Keywords: Schrödinger equation, radial equation, boundary condition, singular potentials.

Transcription

1 What is the bounday condition fo the adial wave function of the Schödinge equation? Anzo A. Khelashvili a) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9, 19, Tbilisi, Geogia and St. Andea the Fist-called Geogian Univesity of Patiachy of Geogia, Chavchavadze Ave. 53a, 16, Tbilisi, Geogia Teimuaz P. Nadaeishvili b) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9, 19, Tbilisi, Geogia Abstact. Thee is much discussion in the mathematical physics liteatue as well as in quantum mechanics textbooks on spheically symmetic potentials. Nevetheless, thee is no consensus about the behavio of the adial function at the oigin, paticulaly fo singula potentials. A caeful deivation of the adial Schödinge equation leads to the appeaance of a delta function tem when the Laplace opeato is witten in spheical coodinates. As a esult, egadless of the behavio of the potential, an additional constaint is imposed on the adial wave function in the fom of a vanishing bounday condition at the oigin. Keywods: Schödinge equation, adial equation, bounday condition, singula potentials. PACS numbes: w, 3.65.Ca, 3.65.Ta I. INTRODUCTION Accoding to the geneal pinciples of quantum mechanics, the wave functions must obey cetain equiements, such as continuity (moe pecisely, two -fold diffeentiability), uniqueness, and squae integability. In many poblems knowledge of the behavio of the wave function is needed at points whee the potential has a singulaity. In this pape we 1

2 conside spheically symmetic potentials fo which sepaation of vaiables is pefomed in spheical coodinates. It is well known that the tansfomation to spheical coodinates is a singula at the oigin. The tansfomation fom Catesian to spheical coodinates is not unambiguous, because the Jacobian of this tansfomation is J sin and is singula at and n n,1,,.... The angula pat is unambiguously fixed by the equiements of continuity and uniqueness 1 m and gives the spheical hamonics, We note that though. is an odinay point in the full Schödinge equation, it is a singula point in the adial equation and thus, knowledge of the behavio at equied. We conside the adial wave function d u d u, which is a solution to ll 1 u m E V u. (1) Equation (1) includes only the second deivative. It is clea fom Eq. (1) that the behavio l is of u at the oigin depends on the behavio of the potential V(), in paticula, whethe it is egula o singula. Although a definite answe exists fo egula potentials, the situation is unclea fo singula potentials. We econside the deivation of the adial equation in moe detail and show that the existence of the adial equation depends on the behavio of u at the oigin. This aticle is oganized as follows. In Sec. II we conside the consequences of some geneal pinciples. We will show that thee is no unambiguous answe. In Sec. III we conside the tansfomation to the adial equation and obtain a additional delta-like tem, elimination of which povides a constaint on the adial wave function at the oigin. Only afte satisfying this constaint does the adial equation take its usual fom, which is Eq. (1). This constaint has the fom of a bounday condition fo the adial wave function at the

3 oigin. In Sec. IV we give concluding emaks, and in the Appendix we discuss the appeaance of the delta function in the adial equation. II. THE BOUNDARY CONDITION AT THE ORIGIN The question is what is the maximal singulaity that the adial function R() o u R can have at the oigin. The complete thee-dimensional wave function is, m R l ; ; ;, () and the equation fo the function R() is d R d dr d m E V l l 1 R R. (3) The taditional change of vaiables eliminates the fist deivative tem fom Eq. (3) by the substitution u( ) R( ), (4) which leads to Eq. (1) fo the adial wave function u. We will efe to u() as the adial wave function. Fom the continuity of R () at it follows that u, insuing a finite pobability at this point. 3 We can weaken this condition by equiing a finite diffeential pobability in the spheical slice, d R d. (5) If s R ~ at the oigin, it follows that 1 s, o u. adius a, Anothe genealization is to equie a finite total pobability inside a sphee of small 3

4 a R d In this case moe singula behavio is pemissible, namely, limu lim 1/. (6), (7) whee is a small positive constant and at the end of the calculation. The same behavio follows fom the finite behavio of the nom. R d. (8) We can also use a stonge agument by Pauli, 4 namely, the time independence of the nom. To exploe it we follow the pocedue in Ref. 5. In quantum mechanics the nom of the wave function is independent of time d dt dv. (9) By using the time-dependent Schödinge equation, we tansfom Eq. (9) to i * * ( H ) H dv The time independence of the pobability means that the Hamiltonian must be a Hemitian opeato. By intoducing the pobability cuent density it is easy to show that h J Re im, (11) i div J H H The equation fo consevation of pobability takes the fom (afte using Gauss theoem) 4

