PHYSICAL REVIEW A VOLUME 49, NUMBER 4 APRIL 1994 Semiclassical quantization in momentum space Bernd Rohwedder' and Berthold-Georg Englertt Sehtion Physih, Unioersitiit Miinchen, Arn Coulornbwall 1, D 8$— 7/8 Garching, Germany (Received 27 September 1993) Three-dimensional, spherically symmetric Hamilton operators, which consist of the sum of an arbitrary effective kinetic energy and an attractive Coulomb potential, are quantized semiclassically. The semiclassical quantization rule that we derive passes the test of estimating the Bohr energies of hydrogenlike atoms successfully. The semiclassical spectrum of Thomas-Fermi atoms, which are described in terms of an effective kinetic energy, is found to agree essentially with the spectrum obtained in the standard formalism that employs an effective potential energy. PAGS number(s): 03.65.Ge, 31.20.Lr, 03.65.Sq I. INTRODUCTION In the functional approach to the many-electron prob- lem of atomic structure calculations, one is led to ef- fective independent-particle pictures. They provide the bases for various approximations, such as the Thomas- Fermi (TF) models and their refinements or the Kohn- Sham (KS) schemes. The standard spatial function- als treat the spatial density as the fundamental quan- tity [1,2); the momentum-space density — momental den- sity, for short — plays this role in the momental function- als introduced recently [3,4]. The independent-particle energies account for the electron-electron interactions by using an effective potential energy in the spatial formal- ism, or by using an effective kinetic energy in the mo- mental formalism [4]. These effective energies are to be determined self-consistently. If one does not wish to evaluate the sum of independent-particle energies with the aid of the highly semiclassical TF approximation, the eigenvalue spectrum of the effective Hamilton operator has to be found. This leads to a standard differential Schrodinger equation if one works in the spatial formalism with its effective po- tential energy. The WKB approximation to the eigenval- ues, which is then available, has proven sufFiciently ac- curate for the analytical calculation of shell effects [5,6]. When, in contrast, one opts for the momental formu- lation with its effective kinetic energy, the Schrodinger eigenvalue equation is an integral equation [7], to which the WKB method in its usual form is not applied easily. The desire to obtain semiclassical eigenvalues of WKB quality nevertheless — be it for the purpose of studying shell effects once more, or to get initial values for the iterative KS scheme — has motivated the investigations reported in this paper. Present address: Facultad de Fisica, Universidad Catolica de Chile, Casilla 306, Santiago 22, Chile. t Also at Max — Planck — Institut fiir Quantenoptik, Hans- Kopfermann-Strasse j. , D-85748 Garching, Germany. Here is a brief outline. Our starting point is a semiclas- sical quantization rule, termed TF quantization, which reproduces WKB results without making use of approx- imate wave functions. Like the WKB method, TF quan- tization is limited to one-dimensional situations to begin with, but can be extended to three-dimensional problems with spherical symmetry if the Langer correction [8] is taken into account. In a first step we find the Langer correction appropriate to Hamilton operators that are the sum of an effective kinetic energy and a potential energy of Coulomb form. Then we employ TF quan- tization to arrive at an expression that determines ap- proximate eigenvalues. A test performed on hydrogenic systems shows very satisfactory agreement with the ex- act eigenvalues. The application to the effective kinetic energy of the momental TF model produces lines of de- generacy that are indiscernible &om the ones computed in the spatial formalism [5]. We shall use atomic units throughout and employ the notational conventions of [9]. II. THOMAS-FERMI QUANTIZATION For a one-dimensional Hamilton operator of the struc- ture H(x, p) = — p' + V(x), [x, p] = i, (2 1) the well known WKB quantization formula n+ —= — dx' 2(E„—V(x')) 1 1 2 (2.2) provides a simple and usually very accurate method for predicting its energy spectrum. The usual textbook derivation of this semiclassical approximation makes ex- plicit use of WKB wave functions, which is facilitated by the simple form of the kinetic potential in (2. 1). The more general Hamiltonians that appear in momental energy-functional theory do not lend themselves to this approach. Nonetheless, a generalization of (2.2) for quite arbi- trary, physically reasonable Hamilton operators is pos- 1050-2947/94/49(4)/2340(7)/$06. 00 2340 1994 The American Physical Society