Critical dipoles in one, two, and three dimensions

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1 Critical ipoles in one, two, an three imensions Kevin Connolly a an Davi J. Griffiths b Department of Physics, Ree College, Portlan, Oregon 970 Receive 1 November 006; accepte 6 January 007 The Schröinger equation for a point charge in the fiel of a stationary electric ipole amits boun states only when the ipole moment excees a certain critical value. It is not har to see why this might be the case, but it is surprisingly ifficult to calculate the critical ipole moment. The analogous problem shoul be simpler in one an two imensions, but a general theorem forbis critical moments in one imension, an explicit calculation shows that there is no critical moment in two imensions. 007 American Association of Physics Teachers. DOI: / I. INTRODUCTION It is a remarkable fact, iscovere by Fermi an Teller in the 1940s an reiscovere in the 1960s, 1 that a stationary electric ipole supports boun states if but only if the ipole moment excees a critical value: p crit = qm, 1 where m an q are the mass an charge of the orbiting particle. For the electron, p crit = Cm. Even more surprising, the critical ipole moment has the same value for a physical ipole see Fig. 1 Vr = Q r Q r as it oes for the point ipole Vr, = 1 p cos 4 0 r 4 where p=q. The purpose of this paper is to explore an explain the occurrence of a critical moment in this system. In Sec. II we recapitulate the stanar arguments an offer some potentially illuminating ways of thinking about the problem. It is natural to woner whether similar consierations apply to the one- an two-imensional analogs, which we consier in Secs. III an IV, respectively. II. CRITICAL MOMENT IN THREE DIMENSIONS We start with the physical ipole: ±Q separate by a istance. If is large, q will form a hyrogenic system at the Q en, 3 with Q too istant to be relevant. In this limit there are obviously boun states, with the lowest energy given by the Bohr formula mq Q E 1 = As we reuce the separation istance keeping the charges constant, the approaching Q charge repels the electron, an it is plausible that at some point it will ionize the atom completely closer than this separation, no boun state is possible. We can estimate the separation at which this ionization will occur. The Coulomb repulsion energy is approximately qq/4 0. When this equals E 1 in magnitue, the total energy is zero an the electron is no longer boun: 1 qq 4 0 = mq Q 4 0, which implies p = Q = qm. 7 This crue estimate is off by a factor of slightly more than 3 see Eq. 1, but it oes account for the existence of a critical ipole moment. 4 Now let us show that if there is a critical ipole moment, it is inepenent of the separation all that matters is the prouct p=q, an hence has the same value for the point ipole as for the physical ipole. 5 Schröinger s equation for a charge q in the ipole potential, Eq. 3, is m qq = E. 8 r r We are intereste in boun states, which is to say solutions with E0. The critical moment occurs when the groun state energy goes to zero. Assume that is fixe, an all lengths are measure in units of : r r/, r ±r ± /, =. Then Eq. 8 assumes the imensionless form where 1 1 =4, r r m qp an me. 11 Suppose we have solve Eq. 9 an obtaine the formula for the groun state energy 6 in the imensionless form as a function of the ipole moment in the imensionless form : g. Now we ecrease by reucing Q, because we are holing fixe until g reaches zero, an the last boun state is squeeze out: g crit =0. This conition tells us the critical ipole moment: 9 54 Am. J. Phys. 75 6, June American Association of Physics Teachers 54

