Radial equation perturbation theory

Scott E. Hoffmann

In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.

Alto Osada, Rekishu Yamazaki, Atsushi Noguchi

Michele Cini

Fan Hong‐Yi

By virtue of the technique of integration within an ordered product of operators we derive the concise normally ordered expansion of the even-power of the radial coordinate operator via the equation = ∫ d 3 x | x ⟩⟨ x | r , where | x ⟩ is the three-dimensional coordinate eigenvector, and r = ( x 2 + y 2 + z 2 ) 1/2 . The application to perturbation theory is briefly discussed.

Kristina Mallory, Robert A. Van Gorder

Stationary solutions for the 2+1 cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BEC) in a small potential are obtained via a form of perturbation. In particular, perturbations due to small potentials which either confine or repel the BECs are studied, and under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of radial BEC solutions. Numerical results are also provided for regimes where perturbative results break down (i.e., the large-potential regime). Both repulsive and attractive BECs are considered under this framework. Solutions for many specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular small potentials on the behavior of the BECs.

Jochen Pade