5 d dt dv di JdV J ds V V S N (13) whee J N is the nomal component of the cuent elative to the suface. If we assume that at the Hamiltonian has a singula point, Gauss theoem in Eq. (13) is not applicable. We must exclude this point fom the integation volume and suound it by a small sphee of adius a. In this case the suface integal is divided into a suface at infinity that encloses the total volume, and the suface of a sphee of adius a: lima a J d J ds. (14) a N In the fist integal in Eq. (14) we have expessed the suface element of the sphee as ds a d, whee d is an element of solid angle. Because the wave function must vanish at infinity, the second tem goes to zeo. If we substitute J a i m a and u ~ assume, whee u ~ is egula at, we obtain s a lim a a s ~ ~ u u u~ u~ a d. (15) Equation (15) is satisfied if 1 s. It follows that R does not divege moe quickly than s 1 /, with 1 s1 s, which means that lim lim u. We see that the diffeent aguments lead to diffeent conclusions fo the wave function behavio at the oigin. A finite nom allows fo divegent behavio of oigin, but the time independence of the nom gives vanishing behavio. u at the 5

6 Does the bounday behavio at the oigin have some physical meaning? To discuss this question we stat fom the Eq. (1) and conside the well known example of a egula potential lim V. (16) Afte we substitute u ~ s nea the oigin, it follows fom the chaacteistic equation that 1 l l 1 l1 l s s, which gives two solutions: u ~ c c. Fo nonzeo l the second tem is not locally squae integable and is usually ignoed. Many authos discuss how to deal with the solution fo l, 6,7 which is squae integable at the oigin. Messiah 7 wites: The foegoing agument does not apply when l. But in that case, the coesponding wave function ( R in ou notation) does not satisfy the Schödinge equation [condition (a)]. In fact, 1/ 4, behaves as / H E 1 1 at the oigin, and since. (17) m One must theefoe keep only the so-called egula solutions, that is, the solutions satisfying the condition u. With such a solution we can be sue that the function m l is a solution of the Schödinge equation eveywhee, including the oigin. Howeve this consideation coesponds only to a egula potential. The analysis changes dastically when the potential is singula. Conside the following singula potential, (18) whee V coesponds to attaction and V coesponds to epulsion. Fo this potential the equation fo the exponent a takes the fom 6

7 Theefoe 1 1 s s l l mv, which has two solutions: 1 1 s l mv. u 1 P 1 1 P 1 ~ c c ; P l mv. (19) Both solutions ae squae integable nea the oigin as long as P 1. This condition is studied in connection with the self-adjoint extension of the adial Hamiltonian. 8,9 It coesponds to the condition in Eq. (6). Fo the condition in Eq. (5), P is esticted to P 1/. The diffeence in the uppe bound is essential. The adial equation takes the fom P 1/ 4 u u meu. () Depending on whethe P is geate then 1/ o not, the sign in font of the faction in Eq. () changes, and we can deive the esults fo an attactive potential using the case of a epulsive potential and vice vesa, wheeas the condition in Eq. (5) fobids this undesiable case with 1/ P 1. III. WHEN IS THE RADIAL EQUATION VALID? It seems that the choice u is pefeable. But this condition does not follow diectly fom Eq. (1). Theefoe we econside the deivation of Eq. (1) in moe detail. We etun to the igoous deivation of the adial equation fo u. Afte substitution of Eq. (3) into Eq. () we obtain 1 d d d d u u d d d d 1 du d d d 1 l l 1 m u E V (1) 7