2 r u rr r u r r = 1 sin sin m sin cos. 0 The left sie of Eq. 0 is a function of r alone an the right sie is a function of alone, so each sie must be a constant call it : u r r u = u; 1 Fig. 1. Point charge q in the fiel of a stationary ipole. p crit = crit 4 0 qm. 1 Note that crit oes not epen on ; it is the value of for which the largest in Eq. 9 reaches zero. So p crit, too, is inepenent of. This being the case, we might as well use the simplest moel, the point ipole, to calculate p crit. Schröinger s equation 7 m qp cos 4 0 r = E 13 is separable in spherical coorinates. Let r,, = 1 r ur, an recall that = 1 r rr r 1 r sin sin 1 r sin. Then Eq. 13 becomes r u rr r r u 1 sin sin 1 sin cos = r, where me/ Multiplying through by sin isolates the epenence: = m, 18 where m is the usual azimuthal separation constant. Eviently = e im, 19 an the perioicity in means that m must be an integer. What remains is cot cos m sin =. Equation 1 is the one-imensional Schröinger equation for the notorious 1/r potential. 8 The force is attractive as long as 0, but it oes not support normalizable boun states unless 1 4. One way to see this9 is to look for solutions by the metho of Frobenius: Let ur = r a n r n, with a 0 0. Equation 1 becomes n n 1 a n r n = a n r n, from which it follows that 1 a 0 =0, 1 a 1 =0, n n 1 a n = a n 3 4 5a 5b n = 0,1,,.... 5c Equation 5a yiels ± =1± 14/, an Eq. 5b gives a 1 =0. There are two solutions, one for an one for. Near the origin they go like u ± ra 0 r ± = a 0 r 1/ e ± 1/4 ln r. 6 As r 0, ln r, sou iverges unless 1 4, in which case the square root is imaginary: u ± ra 0 r 1/ e ±ig ln r, 7 where ig 1 4 ; both solutions go to zero at the origin. The general solution is a linear combination of u an u, but only one combination is normalizable: 10 ur = A rkig r, 8 where K ig is the moifie Bessel function of orer ig. Conclusion: For a normalizable solution the separation constant must satisfy 1 4. Presumably11 the larger becomes, the tighter is the bining, but the critical value, above which no boun states can exist, is = 1 4. Turning now to Eq., we are intereste in the groun state, 1 so m =0: 55 Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 55

3 cot cos =. 9 We expan in normalize Legenre polynomials 13 1 = P cos, 30 =0 which satisfy the ifferential equation P cos cot P cos = 1P cos, 31 an write Eq. 9 as But 14 so =0 1 1 cos P cos =0. 3 cos P cos = = P 1cos P 1 cos, P cos =0. 34 We substitute the critical value = 1 4 an use orthogonality of the Legenre polynomials to show that =0, for =0,1,,... In matrix form, 35 1/4 /3 0 0 / 3 9/4 / / 15 5/4 3/ / 35 49/4 3 0 = Eviently the eterminant of this matrix is zero. By truncating at the, 33, 44,... level, we obtain a sequence of approximations for : = = , 37a 3 = 15 5 = , 37b = = , 37c etc. The sequence converges very rapily from 7 on the first 16 igits are stable at = , 38 an the critical ipole moment, Eq. 1, is p crit = qm, 39 confirming Eq. 1. That s for the point ipole, of course, but we know from the general theorem that the critical moment is inepenent of the separation of the charges, so it applies as well to the physical ipole. Still, it woul be nice to check this inepenently. Equation 8 separates in prolate spheroial coorinates, 15 r r, r r,, 40 where is the usual azimuthal angle. Note that 1, 1, an r ± =±/. The potential energy is qq r 1 r = an Eq. 8 takes the form 4 = 4. qp 4 4 0, 41 4 The Laplacian in prolate spheroial coorinates is For a separable solution,, = XM, 44 an 1 X 1 X 1 M 1 M 1 11 =. 45 The epenence is the same as before: = m, 46 so =e im for integer m. Putting this into Eq. 45 an noting that 11 = , 47 we obtain the orinary ifferential equations for X an M: 56 Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 56

4 1 X 1 M = m = m 1 X, 1 M, where is the separation constant. We are intereste in the groun state so m =0 at the critical point =0 where E crosses from negative to positive, an the boun state isappears: 1 X = X, 1 M M = M. Equation 50 has the general solution X = AP 1/ 1 14 BQ 1/ 1 14, where P an Q are Legenre functions. Q iverges at =1 that is, on the line joining Q an Q, an P iverges as unless It turns out 17 that must be 1, an hence = 1 4. The change of variables cos v transforms Eq. 51 into a form we have encountere before compare Eq. 9: M cot vm cos vm = M. v v 55 We have alreay shown that for = 1 4 this elivers =1.786 an the critical ipole moment Eq. 39. Alternatively, one can solve Eq. 4 by the variational metho, using a trial wave function of the form, = C n n,, 56 with n, p n, where p an C n are ajustable parameters. 