Eigenfunctions parabolic coordinates

In Chapter 3 we investigated the development in time of a decaying state, expressed in terms of the time-independent eigenfunctions satisfying a system of two coupled differential equations, resulting from the separation of the Schrodinger equation in parabolic coordinates. In this analysis we obtained general expressions for the time-dependent wave function and the probability amplitude. 

The choice of parabolic coordinates in [35] and Equation (56) is motivated by our interest in exploiting the connection between the superintegrable harmonic-oscillator and atomic-hydrogen systems [33-35]. For instance, the well-known eigenfunctions and energy eigenvalues for the two-dimensional harmonic oscillators can be written immediately by borrowing them from [33] ... 

Although it is typically solved in spherical and parabolic coordinates, the hydrogenic Schrodinger equation is also separable in prolate spheroidal coordinates, with one focus at the nucleus and the other located along the Lenz vector at a distance R away. These coordinates are ordinarily used for two- center problems such as H. Previously, general features of the spheroidal hydrogen atom have been explored by Coulson and Robinson [6], who noted that the limits R —> 0 and R —> oo yield the spherical and parabolic solutions, respectively. Demkov [7] used the spheroidal eigenfunctions to construct... 

The Graetz problem (heat or mass transfer) in cylindrical coordinates with parabolic velocity profile is solved here. The governing equation for the eigenfunction is [15] [8]... 

The chapter contains a review of the free hydrogen atom eigenfunctions in the spherical, spheroconal, parabolic, and prolate spheroidal coordinates an overview of our own works for confinement by most of the above-mentioned boundaries and a preview of problems on confined atoms and molecules of current and future investigations. 

Problem 7-20. Sphere in a Parabolic Flow. Use the general eigenfunction expansion for axisymmetric creeping-flows, in spherical coordinates, to determine the velocity and pressure fields for a sohd sphere of radius a that is held fixed at the central axis of symmetry of an unbounded parabolic velocity field,... 

For the parabolic and prolate spheroidal coordinates, the changes in the eigenfunctions and eigenvalues are reduced to the replacement of m by the value of iJb. The respective energy parameters become... 

In the limit R —0, each spheroidal eigenfunction n m > reduces to a particular spherical function n/m >, specified by a —> —1(1 -h 1). Likewise, in the limit R —> oo, each nam > becomes a particular parabolic function nrm >, which in turn can be obteuned from Equation (8) as a linear combination of spherical functions with the gi given by the Clebsch-Gordan coefficients.[9] Regardless of the internuclear separation, and, consequently, of the coordinate system chosen, the total number of nodes of the eigenfunctions is conserved at n — m — 1. Other aspects of the transition to these limits have been examined and illustrated by Coulson and Robinson. [6]... 

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