Discrete spectra for confined and unconfined -a/r + b r^2 potentials in d dimensions
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Discrete spectra for confined and unconfined − a/r + br potentials in d -dimensions Richard L. Hall , Nasser Saad , and K. D. Sen Department of Mathematics and Statistics, Concordia University,1455 de Maisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1M8 ∗ Department of Mathematics and Statistics, University of Prince Edward Island,550 University Avenue, Charlottetown, PEI, Canada C1A 4P3. † and School of Chemistry, University of Hyderabad 500046, India. ‡ Exact solutions to the d -dimensional Schr¨odinger equation, d ≥
2, for Coulomb plus harmonicoscillator potentials V ( r ) = − a/r + br , b > a = 0 are obtained. The potential V ( r ) isconsidered both in all space, and under the condition of spherical confinement inside an impenetrablespherical box of radius R . With the aid of the asymptotic iteration method, the exact analyticsolutions under certain constraints, and general approximate solutions, are obtained. These exhibitthe parametric dependence of the eigenenergies on a , b , and R . The wave functions have thesimple form of a product of a power function, an exponential function, and a polynomial. In orderto achieve our results the question of determining the polynomial solutions of the second-orderdifferential equation k +2 X i =0 a k +2 ,i r k +2 − i ! y ′′ + k +1 X i =0 a k +1 ,i r k +1 − i ! y ′ − k X i =0 τ k,i r k − i ! y = 0for k = 0 , , PACS numbers: 31.15.-p 31.10.+z 36.10.Ee 36.20.Kd 03.65.Ge.Keywords: oscillator confinement, confined hydrogen atom, discrete spectrum, asymptotic iteration method,polynomial solutions of differential equations.
I. INTRODUCTIONA. Formulation of the problem in d dimensions The d -dimensional Schr¨odinger equation, in atomic units ~ = µ = 1, with a spherically symmetric potential V ( r ) canbe written as (cid:20) −
12 ∆ d + V ( r ) (cid:21) ψ ( r ) = Eψ ( r ) , (1)where ∆ d is the d -dimensional Laplacian operator and r = P di =1 x i . Following [1], in order to transform (1) to the d -dimensional spherical coordinates ( r, θ , θ , . . . , θ d − ), we separate variables using ψ ( r ) = r ( d − / u ( r ) Y l d − ...l ( θ . . . θ d − ) , (2)where Y l d − ...l ( θ . . . θ d − ) is a normalized spherical harmonic with characteristic value l ( l + d − , l = 0 , , , . . . (theangular quantum numbers), one obtains the radial Schr¨odinger equation as (cid:20) − (cid:18) d dr − ( k − k − r (cid:19) + V ( r ) − E (cid:21) u ( r ) = 0 , Z ∞ u ( r ) dr = 1 , u (0) = 0 , (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] where k = d + 2 l . Assume that the potential V ( r ) is less singular than the centrifugal term so that u ( r ) ∼ r ( k −
1) ( r → . We note that the Hamiltonian and boundary conditions of (3) are invariant under the transformation( d, l ) → ( d ∓ , l ± . Thus, given any solution for fixed d and l , we can immediately generate others for different values of d and l .Further, the energy is unchanged if k = 2 ℓ + d and the number of nodes n is constant. Repeated application of thistransformation produces a large collection of states, the only apparent limitation being a lack of interest in somevalues of d (see, for example [2]). In the present work, we consider the Coulomb plus a harmonic oscillator potential V ( r ) = − ar + br , b > r = k r k denotes the hyper-radius, and the coefficients a and b are both constant. B. Degeneracy in spherically confined d -dimensional quantum model systems Since the early days of quantum mechanics there has been interest in studying the Schr¨odinger equation withmodel systems in higher spatial dimensions [3–6]. The so-called accidental degeneracy of the hydrogen atom andisotropic harmonic oscillator, characterized by different sets of parity conditions, is generally understood in terms of thecorresponding SO (4) and SU (3) symmetry groups [7, 8]. Following the introduction of ‘interdimensional degeneracies’[9, 10] there have been several reports involving arbitrary d -dimensional analyses covering many branches of chemicalphysics which have been briefly reviewed in Refs. [11–14]. It is interesting to note here that the information-theoreticaluncertainty-like relationships in terms of the Shannon entropy [15, 16] and the Fisher measure [17, 18] are also statedin d -dimensional form.Owing to the recent interest in quantum dots and fullerine encapsulated electronic systems there has been anupsurge of interest in studying model quantum systems confined inside an impenetrable sphere of radius R . We shallpresent here a brief description of the new degeneracy-related changes which are known to occur in the d -dimensionalH atom V c = − a/r and the isotropic harmonic oscillator V h = br . The eigenspectrum of the spherically confined Hatom (SCHA) is characterized by three kinds of degeneracy [19]. Two of them are generated from the specific choice ofthe radius of confinement R , chosen exactly at the radial nodes corresponding to the free hydrogen atom (FHA) wavefunctions. In the incidental degeneracy case, the confined ( ν, ℓ ) state with