Bound states of relativistic spinless particles in a mix of circularly symmetric vector and scalar harmonic oscillators

We study the dynamics of relativistic spinless particles moving in a plane when there is circular symmetry. The general formalism for solving the Klein–Gordon equation in cylindrical coordinates for such systems is presented, as well as the conserved observables and the corresponding quantum numbers. We look for bound solutions of the corresponding Klein–Gordon equation when one has vector and scalar circularly symmetric harmonic oscillator potentials. Both positive and negative bound solutions are considered when there is either equal vector and scalar potentials or symmetric vector and scalar potentials, and it is shown how both cases are related through charge conjugation. We compute the non-relativistic limit for those cases, and show that for symmetric scalar and vector potentials the limit does not exist in the first order of an harmonic oscillator frequency, recovering a known result from the Dirac equation with the same kind of potentials.


Introduction
The harmonic oscillator stands out as one of the most important potentials arising in both relativistic and nonrelativistic quantum systems.One particular feature is that it can describe bound quantum states in regions around local minima potentials with complex shapes [1].The fact that in general it allows for analytical solutions for the wave function and the energy spectrum also enables to extract the essential physical features of the quantum system under study.Its applications range from quantum field theory of elementary particles to describing some molecule degrees of freedom and many kinds of physical lattices [2].
Concerning the Klein-Gordon theory, which models the relativistic behavior of spinless quantum particles, the harmonic oscillator problem can be set up in different ways because of the different varieties of Lorentz structures [3].One has the Lorentz four-vector potential V μ = (V v , V), which among other applications, is used to model the electromagnetic field interaction, introduced by the minimal coupling p μ → p μ − V μ ; and the Lorentz scalar potential V s , established by its coupling to the mass term m → m + V s .In the case of vector interactions based on electromagnetic origin, the four-vector components in 3 + 1 dimensions are connected to the electric and magnetic fields, E and B, respectively, by the expressions should note that the denominations ʼscalar' and 'vector' for electromagnetic potentials used in textbooks have often a different meaning than the one used here.In our case, V v (vector) is the time component of a Lorentz four-vector potential whereas V s is a Lorentz scalar potential.In this paper, as explained below, we consider a mix of those scalar and vector potentials, so they cannot per se be related to electromagnetic potentials.If one would have only the V v potential, then the corresponding electric field would be E = − ∇V v which, for a harmonic oscillator potential considered below, would be a linear radial vector field.
In discussions regarding the non-relativistic limits of the harmonic oscillator problem, plenty of attention has also been given to a third possible coupling, which originally appeared in the context of the Dirac theory, the so-called Dirac oscillator [4].This inspired research on the Klein-Gordon oscillator coupling [5,6] or nonminimal vector coupling, given by the prescription p p r p r im im . This interaction model has been employed to study noncommutative spaces [7,8], topological defects [9], the realization of bound states, [10] and violation of the Lorentz symmetry [11], to cite some examples.In [12], it was presented a modified D-dimensional Klein-Gordon equation featuring a non-minimal vector interaction which incorporates the Klein-Gordon oscillator as a particular case.
In recent studies there is a particular interest in relativistic quantum systems in which the potentials involved are the time component of a vector coupling V v and the scalar coupling V s .The realization of bound states for this mix has been explored for a myriad of potentials.However, in the Klein-Gordon theory, analytical solutions have been found only for a restrict group of potentials.By making a convenient approximation to the centrifugal term, the Hulthén potential has analytical bound solutions for s-waves and l ≠ 0 waves [13][14][15].Usage of supersymmetric quantum mechanics and hypergeometric equations yields analytical bound states for the pseudoharmonic oscillator potential [16].The hypergeometric approach also has analytical solutions for the V(r) = Ar −2 + Br −1 potential [17].Potentials given by trigonometric functions such as the ring-shaped harmonic oscillator [18], the ring-shaped Kratzer-type [19], the V r d tanh 0 2 ( ) [20], as well as the s-wave states for the symmetrical double-well model [21] and the Rosen-Morse-type [22] also give rise to bound states.Lastly, there are bound states for exponential type potentials, such as the s-wave solutions for five-parameter exponential-type potential [23].
The majority of these works assume that V v = V s .For the Dirac theory, when the vector and scalar potentials are such that V v = ± V s + C, being C an arbitrary constant (actually zero for potentials going to zero at infinity), two additional SU(2) symmetries arise: the spin symmetry, for V v = V s + C, and the pseudospin symmetry, for V v = − V s + C [24].In the Dirac equation, the spin (pseudospin) symmetry account for the suppression of the spin-orbit coupling term for the upper (lower) component of the Dirac spinor.The resulting equations for these components are actually Klein-Gordon equations.These symmetries play an important role in describing the nucleon single particle structure of heavy nuclei and can describe the suppression of spin-orbit in mesons with a heavy quark.A review of these symmetries and its applications can be found in [25].It can also be shown that for both Klein-Gordon and Dirac theories, solutions for the V v = V s case also provides the solutions for the V v = − V s case, and vice-versa.This can be done using the charge conjugation operation and the chiral transformation as well [26].
In the present work we are interested in the configuration of a mix of vector V v and scalar V s potentials, both having the shape of a two-dimensional harmonic oscillator.There are plenty of physical systems in which the particle dynamics is essentially confined to a fixed plane.Such motion is usually described by considering a 2 + 1 −dimensional world instead of the usual 3 + 1−dimensional space-time.For the Klein-Gordon equation, this procedure is equivalent to a 3 + 1−dimensional system such that its wave function has zero linear momentum (has eigenvalue zero of the corresponding operator) along the direction perpendicular to the plane of motion.The resulting wave function depends only on the coordinates in the plane, exactly like in the 2 + 1−dimensional world.
In reference [27] this setup was studied.However, it only dealt with the particle states with the condition V v = V s > 0, and the normalization condition used is not suitable for the relativistic theory.In the present work, we analyze both particle and antiparticle solutions, discuss also the V v = − V s condition as well as the nonrelativistic limits for both cases.We find that, to first order in the frequency over mass expansion, there is no non-relativistic limit for negative energy states when V v = V s < 0 and to positive energy states when V v = − V s > 0. This result has already been obtained for the Dirac equation with harmonic oscillator potentials [28].In this way, we are able to get new results that can be used in studying two-dimensional relativistic quantum systems for which pseudospin symmetry and the consideration of anti-particles states is important.As referred before, this particular symmetry is relevant to study nuclear systems, where pseudospin has been extensively studied, including for anti-nucleon systems and for both spherical and deformed harmonic oscillator meanfields [24].The use of Klein-Gordon equation for these fermion systems in pseudospin symmetry conditions is warranted for analysing the energy spectrum, as was shown in [29].The application to a two-dimensional problem of these results can be done if the there would be no (or almost no) dynamics in a particular space direction.
The paper is organized as follows.In section 2 we will present the Klein-Gordon equation in 3 + 1 dimensions for circularly symmetric systems in cylindrical coordinates and discuss its properties.In section 3 the harmonic oscillator problem will be completely solved by mapping it to a non-relativistic problem which solutions are known.Section 4 will be devoted to discussing the relation between the V v = + V s configuration and the V v = − V s one and the respective non-relativistic limits.The conclusions will be left to section 5.

