The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II.
This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, () and (), to the standard spherical waves in . The curvature is considered as a parameter and for any we show how the radial Schrödinger equation can be transformed into a -dependent Gauss hypergeometric equation that can be considered as a -deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both -dependent and -independent), and are explicitly obtained.
Keywords: Quantization. Quantum mechanics on spaces of constant curvature. Curvature-dependent orthogonal polynomials.
Running title: The quantum free particle on three-dimensional spaces with curvature.
MSC Classification: 81Q05, 81R12, 81U15, 34B24
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This article can be considered as a sequel or continuation of a previous paper , which was devoted to the study of the quantum free particle on two-dimensional spherical and hyperbolic spaces making use of a formalism that considers the curvature as a parameter. Now, we present a similar analysis but introducing two changes related with the dimension of the space and with the states of the quantum free particle we are looking for. Now we work in a three-dimensional space, and we look for the states analogous to the Euclidean spherical waves, which are determined among all free particle states by the condition of being separable in the geodesic polar coordinate system. We follow the approach of , which contains the fundamental ideas and motivations, and we also use the notation, ideas, and results discussed in some related previous studies -.
There are two articles that are considered of great importance in the study of mechanical systems in a spherical geometry (see  for a more detailed information; we just make here a quick survey in a rather telegraphic way). Schrödinger studied in 1940 the hydrogen atom in a spherical space  and then other authors studied this problem (hydrogen atom or Kepler problem) - or other related questions (as, e.g., the quantum oscillator on curved spaces) -. Higgs studied in 1979 the existence of dynamical symmetries in a spherical geometry  and since then a certain number of authors have considered - the problem of the symmetries or some other properties characterizing the Hamiltonian systems on curved spaces (the studies of Schrödinger and Higgs were concerned with a spherical geometry but other authors applied their ideas to the hyperbolic space). In fact these two problems, the so-called Bertrand potentials, have been the two problems mainly studied in curved spaces (at the two levels, classical and quantum). Nevertheless in quantum mechanics there are some previous problems that are of fundamental importance as, for example, the quantum free particle or the particle in a spherical well.
This article is concerned with the study of the spherical waves for a quantum free particle on spherical and hyperbolic spaces (an analogous problem was studied in  but in three dimensions and making use of -dependent parallel geodesic coordinates, which affords the analogous of plane waves). There are some questions as, for example, (i) analysis of some -dependent geometric formalisms appropriate for the description of the dynamics on the spaces with constant curvature , (ii) transition from the classical -dependent system to the quantum one, (iii) analysis of the Schrödinger separability and quantum superintegrability on spaces with curvature that have been discussed in , and therefore they are now omitted (or revisited in a very sketchy way). Thus, this paper is mainly concerned with the exact resolution of the -dependent Schrödinger equation, existence of bound states, and with the associated families of orthogonal polynomials.
In more detail, the plan of the article is as follows: In Sec. 2 we first study the Lagrangian formalism, the existence of Noether symmetries and Noether momenta and then the -dependent Hamiltonian and the quantization via Noether momenta. In Sec. 3, we solve the -dependent Schrödinger equation and then we analyze with great detail the spherical case, writing explicitly the spherical waves on a 3D-sphere and discussing their Euclidean limit when the curvature of the sphere goes to 0. The study of the hyperbolic case is only sketched, but the details displays several important differences which would require a separate study. Finally, in Sec. 4 we make some final comments.
2 Geodesic motion, -dependent formalism and quantization
We first present a brief introductory comment on some possible approaches to the two-dimensional manifolds with constant curvature : the sphere (), Euclidean plane , and hyperbolic plane (), and then we move to the corresponding three-dimensional spaces: the sphere (), Euclidean space , and hyperbolic space ().
