Hindawi Publishing Corporation Journal of Atomic, Molecular, and Optical Physics Volume 2011, Article ID 573179, 4 pages doi:10.1155/2011/573179

Research Article Bound States in Delta Function Potentials

Sydney Geltman

Department of Physics, University of Colorado, Boulder, CO 80309, USA

Correspondence should be addressed to Sydney Geltman, [email protected]

Received 28 June 2011; Accepted 18 September 2011

Academic Editor: Gregory Lapicki

Copyright © 2011 Sydney Geltman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present a systematic treatment of the bound state structure of a short-range attractive interatomic potential in one, two, and three dimensions as its range approaches zero. This allows the evaluation of the utility of delta function potentials in the modeling of few-body systems such as nuclei, atoms, and clusters. The relation to scattering by delta function potentials is also discussed.

1. Introduction with electromagnetic fields, that is, in multiphoton absorp- tion or ionization processes. While the above difficulties that The delta function has played an enormous role in the devel- arose in the nuclear case would also be present in the three- opment and advancement of quantum mechanics since its dimensional treatment of electron binding in an atom, a introduction by Dirac [1]. From its early use in the momen- feasible situation presents itself in the one-dimensional case, tum representation in scattering theory, it has found many where the existence of a single bound state may serve as a conceptual applications. In the present work, we study its reasonable model for the ground state of an atom, or an even applicability to represent a very short-ranged attractive inter- better model for a negative ion, which typically has only one atomic potential and what bound state structure may be ex- bound state. For linearly polarized radiation fields, the use of pected in the zero-range limit. a one-dimensional atomic model is also physically reasonable In 1935, Thomas [2] published a paper in which he inves- and has been extensively used [3–7]. tigated the possible relationship between the mass defect of Another area where delta function potentials have come the deuteron and the range of the proton-neutron force. into considerable use is in the description of a many-atom The conclusion was that a finite mass defect, attributable system, such as a cluster or a Bose-Einstein condensate. If to a finite binding energy, was not consistent with a zero- the atomic density is small enough such that the mean inter- range potential. He showed that going to the limit of a zero atomic separation is large compared to the range of the inter- range proton-neutron attractive potential could only result atomic force or potential, it is reasonable to represent the in an infinite binding energy, or infinite mass defect, which interatomic potential by a weighted delta function, −Bδ(r − contradicts the experimental finding of a finite mass defect. r ). This often arises where a many-body problem is treated An extension of this argument to the triton would conclude in a mean-central-field approximation and enters into the that, in the limit of zero-range two-body forces, the three- Hartree self-consistent-field nonlinear wave equation as body system would also tend to have an infinite binding 2 energy. This suggests that the use of delta function potentials dr V r − r Ψ r =∼ −B|Ψ(r)|2. (1) may lead to unrealistic binding energies in composite sys- tems. Our present objective is to treat the problem in a more This is a reasonable way of handling the interatomic potential complete and systematic manner by evaluating all the bound in a diffuse many-body system and does not depend on the states rather than using a variational method that only gives bound states in the zero-range limit. However, the evaluation an upper bound to the ground state energy [2]. of the coefficient B (often called U0 in the literature) is not In more recent years, delta function potentials have been obvious and has been the subject of some disagreement used to represent model atomic systems in their interactions [8–11]. 2 Journal of Atomic, Molecular, and Optical Physics