the principal quantum number ν is iso-energic with ( ν + 1 , ℓ ) state of the FHA with energy − / { ν + 1) } atomic units (a.u.), at an R defined by the radialnode in the FHA. For example, the ( ν, ℓ ) state corresponding to the lowest energy value, when confined at the radius R given by the radial node the first excited free state ( ν + 1 , ℓ ), increases in such a way that the confined-state energybecomes the same as excited free-state energy. The specific node in question is given by R = 0 . ℓ + d − ℓ + d +1).Such a degeneracy can be realized at similar choices for R where multiple nodes exist in the second and higher excitedstates of a given ℓ . However, such closed analytical expressions for the radial nodes are not available in the case ofhigher excited states. In the simultaneous-degeneracy case, on the other hand, for all ν ≥ ℓ + 2, each pair of confinedstates denoted by ( ν, ℓ ) and ( n + 1 , ℓ + 2) state, confined at the common R = 0 . ℓ + d − ℓ + d + 1), becomedegenerate. Note that the pair of levels in the free state are nondegenerate. Both these degeneracies have beenshown [19] to result from the Gauss relationship applied at a unique R c by the confluent hypergeometric functionsthat describe the general solutions of the SCHA problem. Finally, the interdimensional degeneracy [9, 10] arises,as in the case of the free hydrogen atom, due to the invariance of the Schr¨odinger equation to the transformation( ℓ, d ) → ( ℓ ± , d ∓ ν → ν + 1. The incidental degeneracy observed in the case of a sphericallyconfined isotropic harmonic oscillator (SCIHO) is qualitatively similar to that of the SCHA. For example, the onlyradial node in the first excited free state of any given ℓ for d -dimensional SCIHO is located at R = p (2 ℓ + d ) /
2. Forthe multiple node states, the corresponding numerical values must be used. However, the behavior of the two confinedstates at a common radius of confinement is found to be interestingly different [20, 21]. In particular, for the SCIHOthe pairs of the confined states defined by ( ν = ℓ + 1 , ℓ ) and ( ν = ℓ + 2 , ℓ + 2) at the common R = p (2 ℓ + d ) / ν , a constant energy separation of exactly ~ ω , with the state of higher ℓ corresponding to the lower energy. It is interesting to note that the two confined states at the common R with ∆ ℓ = 2,considered above contain different numbers of radial nodes. The condition for interdimensional degeneracy[9, 10] dueto the invariance of the Schr¨odinger equation remains the same as before. Recently, the confined systems of the d -dimensional hydrogen atom [22] and harmonic oscillator [23] have been studied. Problems involving short-rangepotentials in d dimensions have recently been considered [24, 25]. In the light of the above discussion, it is interestingto study the various aforementioned degeneracies in the free and spherically confined d -dimensional potential generallygiven by V ( r ) = V c + V h = − a/r + br . C. Organization of the paper
The present paper is organized as follows. In section 2, we discuss some general spectral features and bounds, insection 3 we briefly review the asymptotic iteration method of solving a second-order linear differential equationwhere we discuss the necessary and sufficient conditions for certain classes of differential equations with polynomialcoefficients to have polynomial solutions. In sections 4 and 5, we use the asymptotic iteration method (AIM) to studyhow the eigenvalues depend on the potential parameters a, b, R , repectively for the free system ( R = ∞ ), and for finite R . In each of these sections, the results obtained are of two types: exact analytic results that are valid when certainparametric constraints are satisfied, and accurate numerical values for arbitrary sets of potential parameters. II. SOME GENERAL SPECTRAL FEATURES AND ANALYTICAL ENERGY BOUNDS
We shall show shortly that the Hamiltonian H is bounded below. The eigenvalues of H may therefore be char-acterized variationally. The eigenvalues E dn,ℓ = E ( a, b, R ) are monotonic in each parameter. For a and b , this is adirect consequence of the monotonicity of the potential V in these parameters. Indeed, since ∂V /∂a = − /r < ∂V /∂b = r >
0, it follows that ∂E ( a, b, R ) ∂a < ∂E ( a, b, R ) ∂b > . (5)The monotonicity with respect to the box size R may be proved by a variational argument. Let us consider two boxsizes, R < R and an angular momentum subspace labelled by a fixed ℓ. We extend the domains of the wave functionsin the finite-dimensional subspace spanned by the first N radial eigenfunctions for R = R so that the new space W may be used to study the case R = R . We do this by defining the extended eigenfunctions so that ψ i ( r ) = 0 for R ≤ r ≤ R . We now look at H in W with box size R . The minima of the energy matrix [( ψ i , Hψ j )] are the exacteigenvalues for R and, by the Rayleigh-Ritz principle, these values are one-by-one upper bounds to the eigenvaluesfor R . Thus, by formal argument we deduce what is perhaps intuitively clear, that the eigenvalues increase as R isdecreased, that is to say ∂E ( a, b, R ) ∂R < . (6)From a classical point of view, this