmethods need to be modified to include the spatial dependence of the mass.On the other hand, quantum mechanical systems with position-dpendent mass (PDM) have attracted attentionover the years. Namely, the von Roos Hamiltonian [34] has been extensively investigated in the literature [35–55]. Not only because of its ordering ambiguity associated with the non-unique representation of the kinetic energyoperator, but also because of its feasible applicability in many fields of physics. Recent studies on such PDM chargedparticles in constant magnetic fields [56–60], and position-dependent magnetic fields [61] are carried out (using differentinteraction potentials). To the best of our knowledge, however, no studies have ever been considered to discuss thequantum mechanical effects on PDM neutral particles possessing an electric quadrupole moment. To fill this gap, wediscuss in this work a quantum system that consists of a PDM neutral particle with an electric quadrupole momentinteracting with external fields. We follow the discussion in [18] and extend their idea for PDM systems.This paper is organized as follows. In section II, we start giving a brief description of the quantum dynamicsfor a moving electric quadrupole moment interacting with external fields with a constant mass as done in [18],and extend it to include the PDM case. In so doing, we use the very recent result suggested by [58, 59] for the ∗ Electronic address: [email protected] † Electronic address: [email protected]
PDM-minimal-coupling and the PDM-momentum operator. Furthermore, we discuss the possibility of achieving theLandau quantization for such a system, and the separability of the problem in the cylindrical coordinates ( ρ, φ, z ),under azimuthal symmetrization, by considering that the field configurations and the PDM settings are purely radialdependent as in [54, 55, 59–61]. In section III, we discuss a Landau levels analog for an electric quadrupole momentinteracting with an external magnetic field in the absence of electric field. In the same section we obtain exacteigenfunctions and eigenvalues for different PDM settings. We take into account, in section IV, the effect of anelectric field on the problem at hand, by choosing two models for the radial electric field, a Coulomb-type electric field −→ E = λρ b ρ and a linear-type electric field −→ E = λρ b ρ . Finally, we report exact solutions of the radial Schr¨odinger equationfor both case of an electric field and for the same PDM settings presented in previous section. Our conclusion is givenin section V. II. ANALOGOUS TO THE LANDAU-TYPE QUANTIZATION:
In this section,we start our discussion by describing the quantum dynamics of a moving electric quadrupole momentinteracting with electromagnetic fields as suggested in [17] . By considering an electric quadrupole moment as a scalarparticle, the potential energy of a multipole expansion in the rest frame of a particle can be written as U = q Φ − −→ d . ∇ Φ + P i,j Q ij ∂ i ∂ j Φ (1)where q is the electric charge, −→ d is the electric dipole moment, Q ij is the electric quadrupole moment tensor, and Φis the electric potential.In order to study the dynamics of an electric quadrupole moment, we consider q = 0 , −→ d = 0 and −→ E = −−→∇ Φ,where −→ E is the electric field. Therefore, the equation (1) reads U = − P i,j Q ij ∂ i E j (2)For a moving quadrupole, Lagrangian of this system (a constant mass system) becomes L = 12 mv + P i,j Q ij ∂ i E j (3)Now we must apply the Lorentz transformation of the electromagnetic fields. Therefore, we replace the electric fieldin (3) by −→ E → −→ E + 1 c −→ v × −→ B (4)where −→ E and −→ B are the electric and magnetic fields, respectively. Thus, Lagrangian (3) becomes L = 12 mv + −→ Q . −→ E − c −→ v . ( −→ Q × −→ B ) (5)where we used Q i = P i,j Q ij ∂ j , −→ Q = P i Q i b e i (6)as in [18, 31]. Using the canonical momentum −→ P = m −→ v − c ( −→ Q × −→ B ) (7)the classical Hamiltonian of a constant mass reads H = 12 m (cid:20) −→ P + 1 c ( −→ Q × −→ B ) (cid:21) − −→ Q . −→ E (8)To write the quantum Hamiltonian operator, we replace the canonical momentum −→ P by the operator b P = − i −→∇ for constant mass settings. However, in this work we are interested to study the PDM system. Thus, we rewrite thePDM-non relativistic Hamiltonian (in ~ = 2 m ◦ = c = 1 units) as b H = b P ( −→ r ) + −→ A eff ( −→ r ) p m ( −→ r ) ! − −→ Q . −→ E (9)where the kinetic energy term was proposed by Mustafa and Algadhi [59] along with the definition of PDM- momentumoperator (which resulted from a factorization recipe of Mustafa and Mazharimousavi in [46]): b P ( −→ r ) = − i " −→∇ − −→∇ m ( −→ r ) m ( −→ r ) ! (10)where −→ A eff ( −→ r ) = −→ Q × −→ B , V eff ( −→ r ) = −−→ Q. −→ E (11)In this way, the corresponding time-independent Schr¨odinger equation is written in the form b P ( −→ r ) + −→ A eff ( −→ r ) p m ( −→ r ) ! − −→ Q . −→ E ψ ( −→ r ) = εψ ( −→ r ) . (12)Hence, b P ( −→ r ) p m ( −→ r ) ! − im ( −→ r ) −→ A eff ( −→ r ) · −→∇ − im ( −→ r ) (cid:16) −→∇ · −→ A eff ( −→ r ) (cid:17) + i −→ A eff ( −→ r ) . −→∇ m ( −→ r ) m ( −→ r ) ! + −→ A eff ( −→ r ) m ( −→ r ) − −→ Q . −→ E ψ ( −→ r ) = εψ ( −→ r ) (13)in which the vector potential satisfies the Coulomb gauge −→∇ · −→ A eff = 0. Moreover, using the momentum operator inequation(10) would imply: − m ( −→ r ) −→∇ + −→∇ m ( −→ r ) m ( −→ r ) ! · −→∇ + 14 −→∇ m ( −→ r ) m ( −→ r ) ! − h −→∇ m ( −→ r ) i m ( −→ r ) − im ( −→ r ) −→ A eff ( −→ r ) · −→∇ + i −→ A eff ( −→ r ) . −→∇ m ( −→ r ) m ( −→ r ) ! + −→ A eff ( −→ r ) m ( −→ r ) − −→ Q . −→ E ψ ( −→ r ) = εψ ( −→ r ) . (14)The discussion of the possibility of achieving the Landau quantization for an electric quadrupole moment was donein [18], where they found that the Landau quantization can be achieved by imposing these two conditions: the firstone is that the tensor Q ij must be symmetric and tracless. And the second one is that the field configuration mustbe chosen in such a way that there exists a uniform effective magnetic field given by −→ B eff = −→∇ × −→ A eff = −→∇ × ( −→ Q × −→ B ) = constant vector (15)perpendicular to the plane of motion of the electric quadrupole moment. Thus, it is clear that the field configurationdepends on the choice of the components of the tensor Q ij that describes the electric quadrupole moment. Moreover, −→ E must satisfy the electrostatic conditions (cid:16) −→∇ × −→ E = 0 , ∂ t −→ E = 0 (cid:17) . In the following, we present the field configurations and structures of the tensor Q ij . Thus, we choose the casewhen the tensor Q ij has the non-null components: Q ρρ = Q φφ = Q, Q zz = − Q ( diagonal f orm ) (16)which was studied by [17, 18], where Q is a constant. It is notable that this choice satisfies the properties of the tensor Q ij . Moreover, we consider a magnetic field given by [18, 31] −→ B = 12 B ◦ ρ b z (17)where B ◦ is a constant. By using the the definitions of (6) and the assumption in (16) we obtain the electric quadrupolemoment as −→ Q = ( Q∂ ρ ) b ρ + ( Q∂ φ ) b φ − (2 Q∂ z ) b z (18)At this point, we can find the effective vector potential −→ A eff as −→ A eff ( −→ r ) = −→ Q × −→ B = − QB ◦ ρ b φ (19)Consequently, the effective magnetic field reads −→ B eff ( −→ r ) = −→∇ × −→ A eff ( −→ r ) = − QB ◦ b z (20)which satisfies the second condition (15), where −→ B eff is a uniform effective magnetic field.We may now discuss the separability of the PDM Schr¨odinger equation(14) in the cylindrical coordinates ( ρ, φ, z )and under azimuthal symmetrization. By assuming that the field configurations and that the PDM functions are onlyradially dependent [54, 55, 59–61] (i,e., m ( −→ r ) = M ( ρ, φ, z ) = g ( ρ ) ), the wavefunction can be written as ψ ( ρ, φ, z ) = e imφ e ikz R ( ρ ) (21)where m = 0 , ± , ± , ...., ± ℓ is the magnetic quantum number. Thereby, and with the substitution of (19),(20) and(21) into (14), we obtain the radial equation: R ′′ ( ρ ) R ( ρ ) − (cid:18) g ′ ( ρ ) g ( ρ ) − ρ (cid:19) R ′ ( ρ ) R ( ρ ) − (cid:18) g ′′ ( ρ ) g ( ρ ) − g ′ ( ρ ) ρg ( ρ ) (cid:19) + 716 (cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) + g ( ρ ) ( ε + −→ Q . −→ E ) − m ρ − Q B ◦ ρ + 2 QB ◦ m − k z = 0 . (22)Which is to be solved for no electric field −→ E = 0 and for a different choice of radial electric fields −→ E = 0 , with suitablePDM settings, to find the exact eigenvalues and eigenfunctions of the system. III. PDM PARTICLES WITH AN ELECTRIC QUADRUPOLE MOMENT IN A MAGNETIC FIELD:
In this section,we focus on the discussion of Landau quantization for an electric quadrupole moment interactingwith an external magnetic field, and a vanishing electric field −→ E = 0 ( i.e. V eff = 0) . At this point, equation(22)would read R ′′ ( ρ ) + " − (cid:18) g ′ ( ρ ) g ( ρ ) − ρ (cid:19) R ′ ( ρ ) − (cid:18) g ′′ ( ρ ) g ( ρ ) − g ′ ( ρ ) ρg ( ρ ) (cid:19) + 716 (cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) + g ( ρ ) ε − m ρ − Q B ◦ ρ + 2 QB ◦ m − k z (cid:21) R ( ρ ) = 0 . (23)In the following examples, we use some power-low PDM type in the radial Schr¨odinger equation (23) and report theirexact-solutions. A. Model-I: A linear-type PDM g ( ρ ) = ηρ : Consider a neutral particle with the radial cylindrical PDM setting, g ( ρ ) = ηρ, and the electric quadrupole momentof (18) in presence of the magnetic field in (17). Then, the Schr¨odinger equation(23) would read R ′′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − Q B ◦ ρ + ηρε + 2 QB ◦ m − k z R ( ρ ) = 0 (24)Now, let us make a simple change of variables in equation (24) and use r = √ QB ◦ ρ . Then equation (24) becomes R ′′ ( r ) + " − (cid:0) m − / (cid:1) r − r + ηε ( QB ◦ ) / r + 2 QB ◦ m − k z QB ◦ R ( r ) = 0 , (25)which implies the one-dimensional Schr¨odinger form of the Biconfluent Heun equation ( see, [63, 64]) reads R ′′ ( r ) + " (cid:0) − α (cid:1) r − r δ − βr − r + γ − β R ( r ) = 0 (26)where 14 (cid:0) − α (cid:1) = 3 / − m , β = − ηε ( QB ◦ ) / , δ , γ − β QB ◦ m − k z QB ◦ (27)To solve the above equation (26), we consider the asymptotic behavior for r → and r → ∞ , the function R ( r )can be written in terms of an unknown function u ( r ) as follows R ( r ) = r (1+ α ) / e − ( βr + r ) / u ( r ) (28)that transforms equation(26) into a simpler form ru ′′ ( r ) + (cid:2) α − βr − r (cid:3) u ′ ( r ) + (cid:20) ( γ − − α ) r −
12 ( δ + (1 + α ) β ) (cid:21) u ( r ) = 0 (29)which is the Biconfluent Heun-type equation (BHE) [64], where α, β, γ and δ are arbitrary parameters. The polynomialsolutions of this equation (c.f, e.g., [62–66]) is u ( r ) = H B ( α, β, γ, δ ; r ) (30)where H B ( α, β, γ, δ ; r ) are the Heun polynomials of degree n ρ such that γ − − α = 2 n ρ , where n ρ = 0 , , , ..., and a n ρ +1 = 0 . (31)Here, n ρ is the radial quantum number and a n ρ +1 is a polynomial of degree n ρ + 1 defined by the recurrent relation(see [64–66] for more details). By substituting (27) into (31), we get the exact eigenvalues ε n ρ ,m = (2 QB ◦ ) / η (cid:20) n ρ − m + p m + 1 /