As is well known, square well potentials have been used The two allowed symmetries are Ψg (even), that is, cos(kix), extensively to model bound-state systems since the beginning and Ψu(odd), that is, sin(kix)for|x| 0, e , Their generalization to an arbitrary number of spatial where k0 = i 2M|E|. Joining inside and outside solutions dimensions has also been carried out [12, 13]. Our purpose at |x|=x0 with continuous Ψ /Ψ gives the transcendental in this paper is to present a unified picture of how by letting equations for |E|(E<0 for bound state), the range of the square well approach zero in one, two, and three dimensions, one may deduce the bound-state structure |k0| even: tan(kix0) = , in the delta function limit. ki We will approach the problem for delta function poten- (7) =− ki tials in one, two, and three dimensions in the following odd: tan(kix0) . |k | sections. These potentials are denoted as 0 V (R) =−B δ(R), (2) We now want to use these to find the bound state energies j j → in the x0 0 limit, that is, for the delta function potential. = ffi where Bj (j 1, 2, 3) > 0. The strength coe cient Bj In going to this limit, we note that kix0 = MB1x0(1 − f ), will have varying dimensionalities, since every Vj must where f =|E|/(B1/2x0) is the binding energy expressed as a have the dimensions of energy, while each δ(R) has the fraction of the well depth. Since kix0 → 0asx0 → 0, we may dimensionality of inverse volume, since dR δ(R) = 1. We 3 expand tan(k x ) → k x + ϑ((k x ) ). For the even case, this will take the δ(R)’s for each dimensionality to be the limit of i 0 i 0 i 0 leads to E →−MB2/2 as the only bound state in the limit. a step function, 1 ⎧ The situation is somewhat more complicated for the ⎪ 1 odd symmetry. Again expanding tan(k x ) and dropping ⎨ lim , |x|≤x , i 0 → 0 3 1D: ( ) = x0 0 2x0 ϑ((kix0) ), we have δ x ⎩⎪ 0, |x| >x0, −1 ⎧ E −→ . (8) → 2 ⎪ x0 0 2Mx ⎨ 1 2 0 lim , ρ ≤ ρ0, = → 2 2D: δ ρ, ϕ ⎪ 2π ρ0 0 ρ0 (3) ⎩⎪ For this to be an acceptable bound state energy, its absolute 0, ρ>ρ0, value must lie within the range of the well depth, or ⎧ B /(2x ) > |E| > 0. Thus, the minimum value of x that ⎪ 1 0 0 ⎨ 1 3 lim , r ≤ r0, can support a bound state of odd symmetry is approximately = → 3 3D: δ r, ϑ, ϕ ⎪ 4π r0 0 r0 ⎩⎪ 1/(MB1), and there are no bound states for lower x0,which 0, r>r0. of course includes the delta function limit. For large enough x ,theremaybemanyevenandodd Since −B < 0, our extended potentials will be attractive 0 j symmetry bound states since the singularities of the tangent square wells. We first evaluate the bound states for the functions will give a number of intersections with the rhs of extended square well and then go to the limit of zero range. (7), in fact there will be about MB x /π such intersections. The symmetries that arise in each dimensionality for the 1 0 assumed square wells and the corresponding wave function separabilities are 3. Two Dimensional Ψ − = Ψ Ψ − =−Ψ 1D: even, g ( x) g (x),andodd, n( x) u(x), Adopting the polar coordinates ρ, ϕ in 2D space, the Schrodinger¨ equation is Ψ ρ = imϕ 2D: χm ρ e , 1 ∂2 1 ∂ 1 ∂2 3D: Ψ(r) = Rl (r)Ylm θ, φ . − − Ψ = 2 + + 2 2 +V2 ρ E ρ, ϕ 0, (4) 2M ∂ρ ρ ∂ρ ρ ∂φ (9) 2. One Dimensional where V2(ρ), from (2)and(3), is ⎧ The Schrodinger¨ equation (atomic units used throughout) ⎪ B for the above assumed 1D potential is ⎨− 2 , ρ≤ ρ , = 2 0 V2 ρ ⎪ πρ0 (10) 2 ⎩ 1 d 0, ρ>ρ0. − + V (x) − E Ψ(x) = 0, (5) 2M dx2 1 This central potential allows the separability, Ψ(ρ, ϕ) = where M is the reduced mass and imϕ ⎧ χm(ρ)e ,whereχm(ρ) satisfies the radial equation ⎪ B ⎨− 1 , |x|≤x , = 0 2 2 V1(x) ⎪ 2x0 (6) d 1 d m 2 ⎩ + − + k Xm ρ = 0, (11) 0, |x| >x0. dρ2 ρ dρ ρ2 Journal of Atomic, Molecular, and Optical Physics 3 and the inside and outside ks take the values where k takes the inside and outside values, 1/2 B2 1/2 = 2 −| | , ≤ , 3B3 ki M 2 E ρ ρ0 k = 2M −|E| , r ≤ r , πρ0 (12) i 3 0 4πr0 (16) 1/2 k0 = i(2M|E|) , ρ>ρ0. 