Heisenberg-uncertainty effect is perhaps counter intuitive: if we try to squeeze theelectron into the Coulomb well by reducing R , the reverse happens; eventually, the eigenvalues become positive andarbitrarily large, and less and less affected by the presence of the Coulomb singularity.For some of our results we shall consider the system unconstrained by a spherical box, that is to say R = ∞ . For thesecases, we shall write E dnℓ = E ( a, b ) . If a very special box is now considered, whose size R coincides with any radialnode of the R = ∞ problem, then the two problems share an eigenvalue exactly. This is an example of a very generalrelation which exists between constrained and unconstrained eigensystems, and, indeed, also between two constrainedsystems with different box sizes.The generalized Heisenberg uncertainty relation may be expressed [26, 27] for dimension d ≥ − ∆ > ( d − / (4 r ) . This allows us to construct the following lower energy bound
E > E = min
FIG. 1: The energy E for a = 1, b = , d = 3 as a function of L = ℓ for n = 0 , , . For the unconstrained case R = ∞ , however, envelope methods [29–33, 35] allow one to construct analytical upperand lower energy bounds with general forms similar to (7). In this case we shall write E dnℓ = E ( a, b ) . Upper and lowerbounds on the eigenvalues are based on the geometrical fact that V ( r ) is at once a concave function V ( r ) = g (1) ( r )of r and a convex function V ( r ) = g (2) ( − /r ) of − /r . Thus tangents to the g functions are either shifted scaledoscillators above V ( r ), or shifted scaled atoms below V ( r ). The resulting energy-bound formulas are given bymin r> (cid:20) r − aP r + b ( P r ) (cid:21) ≤ E dnℓ ( a, b ) ≤ min r> (cid:20) r − aP r + b ( P r ) (cid:21) , (8)where (Ref. [36] Eqs.(1.11) and (1.12a)) P = n + ℓ + ( d − / P = 2 n + ℓ + d/ . (9)We shall sometimes use also the convention of atomic physics in which, even for non-Coulombic central potentials, aprincipal quantum number ν is used and defined by ν = n + ℓ + ( d − / , (10)where n = 0 , , , . . . is the number of nodes in the radial wave function. It is clear that the lower energy bound hasthe Coulombic degeneracies, and the upper bound those of the harmonic oscillator. These bounds are very helpful asa guide when we seek very accurate numerical estimates for these eigenvalues.Another related estimate is given by the ‘sum approximation’ [33] which is more accurate than (8) and is known tobe a lower energy bound for the bottom E d ℓ of each angular-momentum subspace. The estimate is given by E dnℓ ( a, b ) ≈ E dnℓ ( a, b ) = min r> (cid:20) r − aP r + b ( P r ) (cid:21) . (11)This energy formula has the attractive spectral interpolation property that it is exact whenever a or b is zero. Theenergy bounds (8) and (11) obey the same scaling and monotonicity laws is those of E dnℓ ( a, b ) . Because of theirsimplicity they allow one to extract analytical properties of the eigenvalues. For example, in Fig. 1 we show fromEq.(11) approximately how the eigenvalue E nℓ (1 , ) depends on ℓ for n = 0 , , . III. THE ASYMPTOTIC ITERATION METHOD AND SOME RELATED RESULTS
The asymptotic iteration method (AIM) was originally introduced [37] to investigate the solutions of differentialequations of the form y ′′ = λ ( r ) y ′ + s ( r ) y, ( ′ = ddr ) (12)where λ ( r ) and s ( r ) are C ∞ − differentiable functions. A key feature of this method is to note the invariant structureof the right-hand side of (12) under further differentiation. Indeed, if we differentiate (12) with respect to r , we obtain y ′′′ = λ y ′ + s y (13)where λ = λ ′ + s + λ and s = s ′ + s λ . If we find the second derivative of equation (12), we obtain y (4) = λ y ′ + s y (14)where λ = λ ′ + s + λ λ and s = s ′ + s λ . Thus, for ( n + 1) th and ( n + 2) th derivative of (12), n = 1 , , . . . , wehave y ( n +1) = λ n − y ′ + s n − y (15)and y ( n +2) = λ n y ′ + s n y (16)respectively, where λ n = λ ′ n − + s n − + λ λ n − and s n = s ′ n − + s λ n − . (17)From (15) and (16) we have λ n y ( n +1) − λ n − y ( n +2) = δ n y where δ n = λ n s n − − λ n − s n . (18)Clearly, from (18) if y , the solution of (12), is a polynomial of degree n , then δ n ≡
0. Further, if δ n = 0, then δ n ′ = 0for all n ′ ≥ n . In an earlier paper [37] we proved the principal theorem of AIM, namely Theorem 1 [37].
Given λ and s in C ∞ ( a, b ) , the differential equation (12) has the general solution y ( r ) = exp − r Z s n − ( t ) λ n − ( t ) dt C + C r Z exp t Z ( λ ( τ ) + 2 s n − λ n − ( τ )) dτ dt (19) if for some n > δ n = λ n s n − − λ n − s n = 0 . (20) where λ n and s n are given by (17). Recently, it has been shown [38] that the termination condition (20) is necessary and sufficient for the differentialequation (12) to have polynomial-type solutions of degrees at most n , as we may conclude from Eq.(18). Thus, usingTheorem 1, we can now find the necessary and sufficient conditions [39] for the polynomial solutions of the differentialequation ( a , r + a , r + a , r + a , ) y ′′ + ( a , r + a , r + a , ) y ′ − ( τ , r + τ , ) y = 0 , (21)where a k,j , k = 3 , , , j = 0 , , , Theorem 2 [[39] Theorem 5].