8 We wite the adial equation in the fom (1) to show the action of the adial pat of the Laplacian explicitly. The fist deivatives of u cancel, and we ae left with 1 d u d u d d d d 1 l l 1 u m E u V. () As we do the deivatives in the second tem naively, we obtain zeo, when.if we take into account that d d 1 d d d is the adial pat of the Laplacian, we conclude that 1 and thus Eq. () becomes (3) d d d , (4) ll 1 u 4 u u m E V 1 d u 3 d (5) It includes an exta thee-dimensional delta-function tem, which is evident fom Eq. (4), and discussed in the Appendix. Its pesence in the adial equation has no physical meaning and thus it must be eliminated. Note that if, this exta tem vanishes due to the natue of the delta function. If and we multiply Eq. (4) by, we obtain the odinay adial equation (1). If, multiplication by is not pemissible and the exta tem emains in Eq. (5). Theefoe we have to investigate this tem sepaately and find a way to discad it. ove d 3 The effect of the thee-dimensional delta function is detemined by integating d sindd. It is evident that 1 8

9 3 1 J (6) whee J sin is the Jacobian. Thus, the exta tem effectively becomes. (7) Its appeaance as a point-like souce at = is not physical. The only easonable way to emove this tem without modifying the Laplace opeato o including a compensating delta function tem in the potentialv, is equie that u. (8) Multiplication of Eq. (5) by and elimination of the delta function due to the popety is not acceptable, because it is equivalent to multiplication of this tem by zeo. Theefoe we conclude that the adial equation (1) fo u is compatible with the full Schödinge equation () if and only if the condition u is satisfied. Equation (1) supplemented by the condition (8) is equivalent to Eq. (). It satisfies the Diac equiement 11 that the solutions of the adial equation must be compatible with the full Schödinge equation. It is emakable that the supplementay condition (8) has the fom of a bounday condition at the oigin. All of these statements can be easily veified by explicit integation of Eq. (9) ove a small sphee with adius a appoaching to zeo at the end of the calculations. We have aleady seen that thee is some ambiguity in the fomulation of the bounday condition fo the adial wave function fom the geneal pinciples of quantum mechanics. Theefoe vaious bounday conditions have been consideed, especially fo singula potentials. We have shown that the adial equation is valid only togethe with condition (8), independently of the potential, whethe it is egula o singula. 9

10 Usually bounday conditions ae deived fom the adial equation fo a given potential. But ou esult means that the adial equation (1) by itself follows fom the total Schodinge equation if and only if the constaint (8) is satisfied. It is cuious that this fact (appeaanc e of delta functions while educing the Schödinge equation) has appaently gone unnoticed. Pevious papes that have exploed this bounday condition ae obviously coect. 1,13 In contast, papes without this bounday condition ae doubtful, because the Eq. (1) is valid only if Eq. (8) is satisfied. Most textbooks conside only egula potentials with this bounday condition and theefoe thei esults ae coect. The only exception is the l state, which has been discussed by Messiah. 7 We poved his assumption, because the second solution must be ignoed fo any l, including l. Moe fa-eaching consequences follow fo singula potentials. Many authos 8,9,14 neglect the bounday condition entiely and satisfy only squae integability. But in this teatment some of paametes of wave functions go out of allowed egions and a selfadjoint extension pocedue can yield unphysical esults. A coesponding example was mentioned afte Eq. () fo 1/ P 1, whee a epulsive potential gives a bound state afte a self-adjoint extension. 9 Othe examples of singula potentials ae consideed in Ref. 15. IV. CONCLUSIONS We have shown that a igoous eduction of the Laplace opeato in spheical coodinates leads to a peviously unnoticed delta function tem. Caeful investigation of this tem gives a constaint on the behavio of the adial wave function at the oigin in the fom of 1

11 a bounday condition, u. A unique bounday condition follows fo both egula and singula potentials. Only the natue of the appoach to zeo depends on the behavio of the potential at the oigin. Since at least the wok of Case, 1 it has been known the impotance of notions of limit-point, limit cycle and self-adjoint extension pocedue fo the adial Schodinge equation and it s Hamiltonian. 16,17 It povides the coect way to undestand the bounday conditions at the oigin fo Eq. (1). Thee is nothing wong with such a teatment, which yields the condition u by applying poweful mathematics. 16,17,18 But as we have shown, the adial equation (1) has nothing in common with physics without the condition (8). A self-adjoint extension, used in many papes that do not satisfy this condition, has only mathematical impotance. Simila issues aise in classical electodynamics, 19 whee the exta delta function appeas in calculations of dipole electic and magnetic fields, but cancels without any physical consequences. The situation in quantum mechanics diffes because the exta delta tem necessitates the estiction of the adial wave function. The same issue holds fo the adial eduction of the Klein-Godon equation, because in thee dimensions it has the fom m E V, (9) and the eduction of vaiables in spheical coodinates poceeds in the same way as fo the Schödinge equation. 11

