aa r X i v : . [ qu a n t - ph ] J u l Studies on some singular potentials in quantum mechanics
Amlan K. Roy
Department of Chemistry, University of New Brunswick,Fredericton, NB, E3B 6E2, Canada ∗ Abstract
A simple methodology is suggested for the efficient calculation of certain central potentials havingsingularities. The generalized pseudospectral method used in this work facilitates nonuniform andoptimal spatial discretization. Applications have been made to calculate the energies, densities andexpectation values for two singular potentials of physical interest, viz., (i) the harmonic potentialplus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear andquadratic term for a broad range of parameters. The first 10 states belonging to a maximum of ℓ = 8 and 5 for (i) and (ii) have been computed with good accuracy and compared with the mostaccurate available literature data. The calculated results are in excellent agreement, especially inthe light of the difficulties encountered in these potentials. Some new states are reported here forthe first time. This offers a general and efficient scheme for calculating these and other similarpotentials of physical and mathematical interest in quantum mechanics accurately. ∗ Electronic address: [email protected] . INTRODUCTION Application of quantum mechanics to many branches of physics and chemistry, e.g., theatomic and molecular physics, nuclear physics, particle physics, solid-state physics, astro-physics, etc., often involves a potential term having singularity and usually some extraperturbation terms characterizing the physical system under study. Almost for all practicalpurposes, the radial Schr¨odinger equation to be solved requires approximations as the exactsolutions can be obtained only for a few idealized cases such as the harmonic oscillator orthe Coulomb potential. Consequently, a large number of analytical and numerical method-ologies have been developed by many workers over the decades to obtain accurate solutionsfor such systems employing a variety of techniques. Although many attractive and elegantapproaches exist in the literature for both the non-singular and singular potentials, the lat-ter remains relatively less explored as one encounters considerable difficulties and challengesbecause of the singularity. Therefore, the search for a general scheme capable of producingaccurate and reliable results, has continued to remain an active area of research.The purpose of this article is to assess the performance of a simple numerical methodologyfor the singular potentials. Although the method would hold good for other singularities, wewill restrict ourselves to two different classes of such potentials which have found applicationsin various branches of physics, viz., (a) the harmonic potential including an inverse quartic and sextic anharmonicity, V ( r ) = a r + X i =2 a i r − i , ( a , a >
0) (1)and(b) the Coulomb potential with a linear and quadratic coupling, V ( r ) = − b /r + X j =1 b j r j (2)In what follows we will adopt a = a, a = b, a = c ; and b = Z, b = g, b = λ for the sakeof consistency with the literature.The potential in Eq. (1) is defined in r ∈ [0 , ∞ ] with Dirichlet boundary conditions. Sucha potential may lead to many interesting and appealing features; both physically and math-ematically. For example, there may be the so-called Klauder phenomenon [1-2] signifyingthat once the singular perturbation is turned on, it cannot be completely turned off. It is2oticed that the effects of perturbation is never small enough to ignore. For example, thecomplete three-dimensional Schr¨odinger equation is exactly solvable without the inverse sex-tic perturbation in Eq. (1) for both the extreme cases a = 0 and b = 0 . However subjectingthe system to an infinitesimal perturbation a = 0 , b = 0 induces such dramatic effects in thewave function near the origin that the perturbation theory fails to give any estimate at all[3]. On the other hand, when either b or c is large, the anharmonic terms compete with theharmonic term and none of the interacting terms can be considered negligible. Several meth-ods are available for the calculation of the eigenvalues and eigenfunctions of this potential.Some of them, for example, include the singular perturbation theory [4-6] specially designedfor this potential, variational methods [7,8], step-by-step analytic continuation method [9],numerical solution using finite-difference scheme [10,11] or the B spline techniques [12], etc.This potential also has the importance in the context of conditional exact solvability; inother words, exact analytic solutions can be obtained for some severely restricted set ofparameters (related to each other through some special relations) for ground states [13] orexcited states [14,15].The potential in Eq. (2) also has many relevance to physics. Thus, the Coulomb plus thelinear potential has been studied extensively for the spherical Stark effects in hydrogenicatoms [16], or in connection with the various non-relativistic quark confinement and othersimilar problems in particle physics [16-18], etc. On the other hand, the Coulomb plus thequadratic potential has been studied in the context of quadratic Zeeman effects in hydrogenicatoms [19], or in plasma physics [20], etc. Again various methodologies have been devisedfor the calculation of eigenvalues of this potential by many workers. For some special valuesof the parameters Z, g and λ, this leads to a quasi exactly solvable potential and analyticsolutions can be obtained [21,22]. Other works include Rayleigh-Schr¨odinger perturbationmodel [23], the Stieltjes moments method [21], the analytic continued fraction theory [24,25],the Hill-determinant method [26], the supersymmetric quantum mechanics coupled with theshifted 1/N method [22,27], the two-point quasifractional approximant method [28], etc.More references on this potential can be found in [28].However, for both these potentials, good accuracy results can be achieved only by a fewof the above mentioned methods. Thus leaving aside a few methods, (like [9] or [11], forexample) it is usually quite difficult to reach beyond a six- or seven-decimal place accuracyfor the potential in Eq. (1) for an arbitrary set of parameters. As another author [8] points3ut, to obtain even this much accuracy by standard numerical methods for the potential likethat in Eq. (1) sometimes requires one to use a mesh of at least 80,000 points for some cases.Also, some of these methods, although can provide good quality results for certain type ofparameters, perform rather poorly or even can not provide any result at all in other occasions(see [22], for example). Finally while majority of the works have remained largely focusedto the ground states, relatively less attention has been paid to the excited states (especiallythose associated with the higher angular momentum). Given these facts, a general methodwhich can offer accurate as well as reliable results for a general set of parameters for bothground and excited states (low as well as high) with equal ease, would be highly desirableand demanding. This work attempts to make a small step in such a direction. The seed ofthe motivation grew from a study of singly- and doubly- and triply excited Rydberg statesof many-electron atomic systems [29,30] within the density functional framework [31] usingthe generalized pseudospectral (GPS) [32,33] scheme which produced good quality resultswithin the bounds of the theory.The purpose of this article is to use this simple numerical scheme for the calculation ofthe above mentioned singular potentials. First we will present in brief the essentials of theGPS method for the solution of the single-particle radial Schr¨odinger equation in sectionII. This has witnessed many successful applications in electronic structure and dynamicscalculations in the recent years involving mainly Coulomb singularities (see, for example,[29-30, 32-35]). Thereafter this has been extended to some other physical systems includingthe spiked harmonic oscillators, the logarithmic and power potentials as well as the Hulthenand Yukawa potentials [36-38] with considerable promise. Section III presents the computedeigenvalues, wave functions, radial densities and the expectation values for the tow cases.Both ground as well as higher excited states are calculated and a thorough comparison withliterature data has been made, wherever possible. Finally few conclusions are drawn insection IV. II. METHODOLOGY
