Schrödinger Equation for a Bound Proton:
Asymptotic Solutions

This is a solution to the Schrödinger Equation for the "Hydrogen-like" system of a nucleus, A=N+Z, in terms of a proton and the "core" nucleus A-1=N+(Z-1). These solutions are relevant to astrophysical S factors.

The Schrödinger Equation can be written:

To arrive at this form, the following substitutions were made:

A more detailed approach to the same equation can be found with the description of the solutions for the Hydrogen atom. Looking at the equation, it should be apparent that α has the dimension of inverse length. To write the equation in terms of a dimensionless variable - a "pure number" - make another substitution.

The differential equation for the wavefunction in terms of this dimensionless variable is

Note that, for a bound proton, γ is negative. Also, the substitution z=>-z would give the equation for opposite charges. Below is a simplified model of the potential for the bound proton.

The potential of the nucleus is drawn as a simple spherical square well, when in reality something like a Woods-Saxon potential would be more accurate, but that won't affect the results obtained here at all. The point is, the desired solutions are for the bound proton at large radius, where it is energetically forbidden. The length scale of the problem is set by α. Discussing protons for radiative capture reactions, the reduced mass can vary from the proton mass to half the proton mass for pp fusion. α is μZ/13.5 (1/fm), with μ being the reduced mass in GeV.

As for the hydrogen atom, in the limit where α=0 and l=0, the solutions would be plane waves or exponential decays. Once again, factor an exponential from the solutions.

The equation becomes

Having battled many an equation, one intuitively sets the value of λ to make life easier.

The goal is to find solutions for large r. The Coulomb barrier should suppress the wave function to be smaller than the exponential decay of a wave function with binding energy E in a region of zero potential. So try a power series substitution in terms of 1/z.

The result is

Matching powers of z gives

These solutions should rapidly approach zero with increasing z. It is necessary to have a minimum value of n with a nonzero coefficient. For some value of n - call it k - this equation must be satisfied:

Now, λ is not an integer. It is a constant that depends on the reduced mass, the charge of the nucleus, and the binding energy of the proton. But for a modified substitution, there is a solution.

The recursion relation for the coefficients in the Laurent series is

This time it's simple to write an expression for the coefficients that does not involve any recursion.

Now we have a solution for the form of the proton's wave function at large radii. Given l and the details of the specific nucleus being studied, the only thing left undetermined is c₀. For a given nucleus, the proton separation energy determines λ. The overall normalization is determined by the details of the physics inside the nucleus. This normalization is also known as the Asymptotic Normalization Coefficient (ANC). While ANCs would be difficult to calculate theoretically (the thoeretical values would be plagued by large uncertainties), it is possible to extract the ANCs by measuring proton-transfer reactions in experiments at accelerator facilities.

ANCs are interesting to study because they determine astrophysical reaction rates. The rate of transition can be described by the overlap of the asymptotic wavefunction of the compound nucleus with the relative wavefunction of a free proton and the core nucleus. The form of the wave function is

For the equation above, c₀=1, c_n is given by the previous equation, and the square root of N is for the Normalization.

Consider the system of ⁸B=⁷Be+p. In this case, Z=4 and the reduced mass, μ, is approximately 800 MeV. In this case, roughly, α=1/(4 fm). The nuclear radius of ⁷Be should be approximately 2 fm. And λ is roughly equal to the square root of the magnitude of the binding energy.

For the astrophysical reaction ⁷Be(p,γ)⁸B, the average relative energy for this reaction at solar temperatures is on the order of 10 keV. But the classical turning point for ⁷Be and a proton at an energy of 10 keV corresponds to a distance of 197/(137*0.01)~150 fm. At this distance, the dimensionless coordinate is very approximately αr=40. Not only should the asymptotic function given above be adequate; the first two terms alone describe the behavior at this distance to better than 0.1%!