Circularly symmetric Klein-Gordon equation
The time-independent Klein-Gordon equation for a particle embedded in scalar V s and vector V v potentials can be expressed as (with where the charge density takes the form . The charge conjugation operation is achieved by the substitutions ε → − ε, V v → − V v and V s → + V s .We will consider the motion essentially restricted to a plane for circularly symmetric potentials, in which case the cylindrical coordinates are a natural choice.The wave function can then be written as where ρ is the radial distance from the z-axis, and f is the angle formed with the reference axis in this plane (the x-axis in Cartesian coordinates).The fixed plane coincides with the xy-plane.The functions Φ l are eigenfunctions of L z = − i∂/∂f with eigenvalues l = 0, ± 1, ± 2, K: Since the potentials V s and V v depend only on ρ, U l obeys the radial equation The normalization condition can be obtained from the charge density and is expressed as where the plus and minus signs on the right-hand side refer to positively or negatively charged states, respectively.
For the formation of analytic bound-state solutions, the role of the sum and the difference of the vector and scalar potentials is crucial, as will be shown.Because of that, it is convenient to analyze the problem in terms of the sum and difference potentials V Σ = V v + V s and V Δ = V v − V s .Then, we can write the radial equation as (respectively when ε > m and ε < m).One the other hand, for potentials going to zero when ρ → ∞ one has bound-state solutions with m | | e < only when V Δ = 0 and V Σ < 0, ε > 0, or V Σ = 0 and For small binding energies, ε ; m, we obtain the Schrödinger-like equation For ε ; − m, the equation we get is