16 + k z QB ◦ (cid:21) / (32)where the cyclotron frequency is ω = (2 QB ◦ ) / η , and the exact normalized eigenfunctuons is R ( ρ ) = N ρ | e ℓ | +1 / e − (cid:18) Q B ◦ ρ − ηερ QB ◦ (cid:19) H B (cid:16) α, β, γ, p QB ◦ ρ (cid:17) (33)where N is the normalization constant, (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) = p m + 1 /
16, and α, β and γ are defined respectively in (27).Comparing with [31], the eigenvalues are changed due to the effect of the PDM of the system, where the spectrumof energy (32) is proportional to n / and removes degeneracies (associated with the magnetic quantum number)similar to the energy levels reported in [31], where they are proportional to n . Furthermore, the frequency is alsomodified. B. Model-II: A harmonic-type PDM g ( ρ ) = ηρ : A PDM neutral particle with g ( ρ ) = ηρ , and an electric quadrupole moment interacting with the external magneticfield (17) would imply that equation (23) be rewritten as R ′′ ( ρ ) − ρ R ′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − (cid:0) Q B ◦ − ηε (cid:1) ρ + 2 QB ◦ m − k z R ( ρ ) = 0 (34)To determine the radial part R ( ρ ) of the wave function and the energy spectrum, we follow the same analysis ofGasiorowicz [68] and start by using the change of variable x = (cid:0) Q B ◦ − ηε (cid:1) / ρ in (34) to obtain R ′′ ( x ) − x R ′ ( x ) − L x R ( x ) − x R ( x ) + µR ( x ) = 0 (35)where L = m − / and µ = 2 QB ◦ m − k z ( Q B ◦ − ηε ) / (36)Next, we consider the asymptotic solutions ( x → and x → ∞ ) of the radial wavefunction R ( x ) to come out with R ( x ) = x | e ℓ | e − x/ G ( x ) (37)with e ℓ = L + 1 = ⇒ (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) = p m + 1 / , with e ℓ > . (38)Substituting (37) in (35) would imply G ′′ ( x ) + (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) x − x G ′ ( x ) + (cid:16) µ − − (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12)(cid:17) G ( x ) = 0 (39)Which, in turn, with y = x yields yG ′′ ( y ) + (cid:16) (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) − y (cid:17) G ′ ( y ) + µ − (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) − G ( y ) = 0 (40)this equation is the confluent hypergeometric equation, the series of which is a polynomial of degree n ρ (finiteeverywhere) when n ρ = µ − (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) − . (41)Consequently, (36) and (38) would give the eigenvalues as ε n ρ ,m = Q B ◦ η − k z QB ◦ − m n ρ + p m + 1 / (42)and the eigenfunctions as R ( ρ ) = N ρ | e ℓ | +1 e − √ Q B ◦− ηε ρ F (cid:16) − n ρ ; (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) + 1; p Q B ◦ − ηερ (cid:17) . (43)In this case, the effect of PDM setting produces a new contribution to the non-degenerate energy levels (42), wherethey are proportional to n − and the frequency is modified as ̟ = Q B ◦ η . IV. PDM-PARTICLES WITH AN ELECTRIC QUADRUPOLE MOMENT AND ELECTROMAGNETICFIELDS:
In this section,we study PDM-particles with an electric quadrupole moment and electromagnetic fields. However,we focus on analysis of the effect of the electric field on a PDM particle with an electric quadrupole moment in thepresence of a magnetic field (on the Landau-type system reported in the previous section). For this purpose, wechoose a sample of radial electric fields (c.f., e.g., [14–16, 24, 31]), in the following illustrative examples.