18 The matrix elements of the Hamiltonian are 19 H mn = m H n = 4p 1 57a n 1 mnp 1 m n pm, m n even, 1 p 1 m n, m n o. 57b The integrals converge for p 1. We are intereste in the crossover point where the energy goes to zero; here the wave function becomes very elocalize, which is to say that p is as small as possible: p 1. Because H=0 for the groun state, the eterminant of the Hamiltonian matrix vanishes. Dropping the constants in front, 1 4/3 1/3 4/5 4/3 3 4/5 9/5 1/3 4/5 7/3 4/7 58 4/5 9/5 4/7 11/5 =0. The solution for which, remember, is twice the coefficient in Eq. 1 converges very rapily with the number of terms: At the level we recover the same as Eq. 37a; 33 yiels , an 3030 no problem for Mathematica gives unchange beyon 77, ientical to what we got using the point ipole Eq. 38. Once again, we obtain the critical ipole moment in Eq III. ONE DIMENSION Calculating the critical ipole moment in three imensions turne out to be surprisingly ifficult, an one woners whether the analogous problem might be simpler in one imension, where the Coulomb potential takes the form 1 Vx = 1 Q 4 0 x. 59 The physical ipole is Vx = 1 Q 4 0 x / Q 60 x /, an the point ipole woul be Vx =± 1 p 4 0 x, 61 with the plus sign for x0 an minus sign for x0. We begin, as before, with the large limit, expecting to fin a one-imensional hyrogen atom at one en an a istant Q at the other. Unfortunately, the groun state of one-imensional hyrogen charge q in the Coulomb potential Eq. 59 has infinite bining energy. One way to see this is to regularize the potential removing the singularity at x=0, 57 Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 57

5 Fig. 3. a The elta-function ipole Eq. 63. b The square-well ipole Eq. 64. Fig.. Groun state energy for the regularize potential Eq. 6. The horizontal axis is, in units of a 0 /mqq, an the vertical axis is a, where me/. The graph suggests that E goes to like ln, as 0. Vx = 1 0 1/, 4 1/x, x, x, 6 solve numerically for the groun state energy, 3 an plot it as a function of the cutoff. Figure inicates that the magnitue of the energy increases without limit as 0. No matter how close we bring the Q en, it cannot ionize the atom, because 1/4 0 Q/= woul require =0. This is harly a proof, but it oes suggest that there may not be a critical ipole moment in one imension. We can avoi the pathologies of the one-imensional Coulomb potential by examining other moels, 4 such as the elta-function ipole Fig. 3a: Ux = x / x /, 63 an the square-well ipole Fig. 3b: = V 0, a x a, Ux V 0, a x a, 0, otherwise. 64 Their groun states can be obtaine by solving the Schröinger equation, but the results are isappointing if we were hoping to fin a critical separation istance: These potentials amit at least one boun state for all, regarless of the values of, V 0, an a. Is there perhaps a general theorem lurking here? There is. If a one-imensional potential Ux, not ientically zero, vanishes outsie some finite region, an if Ux x 0, 65 then Ux supports at least one boun state. 5 Conclusion: There is no critical ipole moment in one imension. 6 IV. TWO DIMENSIONS What about the two-imensional analog? Is there a critical moment in this case, an if so, what is its value? The pointcharge potential is Vr, = 1 Q 4 0 r. 66 As always, we begin with the large limit a twoimensional hyrogen atom at one en an a istant Q at the other. The groun state energy for two-imensional hyrogen is four times its value in three imensions, 7 an hence our crue estimate for the critical ipole moment is one fourth as great see Eq. 7: p crit qm As before, the critical ipole moment if there is one is inepenent of, so we look first at the point ipole limit: Vr, = 1 p cos 4 0 r. 68 Schröinger s equation, m qp cos 4 0 r = E, 69 is separable in polar coorinates, where the Laplacian is rr r 1 r. = 1 r We seek solutions of the form r, = ur, r for which Eq. 69 reuces to r u u r r = cos The left sie is a function of r alone, an the right sie is a function of alone, so each must be a constant call it : u r r u = u, 73 1 cos = Equation 73 is ientical to Eq. 1; we know that the critical value of is 1 4. We substitute this value into Eq. 74 an obtain cos =0. 75 We want a solution 8 that is perioic in with perio an even for the groun state; can therefore be expresse as a Fourier cosine series: 58 Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 58

