Kummer function

Watanabe, S. (1987). Kummer-function representation of ridge travelling waves, Phys. Rev. A36, 1566-1574. 

As discussed in [45], the solution of problem (26) is then obtained in terms of the confluent hypergeometric function, also known as Kummer function iFi[a b z], readily available in the Mathematica system [41], as ... 

Equation (244), in turn, can be expressed in terms of the confluent hypergeo-metric (Kummer) function M(a,b,z) [51], namely,... 

Solutions of this equation are related to the Kummer functions (21). The parameter 6, is the quantum defect and c is an integer chosen to ensure the normalizability of the orbitals and their correct nodal pattern -the number of radial nodes is equal to n-l-c-1. 

Exact Solutions for Confined Model Systems Using Kummer Functions... 

Although these two models have been treated using slightly different transformations the generic analysis is identical. The essential transformations are of the form = exp(—/(r))0, where f(r) is chosen to describe the correct asymptotic form for jr and leads to the Kummer equation for . The different transformations used for the hydrogen and harmonic oscillator problems lead to E being related differently to the parameters of the Kummer functions. 

The main purpose of this article is to review the solutions of the radial equation when the region of space considered is confined and, in such a region, the potential V(r) takes on a particularly simple form so that exact solutions may be found in terms of Kummer functions. The examples considered here are when V(r) = — 1/r, V(r) = r2/2 or V(r) = Vo, a constant potential of course, an elementary scale change enables the examples V(r) = —Z/r and V (r) = co2r2/2 to be treated analogously. 

One model of this would be an infinite spherical potential at r = R so that the boundary condition is simply vJ/(R) = 0. This model has been considered by many authors [9-13] and recently we have found the solution in terms of the Kummer functions [2] and extended the theory to the first-order polarizabilities of such systems [4]. In this case, since we require the solution to be finite at the origin, the energies are found from... 

As an example of both exterior and interior confinement we consider the Schrodinger equation for the 3-d hydrogen atom in the interval L < r < R, where the parameters in the Kummer functions are used to match boundary... 

We have examined exact solutions, in the form of Kummer functions, that may be used to describe various forms of confinement to specific spatial regions. The definition of confinement has been enlarged from that used conventionally, and in all cases the physical aspects of the confinement are modeled by boundary conditions. These boundary conditions in some cases include prescribed energy values, but are mainly conditions on the wavefunction and its derivative on the boundary of the confined region. In this way the main aspects of the physics are modeled by the exact solutions found. However,... 

The functions M ai,bi, ) and U ai,bi, ) are Kummer functions of first and second order with the parameters... 

Then, transforming hack and applying the recurrence relations (9.9) and (9.10) for the derivative of the Kummer function leads to... 

We are here interested in the relationship of the Kummer function to the Boys function. From (9.8.35) we obtain, using (9.8.37),... 

Comparing with (9.8.11), we find that the Boys function is a (renormalized) special case of the Kummer function ... 

Therefore, the standard relations that have been established for the Kummer function hold for the Boys function as well. For example, for the Boys function, the general relations [9]... 

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