The second-order linear differential equation (21) has a polynomial solution ofdegree n if τ , = n ( n − a , + n a , , n = 0 , , , . . . , (22) provided a , + a , = 0 along with the vanishing of ( n + 1) × ( n + 1) -determinant ∆ n +1 given by ∆ n +1 = β α η γ β α η γ β α η . . . . . . . . . . . . γ n − β n − α n − η n − γ n − β n − α n γ n β n = 0 where all the other entires are zeros and β n = τ , − n (( n − a , + a , ) α n = − n (( n − a , + a , ) γ n = τ , − ( n − n − a , + a , ) η n = − n ( n + 1) a , . (23) Here τ , is fixed for a given n in the determinant ∆ n +1 = 0 (the degree of the polynomial solution). The coefficientsof the polynomial solutions y n ( r ) = P ni =0 c i r i satisfies the four-term recursive relation ( i + 2)( i + 1) a , c i +2 + [ i ( i + 1) a , + ( i + 1) a , ] c i +1 + [ i ( i − a , + ia , − τ , ] c i + [( i − i − a , + ( i − a , − τ , ] c i − = 0 . (24)The results of this theorem go beyond the question of finding the polynomial solutions of the second-order lineardifferential equation ( a , r + a , r + a , ) y ′′ + ( a , r + a , ) y ′ − τ , y = 0 . (25)Indeed Eq.(25) has a nontrivial polynomial solution of degree (exactly) n ∈ N (the set of nonnegative integers) if, forfixed n , τ , = n ( n − a , + n a , , n = 0 , , , . . . (26)provided a , + a , = 0 where the polynomials y n , up to a multiplicative constant, may be readily obtained from thethree-term recurrence relation: y n +2 = (cid:20) A n x + B n (cid:21) y n +1 + C n y n , n ≥ A n = ((2 n + 1) a , + a , )(2( n + 1) a , + a , )( na , + a , ) ,B n = ((2 n + 1) a , + a , )(2 n ( n + 1) a , a , + 2( n + 1) a , a , − a , a , + a , a , )( na , + a , )(2 na , + a , ) (cid:21) ,C n = ( n + 1)(2( n + 1) a , + a , )((4 a , a , − a , a , ) n + (4 a , a , a , − a , a , ) n + a , a , − a , a , a , + a , a , )( na , + a , )(2 na , + a , ) , initiated with y = 1 , y = a , x + a , . In the next sections, we shall apply the result of theorem 2 to study the possible quasi-exact analytic solutions for the d -dimension Schr¨odinger equation (3) for unconstrained and constrained Coulomb plus harmonic oscillator potential(4). We shall also apply AIM, theorem 1, to obtain accurate approximations for arbitrary potential parameters, again,for the unconstrained and constrained d -dimensional Schr¨odinger equation (3). IV. EXACT AND APPROXIMATE SOLUTIONS FOR UNCONSTRAINED POTENTIAL V ( r ) A. Exact bound-state solutions of a Coulomb plus harmonic oscillator potential in d -dimensions In this section, we consider the d -dimensional Schr¨odinger equation (cid:20) − (cid:18) d dr − ( k − k − r (cid:19) − ar + br (cid:21) u dnl ( r ) = E dnl u dnl ( r ) , < r < ∞ . (28)In order to solve this equation by using AIM, the first step is to transform (28) into the standard form (12). To thisend, we note that the differential equation (28) has one regular singular point at r = 0 and an irregular singular pointat r = ∞ and, since for large r , the harmonic oscillator term dominates, the asymptotic solution of (28) as r → ∞ is u r →∞ ∼ exp( − p b/ r ); meanwhile the indicial equation of (28) at the regular singular point r = 0 yields s ( s − −
14 ( k − k −
3) = 0 , (29)which is solved by s = 12 (3 − k ) , s = 12 ( k − . The value of s , in Eq.(29), determines the behavior of u dnl ( r ) for r →
0, and only s > / u dnl ( r ) = r ( k − exp( − r b r ) f n ( r ) , k = d + 3 l, (30)where we note that u dnl ( r ) ∼ r ( k − as r →
0. On substituting this ansatz wave function into (28), we obtain thedifferential equation for f n ( r ) as rf ′′ n ( r ) + (cid:16) − r √ b + k − (cid:17) f ′ n ( r ) + h(cid:16) E dnl − k √ b (cid:17) r + 2 a i f n ( r ) = 0 . (31)This equation is a special case of the differential equation (21) with a , = a , = a , = a , = 0, a , = 1, a , = − r √ b, a , = k − τ , = − E dnl + k √ b and τ , = − a . Thus, the necessary condition for the polynomialsolutions of Eq.(31) is E dnl = (2 n ′ + k ) r b , n ′ = 0 , , , . . . . (32)and the sufficient condition follows from the vanishing of the tridiagonal determinant ∆ n +1 = 0, n = 0 , , , . . . ,namely ∆ n +1 = β α γ β α γ β α . . . . . . . . . γ n − β n − α n − γ n − β n − α n γ n β n = 0where its entries are expressed in terms of the parameters of Eq.(31) by β n = − a, α n = − n ( n + k − , γ n = − n ′ − n + 1) √ b, (33)where n ′ = n is fixed by the size of the determinant ∆ n +1 = 0 and represent the degree of the polynomial solution ofEq.(31). We may note that, since the off-diagonal entries α i and γ i of the tridiagonal determinant satisfy the identity α i γ i > , ∀ i = 1 , , . . . , the latent roots of the determinant ∆ n +1 are all real and distinct [41]. Further, we can easily show that the determinant(33) satisfies a three-term recurrence relation∆ i = β i − ∆ i − − γ i − α i − ∆ i − , ∆ = 1 , ∆ − = 0 , i = 1 , , . . . (34)which can be used to compute the determinant ∆ i (and thus the sufficient conditions), recursively in terms of lowerorder determinants. In this case, however, we must fix n ′ for each of the sub-determinants used in computing (34).For example, in the case of n ′ = n = 1 (corresponding to a polynomial solution of degree one), we have∆ = − a − ( k − − √ b − a = β ∆ − γ α ∆ =( − a )( − a ) − ( − √ b )( − ( k − a − √ b ( k − , that is, the condition of the potential parameters reads2 a − √ b ( k −