6 = b n cos n. 76 Putting this into Eq. 75, using the ientity cos x cos y= 1 cos xycos x y, an exploiting the orthogonality of the cosine functions, we fin b 1 =0, 77a Letting u v cos v = cosh u cos v. u,v = UuVv, we obtain the orinary ifferential equations for U an V: b 1 b 0 b =0, 77b U u cosh u U =0, 86a n b n b n 1 b n1 =0 n =,3,4,..., 77c or, in matrix form, 0 / / 0 b 1 0 / 4 / b / 9 b 3 b0 = For a nontrivial solution the eterminant of this matrix must vanish. Evaluating by minors, first along the top row, an then own the first column, we obtain 4 / 0 0 / 9 / 0 0 / 16 / / 5 =0. Either =0 or else the remaining eterminant is zero. Woul =0 be acceptable? Absolutely: In this case Eq. 75 has the perioic solution =const, an r, = AK 0 r. 80 Unfortunately, this means that the critical ipole moment is zero which is to say that there is no critical ipole moment. The two-imensional ipole like the one-imensional always has a boun state. Just to be sure, let s examine what happens to the groun state of the physical ipole, as the separation ecreases with Q hel fixe. In this case Schröinger s equation separates most simply in elliptic coorinates u,v; these are closely relate to the prolate spheroial coorinates in Eq. 40, with 9 = cosh u, = cos v. 81 The potential energy Eq. 41 is qq r r = qp 4 cos v 4 0 cosh u cos v, 8 an the Laplacian 30 is 4 = cosh u cos v u v, 83 so Schröinger s equation takes the form V v cos v cos v V =0, 86b where is the separation constant. We are intereste in the critical point =0 where E crosses from negative to positive: U u = U, V cos vv = V. v 87b Equation 87a has the general solution Uu = Ae u Be u. 87a 88 For the groun state we want real, an as small as possible; 31 the limiting case is =0 for which Eq. 87b reuces to V cos v V =0. v 89 Thus we recover Eq. 75, which we alreay know yiels a critical moment of zero. Alternatively, we can use the variational metho, with a trial wave function of the Pascual form Eq. 56. The Laplacian is , an the area element is = The matrix elements of the Hamiltonian are Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 59

7 H mn = m p p 1/ m n! mn p3 1 m n/! p 1 mn m n even. m n 1, 9 /m n 1, m n o. m n 1/! In this case the limiting value of p is zero. Putting that in an ropping the constants out front, we are left with the conition 0 / 0 3/8 / 1/ 3/8 3/8 0 3/8 1/ 5/16 3/8 3/8 5/16 9/16 =0, 93 to which an obvious solution is =0. Again, there is no ipole moment so small that boun states o not exist. V. CONCLUSION When we began this stuy we expecte to fin critical ipole moments in one an two imensions, matching the well-establishe result in three imensions. We were surprise to fin that they o not exist: The electric ipole potential supports at least one boun state no matter how small the moment. All the more remarkable, then, is the threeimensional case. 33 a Present aress: Department of Physics, University of Washington, Seattle, WA b Electronic mail: griffith@ree.eu 1 The critical ipole moment was publishe without explanation by E. Fermi an E. Teller, The capture of negative mesotrons in matter, Phys. Rev. 7, A erivation was given by A. S. Wightman, Moeration of negative mesons in Hyrogen I: Moeration from high energies to capture by an H molecule, ibi. 77, Itwas reiscovere inepenently by several authors more or less simultaneously, using the physical ipole moel: J. E. Turner an K. Fox, Minimum ipole moment require to bin an electron to a finite ipole, Phys. Lett. 3, ; M. H. Mittleman an V. P. Myerscough, Minimum moment require to bin a charge particle to an extene ipole, ibi. 3, At about the same time it was obtaine by several authors using a point ipole moel: J.-M. Lévy- Leblon, Electron capture by polar molecules, Phys. Rev. 153, ; W. B. Brown an R. E. Roberts, On the critical bining of an electron by an electric ipole, J. Chem. Phys. 46, For a elightful account of the history see J. E. Turner, Minimum ipole moment require to bin an electron molecular theorists reiscover phenomenon mentione in Fermi-Teller paper 0 years earlier, Am. J. Phys. 45, Recently, the critical ipole moment has been interprete as an example of an anomaly the quantum mechanical breaking of a classical symmetry in this case scale invariance: H.E. Camblong et al., Quantum anomaly in molecular physics, Phys. Rev. Lett. 87, ; S. A. Coon an B. R. Holstein, Anomalies in quantum mechanics: The 1/r potential, Am. J. Phys. 70, Thus, for example, an electron shoul bin to the water molecule p= Cm, butnottoh S p= Cm. Data from CRC Hanbook of Chemistry an Physics, 86th e., eite by Davi R. Lie CRC Press, Boca Raton, FL, 005. Of course, these are more complicate structures, an in particular the rotation of the molecule turns out to make a significant contribution to the bining; nevertheless, the essential preiction appears to be confirme in the laboratory: see K. D. Joran an F. Wang, Theory of ipole-boun atoms, Annu. Rev. Phys. Chem. 54, for a comprehensive review. 