12 ACKNOWLEDGEMENTS We wish to thank Pofs. John Chkaeuli, Sasha Kvinikhidze, and Pamen Magvelashvili fo valuable discussions. We ae also indebted to Pof. Bois Abuzov, Ds. Iakli Machabeli, and Shota Vashakidze fo eading the manuscipt. APPENDIX : HOW THE DELTA FUNCTION APPEARS Following Ref. 1 we show how the delta function appeas in the adial equation. Conside the following deivative: (A1) A naive calculation would yield zeo. But the sepaate tems in this expession ae highly singula, and theefoe we must egulaize them. We choose the following egulaization nea the oigin Equations (A1) and (A) lead to (A) (A3) The ight-hand side of Eq. (A3) is well behaved eveywhee fo a, but as a it becomes infinite at and vanishes fo. To make the connection to a delta function we integate the ight-hand side of Eq. (A3) by d 3 dd, which gives 3a 4 d. (A4) 5 / a 1

13 We divide the volume of integation into two pats: a sphee of adius R with cente at the oigin and egion outside the sphee. Because integal fom the exteio of the sphee vanishes as a Rand appoaches zeo, the a as a. We thus need to conside only the contibution fom inside the sphee. We can neglect in the denominato, because the integand vaies vey slowly with. Afte this neglect the integal is equal to 3a a 5 / 3 a 3 a a (A5) Thus we have all the popeties of the 3-dimensional delta function, and we confim Eq.(4). a) Electonic mail: anzo.khelashvili@tsu.ge b) Electonic mail: teimuaz.nadaeishvili@tsu.ge 1 L. Schiff, Quantum Mechanics, 3d ed. (McGaw-Hill, New Yok, 1968). R. Newton, Scatteing Theoy of Waves and Paticles, nd ed. (Dove Publications, [xx city? xx] ), pp See any textbook on quantum mechanics. 4 W. Pauli, Die Allgemeinen Pinzipen de Wellenmechanik, in Handbuch de Physik, Bd. 5, Col. 1 (Aufl, Belin 1958). 5 D. Blokhincev, Foundations of Quantum Mechanics, 6th ed. (Nauka, Moscow, 1983) (in Russian), pp Thomas F. Jodan, Conditions on wave functions deived fom opeato domains, Am. J. Phys. 44 (6), (1976). 7 A. Messiah, Quantum Mechanics (Dove Publications, Mineola, USA,1999), p

14 8 P. Gii, K. Gupta, S. Meljanac, and A. Samsaov, A electon captue and scaling anomaly in pola molecules, Phys. Lett. A 37 (17), (8). 9 H. Falomi, M. A. Muschietti, and P. A. Pisani, On the esolvent and spectal functions of a second ode diffeential opeato with a egula singulaity, J. Math. Phys. 45 (1), (4). 1 J. D. Jackson, Classical Electodynamics, 3d ed. (John Wiley & Sons, New Yok, 1999), p P. A. M. Diac, The Pinciples of Quantum Mechanics, 4th ed. (Oxfod Univesity Pess, Oxfod, 1958), pp K. Case, Singula potentials, Phys. Rev. 8 (5), (195). 13 A. M.Peelomov and V. S. Popov, Collapse onto scatteing cente in quantum mechanics, Teo. Mat. Fiz 4, (197) (in Russian). 14 D. Sinha and P. Gii, A family of non-commutative geometies, axiv: T. Nadaeishvili and A. Khelashvili, Some poblems of self-adjoint extension in the Schodinge equation, axiv: M. Reed and B. Simon, Methods of Moden Mathematical Physics (Academic Pess, New Yok, 1978), Vol T. Kato, Petubation Theoy fo Linea Opeatos, nd ed. (Spinge-Velag, Belin, 1995). 18 E. A. Coddington and N. Levinson, Theoy of Odinay Diffeential Equations (McGaw-Hill, New Yok, 1955). 14

15 19 S. M. Blinde, Delta functions in spheical coodinates and how to avoid losing them: Fields of point chages and dipoles, Am. J. Phys 71 (8), (3). 15