A. The influence of A Coulomb-type electric field on the Landau-type system:
Consider a radial electric field in the form of −→ E = λρ b ρ (44)where λ is a constant [31].Thus, we can see that the interaction between the electric quadrupole moment (18) and the electric field (44) leadsto an effective scalar potential V eff ( ρ ) = −−→ Q. −→ E = Qλρ (45)which plays the role of a scalar potential in the PDM-Schr¨odinger equation (22) to imply R ′′ ( ρ ) − (cid:18) g ′ ( ρ ) g ( ρ ) − ρ (cid:19) R ′ ( ρ ) + " − (cid:18) g ′′ ( ρ ) g ( ρ ) − g ′ ( ρ ) ρg ( ρ ) (cid:19) + 716 (cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) + g ( ρ ) ε − g ( ρ ) Qλρ − m ρ − Q B ◦ ρ + 2 QB ◦ m − k z (cid:21) R ( ρ ) = 0 . (46)Hereby, we again consider the same examples used in the previous section to find exact solutions of equation (46):
1. Model-I : g ( ρ ) = ηρ : The PDM radial Schr¨odinger equation (46) with, g ( ρ ) = ηρ, reads R ′′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − Q B ◦ ρ + ηρε − ηλQρ + 2 QB ◦ m − k z R ( ρ ) = 0 (47)with the change of variable r = √ QB ◦ ρ, equation (47) becomes R ′′ ( r ) + " − (cid:0) m − / (cid:1) r − r + ηε ( QB ◦ ) / r − ηλQ ( QB ◦ ) / r + 2 QB ◦ m − k z QB ◦ R ( r ) = 0 (48)To find its solutions, we define these parameters14 (cid:0) − α (cid:1) = 3 / − m , β = − ηε ( QB ◦ ) / , δ ηλQ ( QB ◦ ) / , γ − β QB ◦ m − k z QB ◦ , (49)and follow the same steps as in (28) to (31). Thus the exact eigenvalues are ε n ρ ,m = (2 QB ◦ ) / η (cid:20)(cid:16) n ρ + − m + p m + 1 / (cid:17) + k z QB ◦ (cid:21) / (50)and the exact eigenfunctions are R ( ρ ) = N ρ | e ℓ | +1 / e − (cid:18) Q B ◦ ρ − ηερ QB ◦ (cid:19) H B (cid:16) α, β, γ, δ ; p QB ◦ ρ (cid:17) (51)It is obvious that these eigenvalues (50) are the same as the eigenvalues in the absence of an electric field for thesame PDM setting given in (32) but with different eigenfunctions. Thus, the effective potential with the PDM settingdoes not effect the eigenvalues of the system.
2. Model-II: g ( ρ ) = ηρ : The substitution of g ( ρ ) = ηρ , in the PDM radial Schr¨odinger equation (46) would yield R ′′ ( ρ ) − ρ R ′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − (cid:0) Q B ◦ − ηε (cid:1) ρ − ηλQ + 2 QB ◦ m − k z R ( ρ ) = 0 (52)We repeat the same procedure as in the previous section and immediately write the corresponding eigenvalues andradial wave functions, respectively, as ε n ρ ,m = Q B ◦ η − ηλQ + k z QB ◦ − m n ρ + p m + 1 / (53)and R ( ρ ) = N ρ | e ℓ | +1 e − √ Q B ◦− ηε ρ F (cid:16) − n ρ ; (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) + 1; p Q B ◦ − ηερ (cid:17) (54)where (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) = p m + 1 / , with e ℓ > . In this case, the influence of the PD-effective potential is appeared by makinga shift in the energy levels of (42) and producing new eigenvalues (53), therefore.