1) = 0 . (35)For n ′ = n = 2 (corresponding to a second-degree polynomial solution)∆ = − a − ( k −
1) 0 − √ b − a − k − √ b − a = β ∆ − γ α ∆ =( − a ) − a − ( k − − √ b − a − ( − √ b )( − k )( − a )= ( − a )[( − a )∆ − ( − √ b )( − ( k − ]+8 ka √ b = 8 a ( − a + 2 √ bk − √ b )Consequently, we must have a ( a − √ bk + √ b ) = 0 . (36)In Table I, we give the conditions on the potential parameters to allow for polynomial solutions, from theorem 2. TABLE I: Conditions on the parameters a and b for the exact solutions of Eq.(28) with E dnl = (2 n + k ) p b/ k = d + 2 l . n ∆ n +1 = 00 a = 01 2 a − ( k − √ b = 02 a ( a − (2 k − √ b ) = 03 4 a − a √ bk − b (1 − k ) = 04 a ( a − √ b (2 k + 1) a + 4 b (8 k + 8 k − a − √ b ( k + 1) a + 4 b ( −
65 + 518 k + 259 k ) a − b √ b ( k − k + 3)( k + 1) = 0 It must be clear that although n , the degree of the polynomial solution, it is not necessarily an indication as to thenumber of the zeros of the wave function (node number): further analysis of the roots of f n ( r ) is usually needed tocompute the zeros of the wavefunction.The polynomial solutions f n ′ ( r ) = P n ′ i =0 c i r i can be easily constructed for each n ′ since, in this case, the coefficients c i satisfy the three-term recurrence relation (see Eq.(24)) c − = 0 , c = 1 , c i +1 = − ac i + 2( n ′ − i + 1) √ bc i − ( i + 1)( i + k − , i = 0 , , . . . , n ′ − , (37)where n ′ is the degree of the polynomial solution. When n ′ = 0, f ( r ) = 1. For the first-degree polynomial solution, n ′ = 1 , i = 0, we have c = − ak − , that is f ( r ) = 1 − ak − r, where 2 a − ( k − √ b = 0 . (38)We may further note for a <
0, there is no root of f ( r ) = 0 and the un-normalized wave function reads u d l ( r ) = r ( d +2 l − exp (cid:18) − a r d + 2 l − (cid:19) (cid:18) − ard + 2 l − (cid:19) , a < d and l . For a >
0, there is only one rootof f and the wave function u d l ( r ) = r ( d +2 l − exp (cid:18) − a r d + 2 l − (cid:19) (cid:18) − ard + 2 l − (cid:19) , a > d and l . The zero of this wave function is located at R = k − a , k = d + 2 l, a > . (41)In both cases, a > a <
0, the exact eigenvalues are given by E d l ≡ E d l ≡ E d ∓ l ± = a ( d + 2 l + 2)(2 d + 4 l − , lim d →∞ E d ∓ l ± = a . (42)For second-degree polynomial solution, n ′ = 2 , i = 0 ,
1, we have for the polynomial solution, f ( r ) = c + c r + c r ,coefficients c = 1 , c = − ak − c = 2( a − √ b ( k − k ( k − f ( r ) = 1 − ar ( k −
1) + 2 a r ( k − k − , (43)where a ( a − (2 k − √ b ) = 0 from which we may conclude that a − √ b ( k − >
0. Therefore, the wave function f ( r ) has either two roots or no root based on the value of a > a <
0, respectively. For a >
0, we have asecond-excited state wave function u d l ( r ) = r ( d +2 l − exp (cid:18) − a r d + 4 l − (cid:19) (cid:18) − ard + 2 l − a r ( d + 2 l − d + 4 l − (cid:19) , a > , (44)which has two zeros at R = 2 k − √ k − a , R = 2 k − − √ k − a , k = d + 2 l, a > . (45)For a <
0, we have a ground-state wave function u d l ( r ) = r ( d +2 l − exp (cid:18) − a r d + 4 l − (cid:19) (cid:18) − ard + 2 l − a r ( d + 2 l − d + 4 l − (cid:19) , a < . (46)In either case, a > a <
0, the exact eigenvalues reads E d l ≡ E d l ≡ E d ∓ l ± = a ( d + 2 l + 4)2(2 d + 4 l − , lim d →∞ E d ∓ l ± = a . (47)For third-degree polynomial solution, n ′ = 3 , i = 0 , ,
2, we have for the polynomial coefficients f ( r ) = c + c r + c r + c r that c = 1 , c = − ak − , c = (2 a − √ b ( k − k ( k − , c = − a (2 a − √ bk + 3 √ b )3( k − k ( k + 1) , and the polynomial solution then reads f ( r ) = 1 − ak − r + (2 a − k − √ b ) k ( k − r − a (2 a − (7 k − √ b )3( k − k ( k + 1) r , (48)where the potential parameters satisfy the condition 4 a − a √ bk + 18 b ( k −
1) = 0 which may be solved in termsof √ b as √ b = 2 a (5 k ± √ k + 9)9( k − . (49)From this we have f +3 ( r ) = 1 − ark − − a k − √ k + 9) r ( k + 1) k ( k −
1) + 49 a (26 k − k + 9 + (7 k − √ k + 9) r k − k ( k + 1) , (50)0and f − ( r ) = 1 − ark − − a k − − √ k + 9) r ( k + 1) k ( k −
1) + 49 a (26 k − k + 9 − (7 k − √ k + 9) r k − k ( k + 1) . (51)The polynomial f +3 ( r ) has two roots if a > a < r >
0; while f − ( r ) has no root for a < a > E d ± l = a d + 2 l + 6) (cid:16) d + 10 l ± p d + 2 l ) + 9 (cid:17) ( d + 2 l − d + 2 l + 1) , a = 0 . (52)We can also show for the fourth-degree polynomial solution, n ′ = 4 , i = 0 , , ,
