Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space
The nonrelativistic quantum dynamics of a spinless charged particle in the presence of the Aharonov–Bohm potential in curved space is considered. We chose the surface as being a cone defined by a line element in polar coordinates. The geometry of this line element establishes that the motion of the particle can occur on the surface of a cone or an anti–cone. As a consequence of the nontrivial topology of the cone and also because of two–dimensional confinement, the geometric potential should be taken into account. At first, we establish the conditions for the particle describing a circular path in such a context. Because of the presence of the geometric potential, which contains a singular term, we use the self–adjoint extension method in order to describe the dynamics in all space including the singularity. Expressions are obtained for the bound state energies and wave functions.
keywords:Self–adjoint extension, Aharonov-Bohm problem, Geometric potential, Bound state
Over the years, condensed matter systems have been a proper environment to test theoretical physics. Advanced ideas such as the concept of torsion in geometry have been used in this context. For instance Cosserat continuum behaves as if it was a Riemann–Cartan manifold endowed with a torsion tensor Hehl and Obukhov (2007). Also it is well known that a particle moving in a solid with topological defects is analogous to a 3D gravity model with torsion Katanaev and Volovich (1992); Katanaev (2005). In addition, fluid systems have been used to verify Hawking radiation which otherwise would be extremely difficult to study Unruh (2014).
Due to the great development of new materials such as graphene Novoselov et al. (2004); Geim and Novoselov (2007); Castro Neto et al. (2009), the use of refined concepts like those in geometry became mandatory to properly describe them. It is interesting to note that these theoretical ideas migrated to a central role in the prediction of the behavior of such systems. This feature is similar to what happened in General Relativity in which the dynamics of a particle is defined by the space-time curvature Carroll (2013). Let us consider an electron moving on a 2D curved graphene surface, such as a carbon nanotube, how does it affects its dynamics? In such a context it is not possible to construct a Classical Lagrangian that reveals the proper dynamics. It is necessary to take into account the curvature and the metric tensor of the surface to which the electron is bounded.
To address the problem of a particle confined to a surface which is embedded on a 3D space, at least two different approaches were developed. The first one is based on a purely 3D geometry DeWitt (1957), whilst the second one, proposed by da Costa da Costa (1981), is constructed as a limit of a 3D space to the curved surface . This latter process is equivalent to embedding the surface into an ordinary 3D Euclidean space. As a consequence, the wave function splits into two parts, one of them working as if there was an effective potential constructed in terms of the mean and Gaussian curvatures. Thus, through this procedure, the particle is subjected to the so called geometric potential da Costa (1981). Recently, the da Costa proposition appeared in various physical contexts as, for instance, in the derivation of the Pauli equation for a charged spin particle confined to move on a spatially curved surface in the presence of an electromagnetic field Wang et al. (2014), in the study of curvature effects in thin magnetic shells Gaididei et al. (2014), in effects of non–zero curvature in a waveguide to investigate the appearance of an attractive quantum potential which crucially affects the dynamics in matter–wave circuits Campo et al. (2014), in the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide Stockhofe and Schmelcher (2014), to derive the exact Hamiltonians for Rashba and cubic Dresselhaus spin–orbit couplings on a curved surface with an arbitrary shape Chang et al. (2013), in the study of high–order–harmonic generation in dimensionally reduced systems Castiglia et al. (2013), to explore the effects arising due to the coupling of the center of mass and relative motion of two charged particles confined on an inhomogeneous helix with a locally modified radius Zampetaki et al. (2013), to study the dynamics of shape–preserving accelerating electromagnetic wave packets in curved space Bekenstein et al. (2014), in the derivation of the Schrö dinger equation for a spinless charged particle constrained to move on a curved surface in the presence of an electric and magnetic field Ferrari and Cuoghi (2008), etc.
In this article, we intend to use the same set of electric and magnetic fields of Ref. Ferrari and Cuoghi (2008) for a spinless charged particle, but now confined to a conical surface and in a presence of an Aharonov-Bohm potential Aharonov and Bohm (1959). We analyze the case of a charged particle describing a circular path and also the general dynamics in all space, including the region. We use the self–adjoint extension method to determine the most relevant physical quantities, such as energy spectrum and wave functions by applying boundary conditions allowed by the system.
The paper is organized as follows. In section 2, we recall the main ideas concerning the dynamics of a spinless charged particle on a curved surface under the influence of an electromagnetic field. We start with the Schrödinger Hamiltonian and couple it to the curvature and the electromagnetic potential. In section 4, we establish the conditions for the particle to describe a circular path. We determine the expressions for the energy eigenvalues, wave functions and discuss the role played by the curvature on them. In section 5, we briefly discuss some concepts of the self–adjoint extension method. We analyze the particle’s dynamics when it lies either on a cone or on an anti–cone. After applying the boundary conditions allowed by the system, we obtain expressions for the bound state energies and wave functions in both cases. Finally, in section 6, we present our concluding remarks. In this work we use units such as .