3 We re thinking of the orbiting particle as an electron, so we take q to be negative. 4 If we take q= Q to be the electron charge, Eq. 7 yiels a separation of a 0, where a 0 is the Bohr raius. In other wors, the critical point occurs when Q is quite close to the hyrogen atom at the Q en, so pretening that the electron is right at Q as we i in Eq. 6 is a poor approximation. We can o a little better using first-orer perturbation theory, but it is still far from exact, because this perturbation is by no means small. 5 This argument is ue originally to Lévy-Leblon an Brown/Roberts Ref. 1. A particularly elegant proof is given by Camblong et al. Ref It is an awkwar fact not really relevant to the present argument that for any 0 the energy E goes to minus infinity as 0. 7 Don t try applying the virial theorem T=r U to this system: It yiels T= U, an hence H=E=0. The problem is that T an U are both infinite, in this case though H is finite. 8 See Coon an Holstein Ref. 1; A. M. Essin an D. J. Griffiths, Quantum mechanics of the 1/x potential, Am. J. Phys. 74, ; H. E. Camblong et al., Renormalization of the inverse square potential, Phys. Rev. Lett. 85, an references therein. 9 A more elegant metho is to factor the Hamiltonian an show that its expectation value is positive for 1 4. See K. S. Gupta an S. G. Rajeev, Renormalization in quantum mechanics, Phys. Rev. D 48, The other solution, riig r, iverges at large r. The normalization factor A, such that 0 u r=1, is sinh g/g. See I. S. Grashteyn an I. M. Ryzhik, Table of Integrals, Series, an Proucts Acaemic, San Diego, 1980, Eqs an Incientally, 1/r 0 1/rK ig r r is infinite, as anticipate in Ref This is not quite as obvious as it looks, because the raw 1/r potential has boun states for every negative energy, as long as 1 4. The pure 1/r potential is pathological in this sense. It can be tame in various ways see Ref. 8; the simplest metho is to introuce a spherically symmetric repulsive core. Brown an Roberts Ref. 1 an O. H. Crawfor, Boun states of a charge particle in a ipole fiel, Proc. R. Soc. Lonon 91, , show that such a regularization of the potential oes not affect the existence or value of the critical ipole moment. 1 Any angular momentum about the symmetry axis can only increase the energy. Crawfor Ref. 11 oes the calculation for general m, obtaining critical ipole moments for the excite states. 13 This metho was use by Lévy-Leblon an Brown an Roberts Ref. 1. In principle, any complete set of functions on the interval =0 to = woul o for example, sin nx or cos nx but Legenre polynomials have the virtue that they are eigenfunctions of L Eq. 31, which makes the calculation much simpler. 14 See, for instance, Ref. 10, Eq M. Abromowitz an I. A. Stegun, Hanbook of Mathematical Functions Dover, New York, 1965, Sec. 1. This separation is carrie out by Turner an Fox, Lévy-Leblon, Mittleman an Myerscough Ref. 1, an others. 16 J. Spanier an K. B. Olham, An Atlas of Functions Hemisphere, New York, 1987, Chap Mittleman an Myerscough Ref. 1 prove this by solving Eq. 48 with the term inclue, in the asymptotic regime very large, an joining this solution to P by matching the logarithmic erivatives in the overlap region. It seems that we shoul be able to o it without recourse to, which isappears from the answer, but we have not foun a convincing waytooso. 530 Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 530

