Calculation of Electron and Nuclei Wave Functions and Energy Levels
Calculation of “B”, the Electron Wave Function, and E*
We have placed the linear momentum operator, d2/dx2, into the linear momentum state wave equation 3-6, giving equation 3-8. Figure 3-4 shows the calculated relation between E* and ro, from the solutions to equation 3-8. Note that it is very similar in form to an electrostatic potential. Also shown, is the full pseudo-potential that the nuclei “feel”. The full potential is the combination of the nuclear repulsion and the attractive kinetic energy of the electron, E*:
The detailed numerical solutions of equation 3-8, to give E* as a function of ro, are given in appendix B. Earlier we made the assumption that “B” was not a strong function of ro, and could be removed from the differential operator with respect to ro . Figure 3-5 shows a graph of the wave function “B” for a high, middle and low value of ro (ro=30, 100 & 300 Fermis=10-5 Angstroms). This is the range of the probability distribution for the wave function “R(ro)”. Note that there is very little change in “B”over an "order of magnitude" range in ro, so the rate of change of “B” with ro will be negligible. Thus, “B” can safely be removed from the differential operator with respect to ro. 
Appendices A and B will not be posted for quit some time as they don’t add materially to the discussion. For those who want to venture into this calculation, 4 criteria should be kept in mind:
1) The coulomb (1/r) relation breaks down near the particle surface. We used the potential value of the surface, throughout the diameter of the nucleus.
2) A relativistic mass adjustment of the electron must be made (especially near the nuclei)
3) Use a step size to capture the powerful nuclear potential affect. This means a step size much smaller than the nucleus.
4) An electron shielding term should be used because of the 1s electrons. This is especially important at the higher values of ro, with the n=2, 3 and 4 states.
Nuclear Reduced Mass Radial Wave Function
We are now ready to solve for “R”, the reduced mass radial nuclei wave function. First, we complete equation 3-7 with the spherical coordinate representation of the energy operator and the pseudo-potential from equation 3-9, giving equation 3-10 (above- and to simplify, neglect the angular component for now: we will have need of it later).
This is very similar to the hydrogen atom wave function. The detailed numerical solution of this equation is shown in appendix A. The graph of the ground state radial wave function “R”, from the calculations, is shown in figure 3-6. Note that the wave function drops very low at ro=0 because of the nuclear repulsion. Note also, that even with the nuclear repulsion, there is a substantial presence of the wave function at ro=0. This small value at ro=0, is swamped by the ro2 term in the radial probability function as seen in figure 3-7. In Figure 3-8 the most probable separation is shown to be ro =~160 Fermis (0.0016 Angstroms). This is about the same spacing calculated for the muon bonded hydrogen molecule from chapter 1. Our first miracle complete.
A graph showing the probability distributions of the nuclei and the electron is shown in figure 3-9. Note how a large fraction of the electron probability lays directly between the two nuclei. This is the origin of the very strong bond, which arises naturaly from the quantum mechanics.
With enough center of mass velocity, the channeled ~100eV "He-d" molecule will continue through the crystal and react with another tetrahedral deuterium, giving Hed2. The stage is then set for the fusion reaction. It would be expected that the 2 deuterons, in Hed2, would react using the standard fusion channels; d+d => (He+n) or (Tr+p). In chapter 4 we will learn how the normal fusion channels are completely suppressed by angular momentum constraints in the Hed2 molecule. The constraints are caused by characteristics of the helium wave function as it enters the activated complex with deuterium. The transition from the crystal electron energies, of tens of electron volts, to the tens of thousands of electron volts will also be addressed.