B. The influence of A linear-type electric field on the Landau-type system:
Now, let us consider another radial electric field (e.g., [14–16, 24]) as −→ E = λρ b ρ (55)Using the same components of the the electric quadrupole moment tensor defined in (16), the effective scalar potentialgiven in the PDM-Schr¨odinger equation(22) becomes V eff ( ρ ) = −−→ Q. −→ E = − Qλ R ′′ ( ρ ) − (cid:18) g ′ ( ρ ) g ( ρ ) − ρ (cid:19) R ′ ( ρ ) + " − (cid:18) g ′′ ( ρ ) g ( ρ ) − g ′ ( ρ ) ρg ( ρ ) (cid:19) + 716 (cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) + g ( ρ ) ε + g ( ρ ) Qλ − m ρ − Q B ◦ ρ + 2 QB ◦ m − k z (cid:21) R ( ρ ) = 0 . (57)In the two examples below, we investigate the influence of this effective potential using the same PDM setting thatused in the previous sections:
1. Model-I: g ( ρ ) = ηρ : With g ( ρ ) = ηρ in (57), we obtain0 R ′′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − Q B ◦ ρ + (cid:18) ηε + ηλQ (cid:19) ρ + 2 QB ◦ m − k z R ( ρ ) = 0 (58)Using the previous technique for the linear-type PDM to get the exact solutions for this case. Hence, this wouldcorrespond to the exact eigenvalues and eigenfunctions given, respectively, ε n ρ ,m = (2 QB ◦ ) / η (cid:20) n ρ − m + p m + 1 /
16 + k z QB ◦ (cid:21) / − λQ R ( ρ ) = N ρ | e ℓ | +1 / e − Q B ◦ ρ − ( ηε + ηλQ ) ρ QB ◦ H B (cid:16) α, β, γ, p QB ◦ ρ (cid:17) (60)where, (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) = p m + 1 /
16 and the parameters α, β, γ and δ are defined as14 (cid:0) − α (cid:1) = 3 / − m , β = − (cid:16) ηε + ηλQ (cid:17) ( QB ◦ ) / , γ − β QB ◦ m − k z QB ◦ , δ . (61)
2. Model-II: g ( ρ ) = ηρ : Considering this radial cylindrical PDM would imply that equation (57) be rewritten as R ′′ ( ρ ) − ρ R ′ ( ρ ) + " − (cid:0) m − / (cid:1) ρ − (cid:18) Q B ◦ − ηλQ − ηε (cid:19) ρ + 2 QB ◦ m − k z R ( ρ ) = 0 (62)Equation (62) is again in the same form of equation (34) and admits the exact solution of the eigenvalues and thecorresponding radial eigenfunctions as ε n ρ ,m = Q B ◦ η − k z QB ◦ − m n ρ + p m + 1 / − λQ R ( ρ ) = N ρ f | ℓ | +1 e − √ Q B ◦− ηλQ − ηε ρ F − n ρ ; (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) + 1; r Q B ◦ − ηλQ − ηερ ! (64)where (cid:12)(cid:12)(cid:12)e ℓ (cid:12)(cid:12)(cid:12) = p m + 1 / , with e ℓ > . It is shown that the effective potential generated by the interaction between the electric quadrupole moment andthe radial electric field given by (55) produces constant potential ( − Qλ ). Thus, the effect of mass settings on theeffective potential in the term ( g ( ρ ) −→ Q . −→ E ) yields only a constant shift in the energy levels given in (32) and (42)creating a new set of energies given in (59) and (63). V. CONCLUDING REMARKS
In this paper, we have started with a quantum system of an electric quadrupole moment interacting with magneticand electric fields of a constant mass, as has been previously reported in the literature [18, 31]. Next, we have extended1this procedure to study PDM systems by using recent results of [58, 59] for the PDM- minimal-coupling and the PDM-momentum operator given by (9) and (10), respectively. We have discussed the possibility of achieving the Landauquantization following [18] and modified this discussion to include a PDM case. Thus, we have recollected the mostimportant and essential relations ( equations (15)-(20) above), that have been reported in [18]. Moreover, we