In this section, we present an overview of the generalized pseudospectral method (GPS)employed to solve the radial eigenvalue problem with the singular potentials. A more detailedaccount can be found in the refs. [29-30, 32-38].4ithout loss of generality, the desired radial Schr¨odinger equation can be written as(atomic units employed unless otherwise mentioned),ˆ H ( r ) φ ( r ) = ε φ ( r ) , (3)where the Hamiltonian includes the usual kinetic and potential energy operators,ˆ H ( r ) = − d dr + v ( r ) , (4)with v ( r ) = V ( r ) + ℓ ( ℓ + 1)2 r (5)and V ( r ) is given by Eq. (1) or (2). The symbols have their usual significances. The usualfinite-difference spatial discretization schemes often require a large number of grid points toachieve good accuracy since majority of these methods employ a uniform mesh (nonuniformschemes are used in a few occasions as well, e.g., in [11]). The GPS method, however, cangive nonuniform and optimal spatial discretization accurately. This allows one to work witha denser mesh at shorter r regions and a coarser mesh at larger r . Additionally the GPSmethod is computationally orders of magnitude faster than the finite-difference schemes.One of the principal features of this scheme lies in the fact that a function f ( x ) definedin the interval x ∈ [ − ,
1] can be approximated by the polynomial f N ( x ) of order N so that, f ( x ) ∼ = f N ( x ) = N X j =0 f ( x j ) g j ( x ) , (6)and the approximation is exact at the collocation points x j , i.e., f N ( x j ) = f ( x j ) . (7)In this work, we have employed the Legendre pseudospectral method using x = − x N = 1,where x j ( j = 1 , . . . , N −
1) are obtainable from the roots of the first derivative of the Legendrepolynomial P N ( x ) with respect to x , i.e., P ′ N ( x j ) = 0 . (8)The cardinal functions, g j ( x ) in Eq. (6) are given by the following expression, g j ( x ) = − N ( N + 1) P N ( x j ) (1 − x ) P ′ N ( x ) x − x j , (9)5beying the unique property g j ( x j ′ ) = δ j ′ j . Now the semi-infinite domain r ∈ [0 , ∞ ] ismapped into the finite domain x ∈ [ − ,
1] by the transformation r = r ( x ). One can makeuse of the following algebraic nonlinear mapping, r = r ( x ) = L x − x + α , (10)where L and α = 2 L/r max may be termed as the mapping parameters. Now, introducingthe following relation, ψ ( r ( x )) = q r ′ ( x ) f ( x ) (11)coupled with the symmetrization procedure [32,33] leads to the transformed Hamiltonian asbelow, ˆ H ( x ) = −
12 1 r ′ ( x ) d dx r ′ ( x ) + v ( r ( x )) + v m ( x ) , (12)where v m ( x ) is given by, v m ( x ) = 3( r ′′ ) − r ′′′ r ′ r ′ ) . (13)Note the advantage that this leads to a symmetric matrix eigenvalue problem which can bereadily solved to give accurate eigenvalues and eigenfunctions. For the particular transfor-mation used in Eq. (10), v m ( x ) = 0. This discretization then leads to the following set ofcoupled equations, N X j =0 (cid:20) − D (2) j ′ j + δ j ′ j v ( r ( x j )) + δ j ′ j v m ( r ( x j )) (cid:21) A j = EA j ′ , j = 1 , . . . , N − , (14)where A j = [ r ′ ( x j )] / ψ ( r ( x j )) [ P N ( x j )] − . (15)and the symmetrized second derivative of the cardinal function, D (2) j ′ j is given by, D (2) j ′ j = [ r ′ ( x j ′ )] − d (2) j ′ j [ r ′ ( x j )] − , (16)with d (2) j ′ ,j = 1 r ′ ( x ) ( N + 1)( N + 2)6(1 − x j ) r ′ ( x ) , j = j ′ , = 1 r ′ ( x j ′ ) 1( x j − x j ′ ) r ′ ( x j ) , j = j ′ . (17)6he performance of the method has been tested for a large number of potentials with manyother works in the literature with respect to the variation of the mapping parameters. Theresults have been reported only up to the precision that maintained stability with respectto these variations. Thus for all the calculations done in this work, a consistent set for thenumerical parameters ( r max = 200 , α = 25 and N = 300) has been used which seemed tobe appropriate for the current problem. III. RESULTS AND DISCUSSION
In order to show the efficacy of the method, we first present some specimen results fora few odd- and even-parity high excited states of the pure three-dimensional quartic oscil-lator corresponding to the large vibrational quantum numbers v = 48 ,