3, we have f ( r ) = 1 − ak − r + (2 a − √ b ( k − k ( k − r + 4 a [ − a + 5 √ bk − √ b ]3( k − k ( k + 1) r + 2[ a − (1 + 8 k ) √ ba + 12 b ( k − k + 2)( k + 1) k ( k − r (53)subject to a ( a − √ b (2 k + 1) a + 4 b (8 k + 8 k − E d ± l = a k + 8) (cid:16) k + 5 ± p (2 k + 1) + 8 (cid:17) k + 1) − , k = d + 2 l, a = 0 (54)and similarly for other cases. Indeed, using the recurrence relation, Eq.(37), it is straightforward to compute explicitlythe polynomial solution of any required degree. B. Approximate solutions for arbitrary potential parameters on half-line
For arbitrary values of the potential parameters a and b that do not necessarily obey the above conditions, we mayuse AIM directly to compute the eigenvalues accurately , as the zeros of the termination condition (20). The methodcan be used, as well, to test the exact solutions we obtained in the above section. To utilize AIM, we start with λ ( r ) = 2 √ b r − ( k − r ,s ( r ) = − (2 E dnl − k √ b ) − ar (55)and computing the AIM sequences λ n and s n as given by Eq.(17). We should note that for given values of thepotential parameters a , b, and of k = d + 2 l , the termination condition δ n = λ n s n − − λ n − s n = 0 yields an expressionthat depends on both r and E . In order to use AIM as an approximation technique for computing the eigenvalues E we need to feed AIM with an initial value of r = r that could stabilize AIM (that is, to avoid oscillations). Forour calculations, we have found that r = 3 stablizes AIM and allows us to compute the eigenvalues for arbitrary k = d + 2 l and n as shown in Table II. There is no magical assertion about r = 3, indeed using an exact solvablecase, say E = 2 . d = 3 , l = 0 , n ′ = 1 for a = 1 and b = 1 /
2, we may approximate r = r by means of E − V ( r ) = 0 which yields r ∼ . Solve of Maple 13 . V. EXACT AND APPROXIMATE SOLUTIONS FOR CONSTRAINED POTENTIALA. Analytic solutions
We now consider the d -dimensional Schr¨odinger equation (cid:20) − (cid:18) d dr − ( k − k − r (cid:19) + V ( r ) (cid:21) u dnl ( r ) = E dnl u dnl ( r ) , < r < R, (56)1 TABLE II: Eigenvalues E d =2 , , , , , n for V ( r ) = − /r + r /
2. The initial value utilize AIM is r = 3. The subscript N refer tothe number of iteration used by AIM. n E d =2 n n E d =3 n − .
836 207 439 051 476 488 N =78 .
179 668 484 653 553 873 N =71 .
576 895 542 024 474 773 N =71 .
500 000 000 000 000 000 N =3 Exact .
828 388 290 161 145 035 N =64 .
631 952 408 873 053 214 N =59 .
963 137 645 125 787 098 N =60 .
712 595 725 661 429 760 N =58 .
052 626 115 348 259 660 N =59 .
769 519 600 328 899 714 N =57 .
118 396 975 257 306 974 N =58 .
812 924 292 726 383 736 N =56 .
169 728 962 611 565 630 N =58 .
847 666 480 105 796 414 N =55 n E d =4 n ≡ E n n E d =5 n ≡ E n .
039 629 453 693 666 062 N =63 .
709 018 091 123 552 219 N =60 .
191 127 807 756 594 984 N =58 .
801 929 609 626 278 046 N =55 .
273 870 062 099 308 315 N =57 .
860 357 172 819 176 603 N =55 .
329 588 502 119 331 779 N =54 .
902 317 748 608 790 676 N =52 .
371 002 235 676 830 145 N =53 .
934 707 216 855 127 521 N =51 .
403 631 794 919 620 038 N =53 .
960 878 210 587 747 585 N =51 .
430 355 223 285 870 950 N =53 .
982 705 338 927 982 296 N =50 n E d =6 n ≡ E n ≡ E n n E d =7 n ≡ E n ≡ E n .
311 633 609 259 797 633 N =56 .
882 228 025 698 769 118 N =52 .
376 287 059 247 773 643 N =52 .
930 673 420 047 524 772 N =50 .
420 575 455 465 976 803 N =52 .
965 837 318 124 071 248 N =48 .
453 864 208 404 376 310 N =49 .
993 183 541 000 280 301 N =47 .
480 309 010 248 775 417 N =50 .
015 405 735 685 306 483 N =47 .
502 107 717 572 917 269 N =48 .
034 024 645 001 717 876 N =47 .
520 559 118 487 876 168 N =48 .
049 980 236 362 344 263 N =47 where V ( r ) = − ar + br , if 0 < r < R ∞ if r ≥ R (57)and u dnl (0) = u dnl ( R ) = 0. We may assume the following ansatz for the wave function u dnl ( r ) = r ( k − ( R − r ) exp − r b r ! f n ( r ) , k = d + 2 l. (58)where R is the radius of confinement, and the ( R − r ) factor ensures that the radial wavefunction u dnl ( r ) vanishes atthe boundary r = R . On substituting (49) into (47), we obtain the following second-order differential equation forthe functions f n ( r ): f ′′ n ( r ) = − (cid:18) k − r − R − r − √ b r (cid:19) f ′ n ( r ) − r ( R − r ) (cid:20) ( − E dnl + ( k + 2) √ b ) r + ( R (2 E dnl − k √ b ) − a ) r − k + 1 + 2 Ra (cid:21) f n ( r ) . (59)We note that this equation reduces to Eq.(31) as R → ∞ . Equation (59) can be written as[ − r + Rr ] f ′′ n ( r ) + [2 √ br − √ bRr − ( k + 1) r + ( k − R ] f ′ n ( r )+ (cid:20) ( − E dnl + ( k + 2) √ b ) r + ( R (2 E dnl − k √ b ) − a ) r + 2 Ra − k + 1 (cid:21) f n ( r ) = 0 (60)2This differential equation cannot be studied using Theorem 2. Consequently a further investigation of the followingclass of differential equations( a , r + a , r + a , r + a , r + a , ) y ′′ + ( a , r + a , r + a , r + a , ) y ′ − ( τ , r + τ , r + τ , ) y = 0 , (61)is needed. Indeed, by using Theorem 1 and a proof along the lines of the proof of Theorem 2, we are able to establishthe following: Theorem 3.