1/2 k0 = i(2M|E|) , r>r0. These are Bessel equations, and we must join the inner As previously done, since higher angular momentum bound regular solution χm(ρ) = Jm(kiρ) to the outer decaying states will be fewer than those for l = 0, we will restrict our solution, Km(|k0|ρ) the modified Bessel function [14]. We restrict our present calculation to the case of zero angular attention to s-states. momentum, that is, m = 0, the one having the largest num- The solutions of (15) are the spherical Bessel functions, ber of bound states. of the first kind inside the potential well [14], We first note that the arguments of the Bessel functions sin(kir) at the edge of the well are R0(r) = j0 (kir) = , (17) kir | | 1/2 = 2MB2 − E kiρ0 1 2 , and modified (of the third kind) outside the well, π B2/πρ0 −| | (13) k0 r 1/2 = (2) = e 2 | | R0(r) h0 (ik0r) . (18) | | = MB2 E |k0|r k0 ρ0 2 . π B2/πρ0 Demanding continuity of the logarithmic derivatives at r0 2 Since the depth of the potential well (from (3)) is B2/πρ0 , gives the transcendental equation for the bound states in we may define the bound states, E =−|E|, as their fractional terms of their fractional amount of the well depth, f = 3 amount of the potential depth (measured from the top of the |E|/(3B3 /4πr ), = 0 well where E 0). The only free variable remaining is the product MB2. We may then obtain numerically the roots for 3MB 1 − f tan 3 1 − f + = 0. (19) the continuity of logarithmic derivative at ρ0, 2 πr0 f k J k ρ |k |K |k |ρ i 0 i 0 = 0 0 0 0 , (14a) Unlike the 2D case, the coefficient of 1 − f in the tangent J0 kiρ0 K0 |k0|ρ0 also depends on r0, so the ratios between various bound | | 2 to give all the roots for E /(B2/πρ0 ) as a function of MB2. state energies do not remain independent of r0. Again, from Again, a good estimate of the number of bound states is the periodicity of the tangent function, we estimate the total = 1/2 given by 2MB√2/π divided by π since asymptotically J0(z) numberofboundstatestobe(3MB3 /2πr0) /π. cos(z − π/4)/ z. Thus, in the delta function limit, there are an infinite Thus, the number of bound states increases as MB2, number of bound states, and they each approach infinite while remaining independent of ρ0. The absolute values of binding energy. each of these bound states will →∞as ρ0 → 0 since All of our present interest in the delta function potential 2 ff the potential depth is increasing as B2/πρ0 . This di ers has been in its bound state structure. It is well known, markedly from the 1D case where both the number of bound however, that there is a close relationship between the states and their relative and absolute values also depended on bound states and the scattering properties of a potential. x0. This was first investigated by Fermi [15] for the delta function case in the study of the scattering of neutrons by 4. Three Dimensional hydrogen atoms. Zero energy scattering is characterized by the scattering√ length, defined as = limk → 0(− tan η0/k), where The Schrodinger¨ equation in the 3D spherical coordinates k = 2ME, M is the reduced mass and η0 is the s-wave phase r, θ, ϕ is shift. For the present 3D case, this may be evaluated exactly and is given by 2 − 1 ∂ 2 ∂ 1 ∂ ∂ 2 + + 2 sin θ 3/2 2M ∂r r ∂r r sin θ ∂θ ∂θ r − a = r − 0 tan cr 1/2 , (20) 0 c 0 1 ∂2 + + V (r) − E Ψ r, θ, ϕ = 0, 2 2 2 3 1/2 r sin θ ∂ϕ where c = (3MB3 /4π) . This is clearly indeterminate in (14b) the r0 → 0 limit. It is further closely tied to our present result that the number of bound states becomes infinite where V3(r)isgivenin(2)and(3). Again with separability, in the delta function r0 → 0 limit. The connection is Ψ(r, θ, ϕ) = Rl(r)Ylm(θ, ϕ), where Rl(r) satisfies the radial Levinson’s theorem [16] that states that η0 → N0π,where wave equation ε → 0 2 N0 is the number of bound s-states. Much theoretical work d 1 d l(l +1) 2 + − + k Rl(r) = 0, (15) [17–21] has gone into the removal of this indeterminacy dr2 r dr r2 by a procedure known as regularization, requiring much 4 Journal of Atomic, Molecular, and Optical Physics more refined advanced mathematical considerations than are in intense laser fields,” Journal of Physics B, vol. 29, no. 23, pp. needed in the present bound state study. Another aspect of 5755–5764, 1996. delta function potentials is that their detailed study is useful [7] S. Geltman, “Comment on “stabilization of a one-dimensional in many aspects of quantum field theory, and an excellent short-range model atom in intense laser fields”,” Journal of survey of this subject is contained in the paper of Gosdzinsky Physics B, vol. 32, no. 3, pp. 853–856, 1999. and Tarrach [22]. [8] E. P. Gross, “Structure of a quantizedvortex in boson systems,” Nuovo Cimento, vol. 20, no. 3, pp. 454–477, 1961. [9] E. P. Gross, “Hydrodynamics of a superfluid condensate,” 5. Summary Journal of Mathematical Physics, vol. 4, no. 2, pp. 195–207, 1963. The objective