2 Equations of motion
In this section, we introduce the equations of motion. We consider the refined fundamental framework for the thin–layer quantization scheme discussed in Yong-Long Wang (2015), where a sound performing sequence in the thin–layer quantization process is addressed. The case we are dealing with will coincide with that which comes from the semiclassical method applied by da Costa to investigate the effective quantum dynamics for a constrained particle da Costa (1981); Ferrari and Cuoghi (2008). Thus, we start with the Schrödinger equation
where the Hamiltonian is given by
where and the index runs from 1 to 3. The coordinate is the one transverse to a thin interface. In the limit , the metric tensor confined on a surface is given by Yong-Long Wang (2015)
and the first part of the above Hamiltonian reads as . Such a behavior determines an immersion of a 2D geometry into a 3D Euclidean space. Thus, we have a 2D effective metric . In Ref. da Costa (1981), the metric in Eq. (3) suggests a separable wave function in the form . This splits the movement into two, one constrained on a surface and another one which takes place on a normal direction of such a surface. In the normal direction, the dynamics is governed by the usual Hamiltonian , where the confining potential is assumed to localize the particle on the surface . However, a fundamental framework for the thin-layer quantization scheme is not explicitly defined in it. Here we follow the explicitly refined fundamental framework for the thin-layer quantization scheme presented in Yong-Long Wang (2015), where the limit must be performed after calculating all curvilinear coordinate derivatives. We also consider the minimal coupling with the electromagnetic field by means of the prescription
where is the charge of the particle and is the potential vector component. Therefore, the Schrödinger equation that describes the dynamics of a spinless charged particle bounded to a thin interface under the effect of electric and magnetic fields is given by
where , is a potential due to the geometry of the surface and is the electric potential on the surface. The Coulomb gauge sets and in the picture described here it gives
where is the mean curvature. Notice that the Lorentz gauge and the effective Schrödinger equation cannot be decoupled from the mean curvature of the surface simultaneously, except when . By considering , we can write
which are the decoupled equations found by da Costa in a semi–classical approach. These equations do not encompass the influence of the interface thickness Yong-Long Wang (2015). This case leads to energy shifts that we ignore in this work.
Here, it is interesting to notice how the break in the isotropy of the space arises. As a matter of fact, the velocity operators became noncommutative usually due to the presence of the magnetic field. Since , we write the commutator
Therefore, we see that does not commutes with , not just because the existence of the magnetic field but also by the geometry of the surface to which the movement is bounded.
3 Motion in a Aharonov–Bohm potential
Now, let us apply Eq. (8) to the Aharonov–Bohm problem. At this point, we can make a connection with the description of continuous distribution of dislocations and disclinations in the framework of Riemann–Cartan geometry of Ref. Katanaev and Volovich (1992). If the particle is now bounded to a surface with a disclination located in the region, the corresponding metric tensor, in cylindrical coordinates, is defined by the line element (see Ref. Filgueiras et al. (2012) for more details.)
with , , which describes a conical surface. For (a deficit angle), the metric (10) describes an actual cone, whilst for the cone turns into a plane, and for (an excess angle), the resulting surface is an anti–cone. It is worthwhile to observe that the line element in (10) can be compared with the metric of the spacetime produced by a thin, infinite, straight cosmic string (for the special case of ) Vilenkin and Shellard (2000)
where the parameter is given in terms of the linear mass density of the cosmic string by , which is smaller than the unity. Moreover, the metric in (11) is locally flat, but globally it is not Vilenkin and Shellard (2000); Vilenkin (1981) and through the metric (11), among others effects Frolov and Serebriany (1987); Bezerra de Mello (2004); Bezerra de Mello et al. (2012); Andrade and Silva (2014); Andrade et al. (2012), we can study the gravitational analog of the Aharonov–Bohm effect Dowker (1967); Ford and Vilenkin (1981); Bezerra (1987).