havestudied this problem within the context of cylindrical coordinates and investigated the exact solvability of the PDMradial Schr¨odinger equation of a neutral particle possessing an electric quadrupole moment interacting with externalfields, where we have considered PDM settings m ( −→ r ) = g ( ρ ), along with the field configurations ( documented in (17)for the magnetic field −→ B ( ρ ) and (44),(55) for electric fields −→ E ( ρ )), which exhibits a pure radial cylindrical dependence.We have shown that the Landau quantization is produced from the interaction between the magnetic field and theelectric quadrupole moment given in (17) and (18) respectively, where this has yielded the PDM radial Schr¨odingerequation of this system in the absence of an electric field (documented in (23) above). However, comparing with[18], the energy levels of the Landau quantization are modified because of the influence of the spatial dependence ofthe mass, where we have obtained different eigenvalues for the two examples of PDM settings (i.e. g ( ρ ) = ηρ and g ( ρ ) = ηρ ) given in (32) and (42), respectively.Furthermore, we have analyzed the effect of the interaction of radial electric fields with the electric quadrupolemoment of a PDM neutral particle by choosing two particular cases of the electric field ( −→ E = λρ b ρ and −→ E = λρ b ρ ) , which have produced the effective potentials. It is shown that the structure of the PD effective potential term, playsthe role of scalar potentials in the PDM radial Schr¨odinger equation (22), producing same or an energy shift in energylevels of the systems with the same mass settings and in the absence of the effective potential. We have observedthat the difference in energy levels depends on the mass structure. The more complex the chosen mass, the morethis difference will be over Landau levels. However, the exact eigenvalues and eigenfunctions for all these cases areobtained.Finally, although the presence of an effective uniform magnetic field produces the Landau quantization, the influenceof the spatial dependence of the mass of the system yields a new contribution to energy levels creating a set of neweigenvalues.This study has investigated, for the first time, that the PDM quantum particle that possesses multipole momentsunder the influence of external fields. Thus, this work opens new discussions regarding the position-dependent conceptand provides a good starting point for future research.2 [1] L. D. Landau, Z. Phys. (1930) 629.[2] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, 3rd edition (Pergamon, Oxford, 1977).[3] J. Hamilton, Aharonov-Bohm and Other Cyclic Phenomena (Springer-Verlag, Berlin, 1997).[4] M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, in: Lecture Notes in Physics, Vol. (Springer- Verlag,Berlin,1989).[5] M.Wilkens, Phys. Rev. Lett. (1994) 5.[6] X.G. He, B. H. J.McKellar, Phys. Rev. A47 (1993) 3424.[7] A. Zeilinger, J. Phys. Colloq. (1984) C3-213.[8] A. Zeilinger, V. Gorini, A. Frigero (Eds.), Fundamental Aspects of Quantum Theory, Plenum, New York, 1985.[9] J. Anandan, Phys. Lett. A138 (1989) 347.[10] J. Anandan, Phys. Rev. Lett. (2000) 1354.[11] K. Bakke, C. Furtado, Phys. Lett. A375 (2011) 3956.[12] K. Bakke, J.Math. Phys. (2010) 093516 .[13] K. Bakke, Internat. J. Modern Phys. A 26 (2011) 4239.[14] C. Furtado et al., Phys. Lett.
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