Quantum unbinding in potentials with 1 Ir tails R. K. P. Zia Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 R. Lipowsky and D. M. Kroll Institutfur Festkorperforschung, Kernforschungsanlage Julich, 51 70 Julich, West Germany (Received 27 October 1986; accepted for publication 13 April 1987) Potentials with repulsive power behavior at large distances may or may not have a zero-energy bound state, depending only on the powerp and, ifp = 2, the strength. For the case of no zero- energy bound state, we further study how the state unbinds as the attractive, small r part of the potential is tuned to let the binding energy go to zero. Although motivated by the physics of wetting, this problem is well suited for senior or graduate level quantum mechanics courses by giving a novel perspective of solutions to one-dimensional Schroedinger equations. I. INTRODUCTION In a typical senior- or first-year graduate textbook'-2 on nonrelativistic quantum mechanics one encounters a chapter on solving the one-dimensional Schroedinger equation with a variety of potentials. In addition to finding specific wavefunctions and energy eigenvalues associated with square wells and simple harmonic potentials, there would be some discussion of general properties such as the number of bound states and the distribution of their ener- gies given certain qualitative features of any potential. The subject is quite old, so that new perspectives or problems, especially ones that are physically rather than mathemat- ically motivated, are rarely seen. Recently, in connection with the phenomenon of wetting transition^,^ such a novel problem appeared. Moreover, we believe it is sufficiently simple to be included in the typical course. For potentials that vanish at large distances, one learns that positive/negative energy solutions correspond to un- boundhound states. The borderline case, a zero-energy solution, is more delicate and usually ignored. In this arti- cle, we will study this delicate threshold and be concerned with how a state becomes unbound if the (short distance parts of the) potential is changed so that a negative energy eigenvalue vanishes. Section I1 is devoted to setting up the problem and specifying the class of potentials and quanti- ties to be used to define "unbinding." Section 111 contains the analysis. A summary of the results and a discussion of the origin and application of this study are found in the See. IV. 11. THE PROBLEM OF UNBINDING Consider the Schroedinger equation for a particle with mass m in a central potential V(r), Concentrating on the zero angular momentum solution, let us look at the radial equation for R =r$. The problem is now one dimensional (on the semi-infinite interval [O, cc ] ). Defining U and by 87? m/h times Vand (-E), respectively, the equation now reads R "(r) - U(r)R(r) = eR(r), (2) where R " = (d R /d? ). Note that e>0 for bound states. In this article we study the class of U's that behave as- ymptotically, at large r, as 160 Am. J. Phys. 56 (2), February 1988 where A is a positive coefficient. These are therefore poten- tials with repulsive tails. Examples include the case ofp = 2 for angular momentum barriers. The small r part of Uis left unspecified except that it must contain'some attractive components so that a bound state may exist. (See Fig. 1.) Since U( oo ) = 0, the energy of this bound state cannot be positive. However, as we will demonstrate below, it may be zero under certain circumstances. If there is no zero- energy bound state, we may ask how the state unbinds as the energy is raised toward zero. One does not normally think of the bound state energy E as a control parameter; one normally thinks of the potential V as controllable. Here, we assume that the small r part of V can be tuned (without changing the asymptotic A r p part) so that E-0. To study quantitative aspects of unbinding we define two characteristic lengths, one associated with the energy ( E or c) and another associated with the wavefunction ($ or R ) : The physical meaning of A is standard: for r^>A, R de- cays exponentially with r/A. To be precise, this behavior really does not set in until values of r for which U(r)A ¥ 1. Since we are interested in potentials with r p tails, this condition can be satisfied for sufficiently large r. To ana- lyze unbinding, we consider the limit A - cc , correspond- ing to E -à 0. The other length 1is nothing but the expectation value of Fig. 1. Sketch of a typical potential with short-range attraction and long- range repulsion. @ 1988 American Association of Physics Teachers 160