where is the generalized two–dimensional –function in flat space. From Eq. (12), it follows that
In other words, when the defect carries a negative curvature, we have an excess of the planar angle, which corresponds to an anti–cone. However, if the defect presents a positive curvature, we have a planar deficit angle, and the result leads to a cone. For the motion of the particle on a cone, the geometric potential , which is a consequence of a two–dimensional confinement on the surface, is found to be da Costa (1981)
where is the mean curvature and is the Gaussian curvature of the surface. For the cone (), these quantities are given by de M. Carvalho et al. (2007):
In this case, the potential reads
However, for the anti–cone (), in order to be consistent with the fact that it has a negative curvature, i.e., the surface takes a saddle like surface form and the mean curvature is now given by Filgueiras et al. (2012)
and the geometric potential becomes
The magnetic flux tube in the background space described by the metric (10), which will be our choice for , is related to the vector potential as (, )
where with ; is the flux parameter with and the magnetic field is
In this manner, assuming a solution of the form , the Schrödinger equation (8) is now written as
We seek eigenfunctions of the form
with . Substituting this solution into Eq. (22), we obtain for :
being the Hamiltonian without the –function and
is the effective angular momentum. The particle, with its motion confined to the conical surface, is therefore subjected to a generalized potential given by
Let us analyze this potential. For , the quantity , and in this case for , or . On the other hand, , in such way that we have a repulsive –function. However, we will see below that, even though the –function being repulsive, the condition guarantees the existence of bound states. For , the quantity , consequently we have for all values of , , and the –function is now attractive. In this manner, the attractive –function potential guarantees at least one bound state. In the Table 1 bellow, we summarize the possible physical scenarios of obtaining and for and . The case is not of interest here because it implies in a flat space. We also summarize the possible physical scenarios of obtaining scattering and bound states in Table 2, based on the signal of in Eq. (25), for .
|anti–cone||Bound and Scattering|
4 Particle describing a circular path
In this section, we analyze a particularity of the present system, which is the simple case when a particle is constrained to move in a circle of radius (for example, bead on a wire ring). In this case, the wave functions in Eq. (22) depend only on the azimuthal angle , so that . This way, the Schrödinger equation yields a linear differential equation with constant coefficients:
where . By assuming eigenfunctions of the form
where is a constant, and substituting it into the Eq. (29), one achieves the following solution for the characteristic equation:
For the wave function to be single–valued, in , the parameter must be an integer. With this condition, we obtain discrete values for the energy, namely,
which depends on the mean curvature . If , we fall into the problem of a charged particle on a circular ring which a long solenoid passing through it leads to the energy levels for the usual AB problem Griffiths (2005),
recovering the lifting of twofold degeneracy of the system due to the presence of the magnetic flux tube.
5 Bound state energy and wave function
In this section, we obtain the bound state energies and wave functions of the system. We know, from Ref. Andrade et al. (2012), that the form of the Hamiltonian (25) requires a procedure of physical regularization because of the presence of the –function. Before we proceed further with this approach, it is important to check what are the criteria revealed by the Hamiltonian (25) to produce physically acceptable results.
We commence by observing that when we deal with singular potentials we need to guarantee that the operator is essentially self-adjoint in order to make sure that the time evolution is unitary. An operator is said to be essentially self-adjoint if and only if and . One observes that even if such operator is Hermitian, i.e., , its domain could be different from its adjoint. Roughly speaking, the self-adjoint extension approach consists, essentially, in extending the domain in order to match . In the present case, if , with denoting the set of functions that is differentiable for all degrees of differentiation, and , should coincide with , in such way that Albeverio et al. (2004); Zorbas (1980). Thus it is reasonable to interpret as an extension of , or more precisely, as a self–adjoint extension of Gesztesy et al. (1987); Dabrowski and Stovicek (1998); Adami and Teta (1998). Using the unitary operator , given by , the operator can be written as
As a result, it is well–known Reed and Simon (1975) that the symmetric radial operator is essentially self–adjoint for . On the other hand, if , it is not essentially self–adjoint, admitting an one–parameter family of self–adjoint extensions. To characterize this one-parameter family of , we will use the approach of Ref. Kay and Studer (1991), which is based on boundary conditions. Basically, the boundary condition is a match of the logarithmic derivatives of the zero–energy solutions for Eq. (24) and the solutions for the problem plus self–adjoint extension.
In this manner, we solve the problem without the –function potential and then we find the boundary condition by invoking the self–adjointness of . For this, we must solve the eigenvalue equation
plus self–adjoint extensions. Here, the label is the self–adjoint extension parameter, which is related to the behavior of the wave function at the origin. In order for to be a self–adjoint operator, its domain has to be extended by the deficiency subspace, which is given by the solutions of the eigenvalue equation
In the next sections, we will use the present approach to determine the energy spectrum for a particle lying on an anti–cone and on a cone.