49 and the angularmomentum quantum numbers ℓ = 0 , , · · · , truncated and not rounded-off .Therefore all the entries in the tables are to be taken as correct up to the place they arereported. The results used for comparison in Table I are chronologically: (a) the linearvariation method involving diagonalization of matrices of large order (800 × ℓ = 0 and 1 as obtained by the GPS method for a selected set of parametersfor the even-power inverse anharmonic potentials. Besides the ground states, some of theexcited states corresponding to ℓ = 0 have been reported in the literature and we quotethem appropriately wherever possible. Exact analytical results are obtainable for parametersfollowing certain relations among them for (i) ground [7,15] and (ii) excited states [15] (theexactly solvable conditionality, e.g., a = 0 . , b = − . , c = 1 . b have been7 ABLE I: Calculated energies (in a.u.) of the three-dimensional quartic oscillator for some high-lying states corresponding to ℓ = 0 , , v = 48 , . v ℓ This work Ref. [39] Ref. [40] Ref. [10] Ref. [41]48 0 250.183358697 250.183351 250.183369 250.183359 250.183358697149 1 256.916238928 256.916220 256.916238 256.916239 256.916238928648 2 250.096690608 250.096679 250.096671 250.096691 250.096690608049 3 256.773728914 256.773732 256.77369 256.773728914648 4 249.894552064 249.894545 249.894505 249.894552064749 5 256.517359165 256.517338 256.517316 256.517359165648 6 249.577151099 249.577138 249.577179 249.577151099149 7 256.147382583 256.147373 256.14750 256.147382583648 8 249.144812457 249.144801 249.1452 249.144812457549 9 255.664161642 255.664146 255.66480 255.6641616427 considered, while the parameter a has been fixed at 0.5 in all the cases. No results could befound for non-zero angular momentum states. For the first set, the numerical integrationresults are available for the first three states of ℓ = 0 [42], while exact analytical result [15]exist for the first two states of ℓ = 0. For the third set, there were some controversy inthe literature regarding the position of the second state corresponding to ℓ = 0. It wasestimated to be at 6.048105 in [14]; later a numerical B-spline basis set calculation [12]re-estimated it at 4.24927125 a.u. The current scheme computes this with a higher accuracy(at 4.24927125613) than before and is more in keeping with the latter result. For the lasttwo sets, no results are available except the exact analytical values for the ground states.A wide range of parameters has been used in this table, and these results may be useful tocheck the performances of other methods.After comparing the GPS results for the harmonic potential with inverse even-poweranharmonicities for ground and higher excited states, we now in Table III present the first10 eigenvalues for the same belonging to ℓ = 0 , , , , a = 0 . ,b = 0 . c = 0 . . The reason for the choice of this particular set lies in the fact that thisis the only set for which we could find some nonzero angular momentum states with goodaccuracy [9]. Reference [9] employed the analytic continuation method and those are quoted.It is evident that for the available states, our results are in excellent agreement with those of[9]. However, these results are available only for the first four states of ℓ = 0 , , ,
3. Theseresults illustrate an advantage of the present method to treat the ground and excited statesat the same footing without any special consideration for excited states as often required by8
ABLE II: Comparison of the energies (in a.u.) with literature data for the singular potential inEq. (1) with a = 0 .
5. First five eigenvalues are presented for ℓ = 0 and 1. b c ℓ = 0 ℓ = 1 − .
625 1.7578125 − . − . a , − . b ) 0.099726562432.99999999999(3.0 a ,3.0 b ) 3.453526409205.48535332842(5.485353 b ) 5.843430776517.79200908589 8.1037987036910.0249058750 10.3072928767 − . c ) 3.751533151145.99788108291 6.202485489248.35808537819 8.5377961851210.6486411431 10.812172301512.8960139536 13.04787322920.02041 0.09 2.04810689953(2.0481069 d ,2.0481069 e ) 2.636808685644.24927125613(4.24927125 d ,6.048105 e ) 4.739816500486.39227593858(6.3922759 d ) 6.828421348008.50708702884(8.507087 d ) 8.9068619165910.6046536820 10.97769534180.5 0.5 2.50000000000(2.5 c ) 2.935834622564.76648152281 5.127137799586.95840432469 7.278293625699.11294921787 9.4059292925711.2443693329 11.517721233222.5 112.5 5.49999999999(5.5 c ) 5.652776061917.97997008055 8.1141805562010.3661382958 10.488263270512.6938328534 12.807254980414.9811074983 15.0878538581 a Exact value, ref. [15]. b Ref. [42]. c Exact value, ref. [7]. d Ref. [12]. e Ref. [14]. some of the methods in the literature.Now results are presented for the perturbed Coulomb potential. The basic strategyof presentation remains