8 18 Turner an Fox Ref. 1 an in greater etail J. E. Turner, V. E. Anerson, an K. Fox, Groun-state energy eigenvalues an eigenfunctions for an electron in an electric-ipole fiel, Phys. Rev. 174, , use a ouble sum,=e /t m,n C mn m n, with variational parameters, t, an C mn. P. Pascual, A. Rivacoba, an P. M. Echenique, Minimum ipole moment require to bin an electron in a polarizable meium, Phys. Rev. B 58, introuce the much simpler form use here. 19 The volume element in prolate spheroial coorinates is =/4 3, integrals run from 1 to, an integrals from 1 to 1. 0 The matrix in Eq. 58 is ientical to the one obtaine much more laboriously by Turner an Fox Ref. 18 which is comforting but also surprising, because their starting wave function was quite ifferent. It is surprising, too, that the variational metho converges so rapily to the correct answer, because we have no reason to suppose that the actual groun state wave function is of either the Pascual or the Turner/Fox form. Eviently the metho is peculiarly robust in the neighborhoo of the crossover point E=0. Presumably the matrices in Eqs. 36 an 58 are similar, but we have not foun the similarity transformation that relates them. 1 It is largely a matter of taste what form we choose for the potential of a point charge in an imaginary one-imensional universe. A referee suggests that we might take the one-imensional analog to a point charge to be an infinite plane, with Vx= x/ 0 for x0 an Vx=x/ 0 for x0. In that case the one-imensional ipole woul be a parallel-plate capacitor, with Vx= / 0 for x/, Vx= x/ 0 for /x /, an Vx=/ 0 for x /, which obviously has no boun states for any. But this is really a three-imensional configuration that happens to have no y or z epenence. In any case, for the purposes of this paper a one-imensional ipole coul be any potential well with a matching hill, but V must go to zero as x. There is some controversy about the legitimacy of this state. See C. V. Siclen, The one-imensional Hyrogen atom, Am. J. Phys. 56, This is easy, using the shooting metho. See N. J. Giorano, Computational Physics Prentice Hall, Upper Sale River, NJ, 1997, Sec. 10. or D. J. Griffiths, Introuction to Quantum Mechanics, n e., Prentice Hall, Upper Sale River, NJ, 005, Problem In these moels Ux is the potential energy, corresponing to qv in the electrostatic versions. 5 A beautifully simple proof using the variational principle is given by K. R. Brownstein, Criterion for existence of a boun state in one imension, Am. J. Phys. 68, The theorem may not apply to the Coulomb potential Eq. 59, because the integral in Eq. 65 is not strictly convergent, an Ux oes not vanish outsie a finite region. This case, therefore, remains problematic. 6 Of course, one coul ask about asymmetric systems, in which the repulsive en is slightly larger than the attractive en, or introuce a thir positive bump at the center, so the integral in Eq. 65 is greater than zero. In this case there is necessarily a critical separation istance, but it s not a ipole any more. 7 B. Zaslow an M. E. Zanler, Two-imensional analog to the Hyrogen atom, Am. J. Phys. 35, Two-imensional hyrogen with a logarithmic potential arguably a more appropriate analog has been stuie by F. J. Asturias an S. R. Arágon, The hyrogenic atom an the perioic table of the elements in two spatial imensions, ibi. 53, ; it leas to a point ipole potential of the form p cos /r, but the critical ipole moment if there is one woul epen on the separation of the charges, an we have not explore this case. 8 The general solution is a linear combination of Mathieu functions, but it is only perioic for certain special values of. What follows is one way of etermining those allowe values. 9 The orinate in Fig. 1 is now the x axis, an x=/ cosh u cos v, y =/ sinh u sin v. 30 Mary L. Boas, Mathematical Methos in the Physical Sciences, n e., Wiley, New York, 1983, pp Normalization requires A=0, of course. As always, the wave function is maximally elocalize as the energy approaches zero. 3 This is not just Eq. 43 without the term rather, it is the Laplacian for elliptic cyliner coorinates,,z without the z term. 33 One might ask whether there is a critical moment for the bining of a point charge to a magnetic ipole. Even in the classical case the orbits of an electron in the magnetic fiel of the earth, for example this problem is enormously more ifficult, with chaotic as well as perioic regimes. Whole books have been written on Størmer s problem, as it is calle. As far as we know, no boun states for the quantum analog have been foun. See J. R. Reitz an F. J. Mayer, New electromagnetic boun states, J. Math. Phys. 41, Am. J. Phys., Vol. 75, No. 6, June 007 Kevin Connolly an Davi J. Griffiths 531