5.1 Quantum dynamics on an anti–cone
According to the Table 1, implies and the particle lies on an anti–cone. In this case, by solving Eq. (36), the only square integrable functions which are solutions are the modified Bessel functions of second kind
with . These functions are square integrable only in the range and, as stated above, in this range is not self–adjoint. The dimension of such deficiency subspace is found to be . Thus, the domain of the extended operator in is given by the set of functions Reed and Simon (1975)
where is the regular wave function with and the parameter represents a choice for the boundary condition. For each different , we have a possible domain for and the physical situation will determine the value of Filgueiras et al. (2010); Filgueiras and Moraes (2008); Andrade et al. (2012, 2013); Andrade and Silva (2013, 2014); Silva (2014) . Thus, in order to find a fitting for compatible with the physical situation, we require a physical regularization for the –function. This is accomplished by replacing Hagen (1990)
with representing the nucleus of a real physical system.
In order to find the energy levels, we need at first to determine a value for compatible with the physics imposed by the regularized –function in Eq. (39). Following Kay and Studer (1991), we consider the zero–energy solutions and for with the regularization in Eq. (39) and , respectively, i.e.,
and the value of is determined by the boundary condition
Using the series expansion for ,
into the Eq. (44), we can get the following asymptotic form ():
Using this result, one finds
This result gives us the parameter compatible with the physics imposed by the problem. In other words, it gives the correct behavior for the wave function when .
We are now in position to determine the bound states for . So, we write Eq.(35) for the bound state. In the present system, the energy of a bound state has to be negative so that is a pure imaginary number, with and . Then, by exchanging , we have
The solution of Eq. (50) is given by the modified Bessel function
A straightforward calculation yields
Solving the above equation for , we find the sought energy spectrum
Notice that there is no arbitrary parameter in the above equation. Moreover, in order to ensure that the energy is a real number, we must have
This inequality is satisfied if and due to the fact taht , it is sufficient to consider . As shown in Table 1, a necessary condition for a –function to generate an attractive potential, which is able to support bound states, is that the coupling constant must be negative.
5.2 Quantum dynamics on a cone
As mentioned above, the only possibility to generate a cone is , implying only if (see Table 1). In this case, the Schrödinger equation reads (with ) as
plus self–adjoint extensions, with
whose solution outside the origin is now given by
which is the modified Bessel function of purely imaginary order Abramowitz and Stegun (1964).
Next, following the same recipe of the previous section, we must solve the following equations:
An asymptotic expansion for the modified Bessel function of the pure imaginary order is obtained by replacing by in Eq. (46) and by writing
where is the Coulomb phase shift Taylor (2006). In this manner, we get at
By using the boundary condition (42), we obtain
By replacing the above expressions in Eq. (65), we get
Solving this equation for , we find
Equation (69) reveals that, while the mean curvature contributes attractively, the Gaussian curvature contributes with a repulsive –function potential. Thus, Eq. (27) implies that the only allowable value for the angular momentum is , meaning that we have a single bound state. In other words, when the –function potential is repulsive () an attractive effective potential potential assures one bound state ().
In this paper, we have studied the dynamics of a spinless charged particle which moves bounded to a 2D surface immersed in an 3D Euclidean space and in the presence of the Aharonov–Bohm potential. In other words, we have solved a spinless Aharonov–Bohm–like problem in curved space. The motion of the particle is decomposed into two, being one on the surface and the other in a normal direction in relation to the surface. The dynamics on the surface is governed by the Schrödinger equation (Eq. (8)) coupled to the potential vector, while on the normal direction the dynamics is given by Eq. (7), which is the equation for an infinity curved quantum well. The surface mean and Gaussian curvatures enter in the Schrödinger equation as a scalar potential, namely, as a geometric potential. We chose a conical surface which is defined by the metric endowed in the line element in Eq. ( 10).
A particularity of this system is that the isotropy of space is broken which means that the velocity operators do not commute with each other. Such a feature is usually due to the presence of a magnetic field. However, in the context, we also have effects of the geometry of the surface since the connection of the space plays an important role. To describe the full dynamics of the system, we have analyzed three situations. First, we have found the energy eigenvalues and wave functions for the simple case of the particle describing a circular path around the solenoid. In the other two cases, we have considered the dynamics in the full space, including the region. For those cases, the geometry of the system dictated by the line element in Eq. (10) establishes that the motion of a particle can occurs on the surface of a cone (for ) or on the surface of an anti–cone (for ). Expressions for the bound state energies and wave functions were obtained for both cases.
We acknowledge some suggestions made by the anonymous referees in order to improve the present work. This work was supported by the CNPq, Brazil, Grants Nos. 482015/2013–6 (Universal), 476267/2013–7 (Universal), 460404/2014-8 (Universal), 306068/2013–3 (PQ), 206224/2014-1 (PDE) and FAPEMA, Brazil, Grants Nos. 00845/13 (Universal), 01852/14 (PRONEM).
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