the same as earlier. First, a few low-lying states along with theliterature results for a fairly large range of parameter sets and then higher states for aparticular case. Table IV shows the first four computed eigenvalues for ℓ = 0 and 1, ofthe five such sets along with other results. In this case the literature data seems relativelyscarce and scanty. Both large and small Z as well as λ regions have been investigated. The9 ABLE III: The first 10 eigenvalues (in a.u.) for ℓ = 0 , , , , a = 0 . , b = 0 . c = 0 . . Numbers in the parentheses denote the valuestaken from Ref. [9]. ℓ = 0 ℓ = 2 ℓ = 4 ℓ = 6 ℓ = 82.46735982710 3.66898315916 5.54021470933 7.51634080120 9.50878342955(2.46735982710) (3.66898315916)4.72473466150 5.76433139697 7.55977095004 9.52189664743 11.5110218564(4.72473466150) (5.76433139697)6.91000701257 7.85154984749 9.58076033037 11.5278374936 13.5133680513(6.91000701257) (7.85154984749)9.05914846383 9.93195050802 11.6028526028 13.5341458494 15.5158206760(9.05914846383) (9.93195050802)11.1859453067 12.0066617412 13.6257807401 15.5408038979 17.518378329313.2973083828 14.0765778031 15.6493304068 17.5477937216 19.521039551215.3972525569 16.1424011188 17.6733301630 19.5550974910 21.523802828517.4883415197 18.2046882685 19.6976430683 21.5626976193 23.526666599119.5723241622 20.2638861627 21.7221597681 23.5705768873 25.529629257821.6504533563 22.3203586599 23.7467929198 25.5787185402 27.5326891600 first three sets keep the Z and g fixed at 1 and 0 respectively varying λ from 0.1 to 1000.There are no direct results for these to compare; however the lower and upper bounds forthe ground-state energies are available [21] from the Stieltje’s moments method and theseare mentioned appropriately. It is gratifying that our results fit very nicely within the smallrange of the bounds. For Z = 8 , g = 1 , λ = 1 /
32, the exact supersymmetric result forthe lowest state of ℓ = 1 is − .
375 (as quoted in [26]). The first four states of ℓ = 1 arecompared with the Hill-determinant results [26]. Table V, now gives the results for first 10eigenvalues corresponding to ℓ = 0 , , Z = 12 , g = 1 , λ = 1 /
32 Afew results are available for ℓ = 2 including the exact supersymmetric result [26] as well asthe Hill-determinant result [26]. The agreement in our results is seen to be excellent.Additionally we now give the expectation values h r − i and h r i for both the potentialsunder consideration in Table VI. Three states belonging to ℓ = 0 and 1 are reported for boththe potentials (one set for each of them). While no results could be found for the former case,a few results are available [43] for the perturbed Coulomb potential with Z = g = λ = 0 . | rR nℓ | for the singular potentials in Eqs. (1) and (2) (in the left andright panel) for the first four states belonging to ℓ = 0 , , ABLE IV: Comparison of the calculated energies (in a.u.) with literature data for the singularpotential in Eq. (2). First four eigenvalues are presented for ℓ = 0 and 1. Z g λ ℓ = 0 ℓ = 11 0 0.1 − . − . a b c / − . − . − . d , − . e − . − . − . e − . e e
10 5 1 − . − . − . a Lower and upper bounds to the eigenvalue are − . − . , from ref. [21]. b Lower and upper bounds to the eigenvalue are 4.1501236 and 4.1501239, from ref. [21]. c Lower and upper bounds to the eigenvalue are 59.3754689 and 59.3754694, from ref. [21]. d Exact supersymmetric result, as quoted in ref. [26]. e Hill-determinant method, ref. [26]. the required number of nodes is clearly manifest.Before passing, a few remarks should be made. In the present method, no unphysical states are obtained. It is worthwhile to note that some of the methods employed for thesekind of potentials in the literature often have the unwanted feature of producing unphysicalroots, e.g., the Ricatti-Pad´e scheme for a class of singular potentials, commonly termed asspiked oscillators. On the other hand, existence of false (unphysical) eigenvalues in the Hill-determinant method for a perturbed oscillator or perturbed Coulomb potential has beensuggested by some authors on mathematical grounds. Later it was pointed out [43] thatthese unphysical or false states manifest themselves by having the negative values for h r i or h r − i . Some of the well-known methods have another undesirable feature that they give11 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (e)l = 0l = 1l = 