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Journal of Physics B Atomic Molecular and Optical Physics

The WKB approximation applied to expectation values of rv

Thomas A. Heim

Published Jan 28, 1994• 9 citations
Publisher

Abstract

Formulae for the calculation of expectation values of arbitrary powers of r for hydrogen-like wavefunctions are derived both in quantum mechanics and within the framework of the WKB approximation. The quantum to classical correspondence is then investigated in the limit n to infinity for finite n r , as well as for finite l. For negative powers and finite l, an asymptotically remaining deviation of the semiclassical formulae from the quantum expressions is found. A modification in the eccentricity of the classical orbit is introduced which corrects the asymptotic behaviour of the WKB approximation.

DOI
10.1088/0953-4075/27/2/001

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Formulae for the calculation of expectation values of arbitrary powers of r for hydrogen-like wavefunctions are derived both in quantum mechanics and within the framework of the WKB approximation. The quantum to classical correspondence is then investigated in the limit n to infinity for finite n r , as well as for finite l. For negative powers and finite l, an asymptotically remaining deviation of the semiclassical formulae from the quantum expressions is found. A modification in the eccentricity of the classical orbit is introduced which corrects the asymptotic behaviour of the WKB approximation.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].
A recent paper by Drake and Swainson [Phys. Rev. A 42, 1123 (1990)] presents a recursion relation that generates a set of coefficients needed to obtain general closed-form expressions for hydrogenic expectation values for rP{\mathit{r}}^{\mathit{P}}rP. By combining this recursion relation with an earlier semiclassical formulation of this calculation [J. Phys. B 14, 1373 (1981)], it is shown that the correct expectation values can also be obtained by a simple substitution of prescribed angular momentum operators into the classical Kepler formula, which provides a transparent connection to the correspondence limit.
For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.