Bound States in Delta Function Potentials

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 systems. 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| <x0,whereki = of quantum mechanics and are discussed in practically every 2M(Bi/2x0 −|E|) inside the well. Outside the potential | | Ψ Ψ | | ik0 x text on it. well, both gand u have the form 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].

Load more