The harmonic oscillator problem
We explore a combination of vector and scalar harmonic potentials V v = K v ρ 2 /2 and V s = K s ρ 2 /2, such that V Σ and V Δ are harmonic potentials as well, with coefficients K Σ = K v + K s and K Δ = K v − K s .To derive precise analytic solutions for the radial equation, we selectively set either K Σ or K Δ to zero, which stands for the cases where V v = − V s and V v =V s , respectively.Consequently, the problem transforms into an exactly solvable scenario, described by a Schrödinger-like equation featuring an effective singular harmonic oscillator potential.By setting V Δ to zero, one gets K v =K s , thus K Σ = 2K v (we could as well have written K Σ = 2K s ), and we achieve the following radial equations, derived from (7): Otherwise, analytical solutions would not be obtainable, due to a ρ 4 term appearing in the equation.Notably, the form of equation ( 10) is akin to that of the nonrelativistic harmonic oscillator equation.Additionally, from equation (6), it is evident that the condition in the case of a V v being an harmonic oscillator potential, indicating that solutions to the nonrelativistic problem can be mapped into the normalizable solutions of the relativistic problem.From the study by Nogueira et al [30], the radial equation for a particle under a harmonic oscillator potential is given by (in their notation): The parameters M, E, ω and S are then related here to m, ε, K v and l in (10) through the following relations: Thus, we have Relations (14) provide additional conditions for the eingenvalues.The first relation implies that K v is greater or less than zero if and only if ε is greater or less than −m, while the third relation restricts the energies to m | | e > .Combining these two conditions into a single expression, we conclude that sgn (ε) = sgn (K v ).In addition, it should be noted that the negative sign in the left-hand side of equation ( 16) is extraneous and should be disregarded.To derive an explicit expression for the energy spectrum, we begin by squaring equation (16) and making the following change of variables: which are subject to the condition Our objective now is not only to solve this equation but also to determine which solutions are relevant to our original problem.It should be noted that squaring the energy spectrum equation leads to double the number of solutions.
To gain insight into the possible solutions of equation (21), we can employ Descartes' rule of signs [32].For ò > 3, there is only one feasible value with A > 0, while ò < − 3 supports also one possible value with A < 0. Now it is suitable to cast ò as We are going now to analyze χ.We can start our search by studying the modulus as well as the argument of χ as a function of A for the principal cubic root, as displayed in figures 1 and 2 respectively.Note that the other two roots are obtained by adding 2π/3 and 4π/3 to the principal value argument.
As suggested by the figures 1 and 2, we have divided our analysis into three intervals: A 1, 0 A < 1 and A < 0. In the first case, the principal argument is always null, which leads us to conclude that only by setting this value, one can obtain real energy values.Therefore, for A 1, we can express this as  Moving on to the interval 0 A < 1, the modulus will always be equal to one, enabling us to state that To fulfill the condition (20) it is necessary to consider the principal argument, which falls within the interval ] 0, π/3].
For the remaining case, in which A < 0, we write as only the second argument, which is equal to π, leads to real energy values satisfying the condition (20).
To summarize the above analysis, we obtain the energy spectrum, returning to the original variables and coefficients: Applying (6), the normalized eigenfunctions can be expressed as The degeneracy of the n-th energy level is n + 1. Figure 3 shows the first three levels, corresponding to n = 0, 1, 2 of the obtained spectrum as a function of the interaction strength.Although the spectrum for different signs of K v may look anti-symmetrical around the origin, the change of sign of K v in (28) reveals that this is not the case.
By considering the condition  K m v 3 | | , we will expand the spectrum (27) to obtain the non-relativistic limit (zeroth order term) and the first order relativistic correction.Thus, we get The non-existence of the first order term in the non-relativistic approximation when K v < 0 also appears in the Dirac equation [33].Indeed, as can be seen from (8), if K v < 0, the potential is not binding in the nonrelativistic limit, and therefore we only see higher relativistic corrections for the spectrum.