of the present paper is to present a unified [10] L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Soviet treatment for the one, two, and three dimensional cases that Physics—JETP, vol. 13, p. 451, 1961. might have application to actual physical systems and look [11] S. Geltman, “A critique on the misuse of the Gross-Pitaevskii equation,” EPL, vol. 87, no. 1, p. 13001, 2009. at their bound state structure as the potential well radius is [12] A. Farrell and B. P. van Zyl, “S-wave scattering and the zero- shrunk to zero, thus approaching delta function potentials. range limit of the finite square well in arbitrary dimensions,” We have shown that the bound state structure of attractive Canadian Journal of Physics, vol. 88, no. 11, pp. 817–824, 2010. delta function potentials differ considerably depending on [13] M. M. Nieto, “Existence of bound states in continuous 0 < D their dimensionality. < ∞ dimensions,” Physics Letters, Section A, vol. 293, no. 1-2, In the 1D case for the even symmetry, there is a single pp. 10–16, 2002. =− 2 well defined bound state at E MB1/2, while there are [14] M. Abramowitz, I. A. Stegun et al., Handbook of Mathematical none for the odd symmetry. This does not occur for the Functions, U S Government Printing Office, 1964. higher dimensionalities, where all bound energies become [15] E. Fermi, Ric. Sci., vol. 7, p. 13, 1936. infinite in the zero range limit, and thus provides the unique [16] N. Levinsonson, Det kongelige Danske Videnskabernes Selskab opportunity to represent a model bound atomic system. Matematisk-Fysiske Meddelelser, vol. 25, no. 9, 1949. [17] K. Wodkiewicz, “Fermi pseudopotential in arbitrary dimen- Indeed, it has been used to investigate photoabsorption sions,” Physical Review A, vol. 43, p. 68, 1991. processes with linearly polarized radiation or tunnelling [18] R. Jackiw, Beg Memorial Volume, World Scientific, 1991. ionization due to a static electric field [3–7]. [19] A. Cabo, J. L. Lucio, and H. Mercado, “On scale invariance In the 2D case, there is a finite number of bound states, and anomalies in quantum mechanics,” American Journal of 3 1/2 givenapproximatelyby(2MB2/π ) , each of which grows Physics, vol. 66, no. 3, pp. 240–246, 1998. 2 → as 1 /ρ0 as ρ0 0, or each of them approaching infinite [20] I. Mitra, A. DasGupta, and B. Dutta-Roy, “Regularization and binding energy. This makes the 2D delta function potential renormalization in scattering from Dirac delta potentials,” also not usable to represent bound atomic systems, since the American Journal of Physics, vol. 66, no. 12, pp. 1101–1109, latter have finite binding energies. 1998. In the 3D case, the delta function limit produces both an [21] S.-L. Neyo, “Regularization methods for delta-function poten- ∼ 3 1/2 tial in two-dimensional quantum mechanics,” American Jour- infinite number of states, (3MB3/2π r0) ,aswellaseach nal of Physics, vol. 68, p. 571, 2000. − 3 state approaching infinite binding energy as B3/r0 .Thus, [22] P. Gosdzinsky and R. Tarrach, “Learning quantum field theory a 3D delta potential is also unsatisfactory to represent any from elementary quantum mechanics,” American Journal of stable two-body (or more) nuclear, atomic, or cluster system. Physics, vol. 59, p. 70, 1991. This confirms the conclusion reached by Thomas [2]76years ago and hopefully provides a more complete understanding of it. The Thomas result was obtained variationally as an upper bound, and it referred only to the ground state, while our present treatment covers all the bound states.

References

[1]P.A.M.Dirac,The Principles of Quantum Mechanics,Oxford University Press, 3rd edition, 1940. [2] L. H. Thomas, “The interaction between a neutron and a proton and the structure of H3,” Physical Review, vol. 47, no. 12, pp. 903–909, 1935. [3] S. Geltman, “Ionization of a model atom by a pulse of coherent radiation,” Journal of Physics B, vol. 10, no. 5, article 019, pp. 831–840, 1977. [4] S. Geltman, “Short-pulse model-atom studies of ionization in intense laser fields,” Journal of Physics B, vol. 27, p. 1497, 1994. [5] J. Z. Kaminski,´ “Stabilization and the zero-range models,” Physical Review A, vol. 52, no. 6, pp. 4976–4979, 1995. [6] Q. Su, B. P. Irving, C. W. Johnson, and J. H. Eberly, “Stabilization of a one-dimensional short-range modelatom Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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