If we make use of the following -dependent trigonometric (hyperbolic) functions
then the expression of the differential element of distance in geodesic polar coordinates on can be written as follows
so it reduces to
in the three particular cases of the unit sphere, the Euclidean plane, and the ‘unit‘ Lobachewski plane. If we make use of this formalism then the Lagrangian of the geodesic motion (free particle) on is given by [5, 28, 29]
Now if we consider the -dependent change then the Lagrangian becomes
and, if we change to ‘cartesian coordinates for ’ defined as , we arrive to
We notice that in the sphere case, in addition to the usual geodesic polar coordinate singularity at the origin (the ‘North pole’) , which passes to the coordinate at , the chart covers only the upper hemisphere as the coordinate ceases to be related to on a one-to-one basis at the equator , where reaches its maximum; however the lower hemisphere can be also covered by another chart, with still given by with a singularity at , which on the lower hemisphere corresponds to , the point antipodal to the origin (the ‘South pole’).
2.1 Lagrangian formalism, Noether symmetries and Noether momenta
Let us start with the following expression for the differential element of distance in the family of three-dimensional spaces with constant curvature written in coordinates (recall is not the geodesic radial coordinate):
reducing in the particular cases of unit sphere, Euclidean plane, and ‘unit’ Lobachewski plane to
Then the following six vector fields
are Killing vector fields, that is, the infinitesimal generators of isometries of the -dependent metric .
The Lie brackets of the vector fields are given by
with given by . The other Lie brackets are -independent and similar to the Lie brackets of the Euclidean case; that is
and so on. All these Killing vector fields close a Lie algebra that is isomorphic to the Lie algebra of the group of isometries (either ) of the spherical, Euclidean or hyperbolic spaces depending of the sign of . Notice that only when (Euclidean space), the vector fields , , will commute between themselves.
Now, let us consider the geodesic motion on , that is, the dynamics determined by a Lagrangian , which reduces to the -dependent kinetic term without a potential
where the parameter can take both positive and negative values. We already mentioned that in the spherical case the coordinate chart we are dealing with covers only the ‘upper’ half-sphere; we see that this Lagrangian becomes singular at the ‘equator’ where , and hence , so in this case the study of the dynamics will be restricted to the interior of the interval which corresponds to the upper half sphere.
The Lagrangian is invariant under the action of the the -dependent vector fields and , , in the sense that, if we denote by and the natural lift to the tangent bundle (phase space with representing , , or ) of the vector fields and , , then the Lie derivatives of vanish, that is
They represent six exact Noether symmetries for the geodesic motion. If we denote by the Lagrangian one-form
then the associated Noether constants of the motion are given by the following:
The three functions , , and , defined as
that are -dependent and given by
The three functions , , and , defined as
that are -independent functions and given by
2.2 -dependent Hamiltonian and Quantization
The standard method for the quantization of a Hamiltonian on a Riemannian manifold is to make use of the Laplace-Beltrami operator for the free part (kinetic energy) of the Hamiltonian. Nevertheless we recall that the standard procedure in a Euclidean space is to first quantize the momenta as self-adjoint operators and then, making use of the quantum momenta, to obtain the quantum version of the Hamiltonian. Our idea is to translate this momentum-approach to the case of spaces with curvature but changing the quantization of the canonical momenta by the quantization of the Noether momenta which are taken as the basic objects (this is one of the reasons why we have studied the properties of the Killing vectors and Noether momenta with great detail). So, we present the quantization of the system in two steps: (i) quantization of the Noether momenta as self-adjoint operators and then (ii) quantization of the Hamiltonain making use of the quantum Noether momenta.
The Legendre transformation is given by
so that the expression of the -dependent Hamiltonian turns out to be
The six Noether momenta become
with Poisson brackets
The other Poisson brackets are similar to the Poisson brackets of the Euclidean case; that is,
and so on. Note the change of the order in the Poisson brackets. This is motivated because of the property ; that is, the map is a Lie algebra isomorphism but with a change of the sign.
Making use of this formalism, the Hamiltonian of the -dependent oscillator can be rewritten as follows
The only measure that is invariant under the action of the three vector fields and the three vector fields , is given in coordinates and up to a constant factor by
This property is proved as follows. The most general expression for a volume three-form is given by
where is a differentiable function to be determined. Then the conditions
lead to the following value for the function :
where is an arbitrary constant. Assuming we obtain .