Charge conjugation
Applying to solutions (27) and (29) the charge conjugation operation [3], which, as mentioned before, performs the changes V Σ → − V Δ , V Δ → − V Σ and ε → − ε, yields the corresponding results to a system in which K Σ is taken to be null and V Δ = K Δ ρ 2 /2, K Δ = 2K v having the opposite sign of the former K Σ , which in practical terms, reverses the sign of K v .We obtain Furthermore, we also obtain the non-relativistic limits to be Here we see that for K v > 0 the non-relativistic limit does not yield bound-state solutions, which is evident from (9).
Referring again to the sum and difference potentials coefficients, we can say in general terms that the charge conjugation for K Σ < 0, yielding the solution with K Δ > 0, is exactly the same as changing K Σ > 0 into K Δ > 0 and m→ − m in equations ( 27) and (28).This is similar to the chiral transformation applied in the Dirac equation (see [26] for details).
If we make the correspondence between the positive energy charge conjugated solutions (K v > 0) to the pseudospin symmetry solutions for the harmonic oscillator potential in 3 + 1 Dirac equation [28] we see that most of the features of the 3 + 1 Dirac spectrum are reproduced in the 2 + 1 Klein-Gordon equation with the same potential, namely the fact that one does not have a non-relativistic limit to the lowest order of the oscillator frequency (proportional to the square root of K v ).

Conclusions
In this work we have presented a complete study of the Klein-Gordon equation in a mix of vector and scalar circularly symmetric harmonic oscillator potentials.We cast the equation in terms of the sum and difference potentials V Σ = V v + V s and V Δ = V v − V s and analyzed the possible configurations in which bound-state solutions could be analitically achieved.Furthermore we discussed the relevant quantum numbers and the correct normalization of these states.
We developed the complete resolution for the case in which V Δ = 0 and V Σ = K v ρ 2 .The eigenfunctions and eigenvalues were obtained by mapping the radial equation in a nonrelativistic framework (Schrödinger equation), whose general solutions are known, to the relativistic case.We were able to show that the eigenfunctions are given in terms of the generalized Laguerre polynomials and solved the irrational equation for the energy eigenvalue which was provided by the mapping.The non-relativistic expansion was shown for both K v > 0 and K v < 0 cases.In the latter case, the first order term in the expansion in the oscillator frequency vanishes, showing that in this order the potential is not binding.
By performing the charge conjugation operation on the results of the V Δ = 0 (K v < 0) configuration we obtain the solutions of the radial two-dimensional Klein-Gordon equation for V Σ = 0 and V Δ = K v ρ 2 (K v > 0).This configuration of harmonic oscillator potentials was not previously discussed in the literature, and has a striking similarity with the solution of the radial 3 + 1 Dirac equation for the lower component radial function in pseudospin conditions (i.e.V Σ = 0).In particular, also in that case there is no lowest order term in the nonrelativistic expansion of the binding energy.In reference [29] it was shown that in spin or pseudospin conditions and for central potentials, the 3 + 1 Dirac equation and the corresponding Klein-Gordon equation with the same potentials have the same energy spectrum.The results obtained in this paper strongly suggest that the same is also true for radial (circularly symmetric) solutions of the 2 + 1 Dirac and Klein-Gordon equations.

Figure 1 .
Figure 1.Modulus of χ as a function of A.

Figure 2 .
Figure 2. Argument of χ as a function of A taking the principal cubic root.

Figure 3 .
Figure 3. Spectrum as a function of the interaction strength for the first three principal quantum numbers.The fine line corresponds to n = 0, the thick line to n = 1, and the discontinuous line to n = 2.