The second-order linear differential equation (61) has a polynomial solution y ( r ) = P nk =0 c k r k if τ , = n ( n − a , + n a , , n = 0 , , , . . . , (62) provided a , + a , = 0 where the polynomial coefficients c n satisfy the five-term recurrence relation (( n − n − a , + ( n − a , − τ , ) c n − + (( n − n − a , + ( n − a , − τ , ) c n − + ( n ( n − a , + na , − τ , ) c n + ( n ( n + 1) a , + ( n + 1) a , ) c n +1 + ( n + 2)( n + 1) a , c n +2 = 0 (63) with c − = c − = 0 .In particular, for the zero-degree polynomials c = 0 and c n = 0 , n ≥ , we have τ , = 0 , τ , = 0 , τ , = 0 . (64) For the first-degree polynomial solution, c = 0 , c = 0 and c n = 0 , n ≥ , we must have τ , = a , (65) along with the vanishing of the two × -determinants, simultaneously, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − τ , a , − τ , a , − τ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − τ , a , − a , a , − τ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (66) For the second-degree polynomial solution, c = 0 , c = 0 , c = 0 and c n = 0 for n ≥ , we must have τ , = 2 a , + 2 a , (67) along with the vanishing of the two × -determinants, simultaneously, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − τ , a , a , − τ , a , − τ , a , + 2 a , − a , − a , a , − τ , a , + 2 a , − τ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − τ , a , a , − τ , a , − τ , a , + 2 a , − a , − a , a , + 2 a , − τ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (68) and so on, for higher-order polynomial solutions. The vanishing of these determinants can be regarded as the conditionsunder which the coefficients τ , and τ , of Eq.(61) are determined. Using Theorem 3, we may note, with a , = a , = a , = 0 , a , = − , a , = R, a , = 2 √ b, a , = − √ bR, a , = − ( k + 1) , a , = ( k − R, τ , = 2 E dnl − ( k + 2) √ b, τ , = − ( R (2 E dnl − k √ b ) − a ) , τ , = − Ra + k −
1, that thenecessary condition for polynomial solutions f n ( r ) = P nk =0 c k r k of Eq.(60) is E dnl = 12 (2 n + k + 2) √ b, k = d + 2 l, (69)where n refers to the degree of the polynomial solution and not necessarily to the number of zeros for the exact wavefunction. For sufficient conditions, we have for the zero-degree polynomial solution n = 0, that Eq.(64) yields f ( r ) = 1 , E d l = 12 ( k + 2) √ b, if a = R √ b and Ra = 12 ( k − , (70)where, again, k = d + 2 l . For example, if a = 3 , b = 4 .
5, we have R = 1 and for k = d + 2 l = 7, we have the exactsolution E = E = E = 13 . a = 4 , b = 8, we have R = 1 and for k = d + 2 l = 9, we have the exact solution E = E = E = E = 22 . Thus, for the values of the potential parameters a, b and R as given by( a, b, R ) = (cid:18) R (2 l + d − , R (2 l + d − , R (cid:19) , (71)we have the exact solutions E d l = R ( d + 2 l − d + 2 l + 2) ,u d l ( r ) = r (2 l + d − ( R − r ) exp( − d +2 l − R r ) . (72)We may note that the confinement size R = ( k − / (2 a ) represent the root of the unconfined wave function, (40),with the same energy (compare (42 with (70)).For first-degree polynomial solution n = 1, we have using (69), or τ , = 4 √ b , E d l = 12 ( k + 4) √ b, (73)along with the two conditions, obtained using (66), which relate the potential parameters by kRa − k ( k − − R a + 2 R √ b ( k −
1) = 0 , √ bR − Ra + k − . (74)where, in this case, the polynomial solution reads f ( r ) = 1 − (2 Ra + 1 − k ) R ( k − r. (75)Thus, for the relations ( a, b, R ) = (cid:18) k − √ k − R , ( k + √ k − R , R (cid:19) , k = d + 2 l (76)we have the exact solutions E d l = R ( k + 4)( k + √ k − ,u d l ( r ) = r ( k − ( R − r ) exp( − k + √ k − R r ) (cid:16) − ( k + √ k − R ( k − r (cid:17) . (77)and for ( a, b, R ) = (cid:18) k − − √ k − R , ( k − √ k − R , R (cid:19) , k = d + 2 l, (78)we have the exact solutions E d l = R ( k + 4)( k − √ k − ,u d l ( r ) = r ( k − ( R − r ) exp( − k −√ k − R r ) (cid:16) − ( k −√ k − R ( k − r (cid:17) . (79)We note that these exact-solutions cases (76) and (78) represent the nodes of the wavefunction in the infinite case(44).4For second-degree polynomial solutions n = 2, we have the exact eigenvalues E dnl = 12 ( k + 6) √ b (80)where k = d + 2 l and the potential parameters a , b and R are related by the following two conditions (obtained fromthe two determinants in (68)4 R a − k + 1) R a − R ( R √ b (7 k − − k ( k + 1)) a + 3( k − k + 1)(3 √ bR − k ) = 0 , (81)and 2 R a − R ( R √ b + k ) a − ( k − √ bR − k ) a + 6 b ( k − R = 0 . (82)In this case the exact solution reads u d l = r ( k − ( R − r ) exp − r b r ! − Ra − k + 1 R ( k − r + (2 R a − Rak + k ( k − − √ bR ( k − R k ( k − r ! . (83)Again in this case we can show that these exact solutions correspond to the zeros of the wavefunction in the infinitecase (50) and (51).Similar results can be obtained for higher n (the degree of the polynomial solutions). It is important to note thatthe conditions reported here are for the mixed potential V ( r ) = − a/r + br , where a = 0 and b = 0 (that is to say,neither coefficient is zero). B. Approximate solutions for confined potential with arbitrary parameters