This property suggests the appropriate procedure for obtaining the quantization of the Hamiltonian . The idea is to work with functions and linear operators defined on the space obtained by considering the three-dimensional space endowed with the measure . This means, in the first place, that the operators , , and , representing the quantum version of of the Noether momenta , , an must be self-adjoint not in the standard space but in the space . If we assume the following correspondence:
then we have
in such a way that the quantum Hamiltonian
is represented by the following differential operator:
We note that this operator is self-adjoint with respect the measure and also that it satisfies the appropriate Euclidean limit (in this limit goes to the Euclidean radial coordinate , so to conform with the standard Euclidean usage we write in this expression):
Note that if we write with , , then , , because of the -dependent terms. Nevertheless , , , so that can be written as sum of two operators that conmute in several different ways. Finally we also note that can also be written as
that corresponds to the approach considered in this paper.
We close this section with the following observations:
Only for reference, we mention that had we used the polar geodesic coordinates , then the Hamiltonian would have be represented by the following differential operator:
The measure was introduced as the unique measure (up to a multiplicative constant) invariant under the Killing vectors. We have verified that it coincides with the corresponding Riemann volume in a space with curvature .
The Noether momenta quantization procedure was motivated in the first paragraph of section 2.2. At this point we must clearly state that the final result (that is, the expression of the Hamiltonian operator) coincides with the one that would be obtained making use of the Laplace-Beltrami quantization. In fact, the fundamental point for the validity of our approach was that the Hamiltonian is the quadratic Casimir of the isometry algebra; this leads to the Laplace-Beltrami operator.
3 -dependent Schrödinger equation
The Schrödinger equation:
leads in the coordinates we are using to the following -dependent differential equation:
Thus, if we assume that can be factorized in the form
where is a function of and are the standard -independent spherical harmonics
then we arrive to the following -dependent radial equation:
that can be rewritten in the form
where is defined for any value of the curvature so that it bears with the energy the same relation as the modulus of the wave vector has with in the Euclidean case:
If we adimensionalyze the radial variable and the curvature to the new adimensional variables
then we arrive to the following equation for the function :
that represents a -dependent deformation of the spherical Bessel differential equation
This ‘deformed’ spherical Bessel equation can be solved in power series using the method of Frobenius. First the function must be written as follows
and then it is proved that must take one of the two values or . Choosing , in order to translate to the unknown function the condition has to satisfy to be well defined at the origin in a simpler form, we arrive at
4 Spherical case
Let us now consider the spherical case . Before starting, we recall that the coordinates we are using cover only the ‘upper’ half the the sphere (where ranges in the interval ), so the range of is , and the range of the new variable is . A quite similar coordinate chart (with the same relation with ) covers the other half, so at the end our results will cover the whole sphere.
We will prove that the equation (15) admits two different types of solutions.
4.1 Solutions of type I
Assuming a -dependent power series for
then the -dependent recursion relation leads to the vanishing of all the odd coefficients, , so that is a series with only even powers of and a radius of convergence given by (determined by the presence of the second singularity). The even powers dependence suggests to introduce the new variable so that the equation becomes
As we have , it is convenient to complement the previously suggested variable change with a further last change , with the range for . Then the equation (16) reduces to
that is, a Gauss hypergeometric equation
and the solution regular at is the hypergeometric function
with and given by
The equation has a singularity at that corresponds to (this is, to or to ). If the origin is placed in the ‘north pole’ of the sphere then this singularity is just at the equator, which is the boundary of the domain covered by the coordinate chart . The property of regularity of the solutions leads to analyze the existence of particular solutions well defined at this point. The polynomial solutions appear when one of the two -dependent coefficients, or , coincide with zero or with a negative integer number
Then, in this case, we have
that can be writen as
Therefore the coefficient that represents the sphere analogue of the modulus of the wave vector of the spherical wave is restricted to one of the values given in the discrete set
and then the hypergeometric series reduces to a polynomial of degree in the variable .
Coming back to the equation (12), that was written making use of the radial variable , then the recurrence relation leads to the following values for the even coefficients