2 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (e)l = 0l = 1l = 2 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (d)l = 0l = 1l = 2 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (d)l = 0l = 1l = 2 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (c)l = 0l = 1l = 2 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (c)l = 0l = 1l = 200.20.40.60.811.21.41.6 0 1 2 3 4 5 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (b)l = 0l = 1l = 2 0123456 0 0.5 1 1.5 2 2.5 3 3.5 R ad i a l d i s t r i bu t i on f un c t i on ( a . u . ) (b)l = 0l = 1l = 2 -10-5051015 0 1 2 3 4 5 V (r) r(a.u.) (a) -140-120-100-80-60-40-2002040 0 0.5 1 1.5 2 2.5 3 3.5 V (r) r(a.u.) (a) FIG. 1: The radial probability distribution functions, | rR nℓ | for the potentials in Eqs. (1) and (2)in left and right panel respectively. The first four states corresponding to ℓ = 0 , , a = 0 . , b = − . , c = 1 . . and Z = 10 , g = 5 , λ = 1 . ABLE V: The first 10 eigenvalues (in a.u.) for ℓ = 0 , Z = 12 , g = 1 , λ = 1 / ℓ = 0 ℓ = 2 ℓ = 5 − . − . − . a , − . b − . − . − . b − . − . − . b − . b a Exact supersymmetric result, as quoted in ref. [26]. b Hill-determinant result, ref. [26].
TABLE VI: Calculated expectation values (in a.u.) along with literature data for comparison forthe two singular potentials in Eqs. (1) and (2) for the first three states corresponding to ℓ = 0 and1. The numbers in the parentheses denote the reference values from [43]. a b c ℓ h r − i h r i − .
625 1.7578125 0 1.037245259 1.0404045410.6106362329 1.8833801790.5127801051 2.3907451261 0.9798264559 1.1090654170.5975951150 1.9469348750.5093809276 2.427714809
Z g λ ℓ poor or erroneous results for potentials having multiple wells, for certain parameter setsor sometimes even for certain states within a particular parameter set. As an example,the shifted 1/N method for the perturbed Coulomb potential [22] fails to give any resultwhen any four of the five quantities Z , λ , g , ℓ, n r are kept fixed and only the fifth one isvaried. The present method is general in the sense that it is applicable to a wide spectrum of13arameters without requiring any special relation among them, irrespective of the strengthof the coupling involved and therefore lifts any such restrictions. In view of the simplicityand accuracy offered by this method for the polynomial as well as the singular potentialsconsidered in this work for both ground and higher excited states, it may be hoped thatthis prescription may be useful for a wide range of other similar potentials of interest. Thusother singular potentials having higher order terms in Eqs. (1) or (2) or the nonsingularpolynomial potentials with higher order anharmonicities like quartic, sextic, octic, etc., mayas well be treated by this method very well. Finally, we mention that while practically allof the common methodologies in quantum mechanics for such systems involve the solutionof the time-independent Schr¨odinger equation, recently a time-dependent (TD) formalism[44-47] has been proposed as well for the accurate calculation of static properties, which is,in principle, exact. Applications have been made for the ground and excited states of doublewell and anharmonic oscillators having higher order terms, multiple-well and self-interactingoscillators in one dimension [44-46], as well as the ground-state electronic properties ofnoble gas atoms [47] through an amalgamation of the density functional and quantum fluiddynamical treatment. It may be worthwhile to study the performance of such TD methodsfor the singular potentials dealt in this work. IV. CONCLUSION An accurate and simple methodology has been proposed for certain singular potentialsemploying the generalized pseudospectral method. The prescription is general and reliable .The energies, densities and expectation values presented for ground and excited states of the(i) harmonic potential with inverse quartic and sextic perturbation and (ii) the perturbedCoulomb potential with linear and quadratic terms are reported for a broad spectrum ofpotential parameters. This scheme is capable of producing very good quality results forboth ground and higher excited states. The v = 48 , , · · · , ,
49 states corresponding tothe angular momentum quantum numbers ℓ = 0 , , · · · , , Acknowledgments
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