For the arbitrary values of a, b and R , not necessarily satisfying the above conditions, we may use AIM directly tocompute the eigenvalues with a very high degree of accuracy. This also allows us to verify the exact solutions weobtained in the perevious sections. Similarly to the unconfined case, we start the iteration of the AIM sequence λ n and s n with λ ( r ) = − (cid:16) k − r − R − r − √ b r (cid:17) ,s ( r ) = − ( − E dnl +( k +2) √ b ) r +( R (2 E dnl − k √ b ) − a ) r +2 Ra − k +1 r ( R − r ) . (84)where 0 < r < R . It is interesting to note in this case, that, unlike the unconfined case, the roots of the terminationcondition δ n = λ n s n − − λ n − s n = 0 are much easier to handle in the present case. This is due to the fact that r is now bound within (0 , R ) for every given R . Thus, it is sufficient to start our iteration process with initial value r = R/
2. In table III, we reported the eigenvalues we have computed using AIM for a fixed radius of confinement R = 1, with r = 0 . N in Tables III. The same procedure can be applied to computethe eigenvalues for other values of a , b R , and arbitrary dimension d . The results of AIM may be obtained to anydegree of precision, although we have reported our results for only the first eighteen decimal places. It is clear fromthe table that our results confirm the invariance of the eigenvalues under the transformation ( d, l ) → ( d ∓ , l ± . VI. CONCLUSION
We study a model atom-like system − ∆ − a/r which is confined softly by the inclusion of a harmonic-oscillatorpotential term b r and possibly also by the presence of an impenetrable spherical box of radius R. For b >
R < ∞ , the entire spectrum E dn,ℓ ( a, b, R ) is discrete. We have studied these eigenvalues and we present an approximate spectralformula for the ‘free’ case, R = ∞ . For the general case of R ≤ ∞ , AIM has been used to provide both a large numberof exact analytical solutions, valid for certain special choices of the parameters { a, b, R } , and also very accuratenumerical eigenvalues for arbitrary parametric data. In the cases where we have found analytic solutions for R = ∞ ,the exact wave functions are no longer expressed in terms of known special functions, as is possible for the hydrogenatom. However, the exact solutions we have found for confining potentials correspond to confinement at the zeros5 TABLE III: Eigenvalues E d =2 , nl ( a, b ; R ) for V ( r ) = − a/r + br , r ∈ (0 , R ), where a = ± b = 0 . R = 1 and different n and l .The subscript N refers to the number of iteration used by AIM. n l E d =2 nl (1 , /
2; 1) n l E d =2 nl (1 , /
2; 1)0 0 − .
275 615 599 206 285 795 N =32 − .
275 615 599 206 285 795 N =32 .
400 467 192 272 980 536 N =26 .
924 630 155 130 440 587 N =27 .
652 661 600 597 110 050 N =24 .
734 045 433 763 800 052 N =32 .
010 259 174 813 201 428 N =24 .
522 506 980 951 712 401 N =40 .
565 689 679 299 850 255 N =25 .
243 804 852 673 032 613 N =47 .
324 795 658 776 520 956 N =28 .
875 359 994 443 580 341 N =53 n l E d =2 nl ( − , /
2; 1) n l E d =2 nl ( − , /
2; 1)0 0 6 .
107 045 323 129 696 121 N =27 .
107 045 323 129 696 121 N =27 .
530 081 242 027 809 913 N =24 .
534 700 629 074 427 546 N =27 .
106 527 319 660 138 719 N =24 .
295 175 016 376 479 090 N =35 .
149 694 772 638 116 456 N =24 .
728 314 736 027 030 722 N =43 .
518 339 762 183 359 381 N =27 .
935 489 978 445 435 860 N =47 .
151 787 835 702 316 786 N =29 .
956 853 183 684 793 313 N =55 n l E d =4 nl (1 , /
2; 1) n l E d =4 nl (1 , /
2; 1)0 0 5 .
400 467 192 272 980 536 N =26 .
400 467 192 272 980 536 N =24 .
652 661 600 597 110 050 N =24 .
123 225 647 087 677 088 N =25 .
010 259 174 813 201 428 N =24 .
910 542 938 654 909 374 N =35 .
565 689 679 299 850 255 N =25 .
660 358 190 161 408 159 N =43 .
324 795 658 776 520 956 N =28 .
333 925 295 766 891 686 N =48 .
278 874 241 139 597 779 N =31 .
912 601 544 108 537 448 N =54 n l E d =4 nl ( − , /
2; 1) n l E d =4 nl ( − , /
2; 1)0 0 9 .
530 081 242 027 809 913 N =24 .
530 081 242 027 809 913 N =24 .
106 527 319 660 138 719 N =24 .
374 386 080 371 192 265 N =27 .
149 694 772 638 116 456 N =24 .
884 084 396 689 521 442 N =36 .
518 339 762 183 359 381 N =27 .
165 200 694 649 737 766 N =44 .
151 787 835 702 316 786 N =29 .
258 753 086 612 471 603 N =49 .
014 646 696 330 218 668 N =32 .
184 615 317 703 801 052 N =54 of the unconfined case. An interesting qualitative feature seems to be that E dn,ℓ ( a, b, R ), for large R , is concave withrespect to n , ℓ , or d , but becomes convex as R is reduced; this may arise because the reduction in R perturbs thehigher states more severely since, when free, they are naturally more spread out. It is hoped that the work reportedin the present paper will provide a useful addition to the growing body of results concerning the spectra of confinedatomic systems in d dimensions. VII. ACKNOWLEDGMENTS
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